🧠:[<think>Okay, so I need to figure out which quadrant the complex number z is located in, given that (1 + i)z = 2i. Let's break this down step by step.First, I remember that complex numbers can be represented in the form a + bi, where a is the real part and b is the imaginary part. The quadrants in the complex plane work just like the regular coordinate plane: the first quadrant is where both real and imaginary parts are positive, the second quadrant has a negative real part and positive imaginary part, the third quadrant has both negative, and the fourth quadrant has positive real and negative imaginary. So, the answer will depend on the signs of the real and imaginary parts of z.The equation given is (1 + i)z = 2i. To find z, I need to solve for it. Since we have a complex equation, I can solve for z by dividing both sides by (1 + i). That is, z = 2i / (1 + i). But dividing complex numbers can sometimes be tricky. I remember that to divide by a complex number, you multiply the numerator and denominator by the conjugate of the denominator. The conjugate of (1 + i) is (1 - i). So, multiplying both numerator and denominator by (1 - i) should help simplify the expression.Let me do that step by step. So:z = (2i) / (1 + i) * (1 - i)/(1 - i)First, multiply the numerator: 2i * (1 - i) = 2i - 2i^2And the denominator: (1 + i)(1 - i) = 1 - i^2 (since it's a difference of squares)Simplify both numerator and denominator. Remember that i^2 = -1.So numerator: 2i - 2i^2 = 2i - 2(-1) = 2i + 2 = 2 + 2iDenominator: 1 - (-1) = 1 + 1 = 2Therefore, z = (2 + 2i)/2 = (2/2) + (2i/2) = 1 + iWait, so z is 1 + i? Let me check that again. If z is 1 + i, then multiplying by (1 + i) gives (1 + i)(1 + i) = 1 + 2i + i^2 = 1 + 2i -1 = 2i. That's exactly the right-hand side of the equation. So that checks out. So z = 1 + i.Now, the complex number 1 + i has a real part of 1 and an imaginary part of 1. Both are positive, so in the complex plane, that would place it in the first quadrant. But the options given are A: first quadrant, B: second, etc. So the answer should be A?Wait, but let me make sure I didn't make a mistake. Let's go through the steps again. Starting from (1 + i)z = 2i. Solving for z, divide both sides by (1 + i). So z = 2i/(1 + i). Multiply numerator and denominator by (1 - i):Numerator: 2i*(1 - i) = 2i - 2i^2 = 2i - 2*(-1) = 2i + 2 = 2 + 2i.Denominator: (1 + i)(1 - i) = 1 - i + i - i^2 = 1 - i^2 = 1 - (-1) = 2.Then, z = (2 + 2i)/2 = 1 + i. Yep, that's correct.So z is 1 + i, which is in the first quadrant. Therefore, answer A.But wait, the options are A: first quadrant, B: second quadrant, etc. Hmm. But I recall that in the complex plane, the real part is on the x-axis and the imaginary part on the y-axis. So a complex number a + bi is plotted as (a, b). So if both a and b are positive, it's in the first quadrant. So if z = 1 + i, then a = 1 and b = 1, so (1,1) in the coordinate plane, which is indeed the first quadrant. Therefore, the answer is A.But wait, hold on. Let me double-check because sometimes there can be confusion about quadrants. For instance, maybe the problem defines quadrants differently? But no, standard complex plane uses the same quadrants as the Cartesian plane. First quadrant: x > 0, y > 0; second quadrant: x < 0, y > 0; third: x < 0, y < 0; fourth: x > 0, y < 0.Given z = 1 + i, which is (1,1), so first quadrant. So answer is A. However, the original problem might have a typo or maybe I misread something. Let me check again.Original equation: (1 + i)z = 2i. Solved for z, got 1 + i. If that's correct, then answer is A. But wait, maybe there's a mistake in the calculation? Let me confirm.Compute (1 + i)*(1 + i) = 1*1 + 1*i + i*1 + i*i = 1 + i + i + i^2 = 1 + 2i + (-1) = 0 + 2i = 2i. Which matches the right-hand side. So yes, z = 1 + i is correct. Therefore, the point is in the first quadrant. So answer A. So why does the problem even list other options? Maybe to test if someone miscalculates.Alternatively, maybe the user has a different approach. Let's try solving it another way. Let me represent z as x + yi, where x and y are real numbers. Then, (1 + i)(x + yi) = 2i.Multiply out the left-hand side: 1*(x + yi) + i*(x + yi) = x + yi + xi + y i^2 = x + yi + xi - y (since i^2 = -1)Combine like terms: (x - y) + (y + x)i. So, the real part is (x - y) and the imaginary part is (x + y).This equals 2i, which is 0 + 2i. Therefore, equating real and imaginary parts:Real part: x - y = 0 Imaginary part: x + y = 2So, we have the system of equations:1. x - y = 0 2. x + y = 2From equation 1: x = ySubstitute into equation 2: x + x = 2 => 2x = 2 => x = 1. Therefore, y = 1.Therefore, z = x + yi = 1 + i. Same result. So, this confirms that z = 1 + i, which is in the first quadrant. So answer A.Therefore, the correct answer should be A. Wait, but the options are A: first, B: second, etc. So why is this a problem? Maybe the user is expecting another answer? Hmm.Wait, perhaps there's a mistake in my calculation. Let me check again. If z = 1 + i, then (1 + i)z = (1 + i)(1 + i) = 1 + 2i + i^2 = 1 + 2i -1 = 2i. Which is exactly the given right-hand side, so it's correct. Therefore, z is in the first quadrant.But the problem might have been presented differently. Wait, the user wrote "The point corresponding to z in the complex plane is located in..." with options A to D. So unless there's a typo in the problem statement, like maybe the equation was supposed to be (1 - i)z = 2i, but the user wrote (1 + i). No, according to the problem, it's (1 + i)z = 2i. So unless I misread the problem, the answer is A.But let me check again. If someone does the division incorrectly, they might think it's in another quadrant. For example, if they forgot to multiply by the conjugate, or miscalculated the signs. Let me see.Suppose someone does z = 2i / (1 + i). If they just divide each part:2i divided by 1 is 2i, and 2i divided by i is 2. So maybe they incorrectly write z = 2i - 2 or something. Wait, that's not how complex division works. Alternatively, maybe someone does:Multiply numerator and denominator by i, which sometimes is another technique.So z = 2i / (1 + i) * (i/i) = (2i * i) / ( (1 + i)*i ) = (2i^2) / (i + i^2) = (2*(-1)) / (i -1) = (-2)/(-1 + i) = 2/(1 - i). Then, multiply numerator and denominator by (1 + i):2*(1 + i) / ( (1 - i)(1 + i) ) = (2 + 2i)/ (1 - i^2) = (2 + 2i)/2 = 1 + i. Same result. So regardless of the method, z = 1 + i.Therefore, the answer is A.But perhaps the problem is written in Chinese or another language, and there's a mistranslation? The user wrote in English, so probably not. Hmm.Alternatively, maybe the user intended (1 + i)z = 2, instead of 2i? But according to the problem, it's 2i. If it were 2, then:(1 + i)z = 2. Then z = 2/(1 + i). Multiply numerator and denominator by (1 - i):2*(1 - i)/ (1 +1) = (2 - 2i)/2 = 1 - i. Then z = 1 - i, which is in the fourth quadrant. But the problem states 2i, so that's not the case.Therefore, unless there's a miscalculation somewhere, z is 1 + i, first quadrant. So answer A.Wait, but let me check once more. If I substitute z = 1 + i into the original equation:Left-hand side: (1 + i)(1 + i) = 1*1 + 1*i + i*1 + i*i = 1 + i + i + i² = 1 + 2i + (-1) = 0 + 2i = 2i. Which matches the right-hand side. So yes, correct.Therefore, the answer must be A. The first quadrant.But I just want to be absolutely sure because sometimes these problems can trick you. For example, if the equation was (1 + i)z = -2i, then z would be -1 - i, which is in the third quadrant. But in our case, it's 2i, so the answer is A.Wait, just thinking through possible variations. But according to the problem statement, it's (1 + i)z = 2i, so unless there's a typo, the answer is A.Therefore, the correct answer is A: The first quadrant.Final Answerboxed{A}</think>To determine the quadrant in which the complex number ( z ) lies, given the equation ((1 + i)z = 2i), we need to solve for ( z ).First, we isolate ( z ) by dividing both sides by ( 1 + i ):[z = frac{2i}{1 + i}]To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator, ( 1 - i ):[z = frac{2i}{1 + i} cdot frac{1 - i}{1 - i} = frac{2i(1 - i)}{(1 + i)(1 - i)}]Calculating the numerator:[2i(1 - i) = 2i - 2i^2 = 2i - 2(-1) = 2i + 2 = 2 + 2i]Calculating the denominator:[(1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 1 + 1 = 2]Thus, we have:[z = frac{2 + 2i}{2} = 1 + i]The complex number ( z = 1 + i ) has a real part of 1 and an imaginary part of 1, both of which are positive. Therefore, the point corresponding to ( z ) in the complex plane is located in the first quadrant.The answer is (boxed{A}).,<think>Okay, so I have this problem here: The complex number z satisfies (1 + i)z = 2i. I need to figure out in which quadrant the point corresponding to z is located in the complex plane. The options are the first, second, third, or fourth quadrant. Hmm, let's see.First, I remember that complex numbers can be represented in the form a + bi, where a is the real part and b is the imaginary part. The quadrants in the complex plane are determined by the signs of a and b. The first quadrant is where both a and b are positive, the second quadrant has a negative a and positive b, the third quadrant both negative, and the fourth quadrant a positive a and negative b. So, I need to find the real and imaginary parts of z and check their signs.The equation given is (1 + i)z = 2i. To find z, I should solve for z by dividing both sides by (1 + i). So, z = 2i / (1 + i). But dividing complex numbers can be a bit tricky. I think the standard method is to multiply the numerator and denominator by the complex conjugate of the denominator to rationalize it. The complex conjugate of (1 + i) is (1 - i). Let me try that.Multiplying numerator and denominator by (1 - i):z = [2i * (1 - i)] / [(1 + i)(1 - i)]Let me compute the denominator first. (1 + i)(1 - i) is a difference of squares, so it should be 1^2 - (i)^2 = 1 - (-1) = 1 + 1 = 2. That simplifies the denominator to 2.Now the numerator: 2i * (1 - i). Let's distribute that:2i * 1 = 2i2i * (-i) = -2i^2So the numerator becomes 2i - 2i^2. But i^2 is -1, so -2i^2 is -2*(-1) = 2. Therefore, the numerator is 2i + 2, which can be written as 2 + 2i.So putting it all together, z = (2 + 2i)/2 = (2/2) + (2i/2) = 1 + i.Wait a second, so z is 1 + i? Let me check that again. If I substitute back into the original equation: (1 + i)(1 + i) = (1)(1) + (1)(i) + (i)(1) + (i)(i) = 1 + i + i + i^2 = 1 + 2i + (-1) = (1 - 1) + 2i = 0 + 2i = 2i. Oh, that works out! So z is indeed 1 + i.So in the complex plane, the real part is 1 and the imaginary part is 1. Both are positive, so the point (1, 1) is in the first quadrant. Therefore, the answer should be A: The first quadrant.But wait, the options given include first, second, third, fourth. Let me just confirm my steps once more to be sure. Starting with (1 + i)z = 2i. Solving for z, division by (1 + i). Correct. Multiplying numerator and denominator by (1 - i) to rationalize. Denominator becomes (1)^2 - (i)^2 = 1 - (-1) = 2. Numerator is 2i*(1 - i) = 2i - 2i^2 = 2i - 2*(-1) = 2i + 2. So z = (2 + 2i)/2 = 1 + i. Yep, that seems right.So real part 1, imaginary part 1. Both positive. So first quadrant. Option A. Hmm. But let me check again because sometimes when dealing with complex numbers, especially with quadrants, people can mix up the real and imaginary axes. Wait, in the complex plane, the x-axis is the real part and the y-axis is the imaginary part. So if z = a + bi, the coordinates are (a, b). So in this case, a = 1 and b = 1, so (1,1) is indeed in the first quadrant. So the answer is A.Alternatively, maybe I can represent 1 + i in polar form to see its angle. The angle θ of 1 + i is arctangent(b/a) = arctan(1/1) = 45 degrees, which is in the first quadrant. So that's another way to confirm.Wait, but just to make sure I didn't make a mistake in solving for z. Let's try substituting z = 1 + i into the original equation: (1 + i)(1 + i) = (1 + i)^2. Expanding that, as I did before, gives 1 + 2i + i^2 = 1 + 2i -1 = 2i. Which matches the right-hand side. Perfect. So z is definitely 1 + i.Therefore, the answer is A: The first quadrant.Final Answerboxed{A}</think>To determine the quadrant in which the complex number ( z ) lies, we start with the given equation ((1 + i)z = 2i). Solving for ( z ), we divide both sides by ( 1 + i ):[z = frac{2i}{1 + i}]To simplify this, we multiply the numerator and the denominator by the complex conjugate of the denominator, which is ( 1 - i ):[z = frac{2i cdot (1 - i)}{(1 + i)(1 - i)}]Calculating the denominator:[(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2]Calculating the numerator:[2i cdot (1 - i) = 2i - 2i^2 = 2i - 2(-1) = 2i + 2]Thus, we have:[z = frac{2 + 2i}{2} = 1 + i]The complex number ( z = 1 + i ) has a real part of 1 and an imaginary part of 1. Both parts are positive, so the point corresponding to ( z ) in the complex plane is located in the first quadrant.[boxed{A}]]

🔑:Since the complex number z satisfies (1+i)z=2i, we have z= dfrac {2i}{1+i}= dfrac {2i(1-i)}{(1+i)(1-i)}=1+i. The coordinates of the corresponding point in the complex plane are (1,1).Therefore, the correct answer is boxed{text{A}}.This problem involves calculating the complex number z using the rules of multiplication and division in algebraic form of two complex numbers, determining the coordinates of its corresponding point in the complex plane, and thus drawing a conclusion. It mainly examines the multiplication and division of two complex numbers in algebraic form, the properties of the power operation of the imaginary unit i, and the relationship between complex numbers and their corresponding points in the complex plane, making it a basic question.

🧠:[<think>Okay, let me try to figure out these three problems step by step. First, I need to understand the setup. The factory plans to produce 700 bicycles in a week, with an average daily production of 100 bicycles. That makes sense because 700 divided by 7 days is 100 per day. Now, each day they either overproduce or underproduce, and the deviations are given in the table. Starting with question (1): "According to the records, the total production for the first four days is ______ bicycles." Alright, so the first four days are Monday, Tuesday, Wednesday, and Thursday. The deviations for these days are +5, -2, -4, and +13. To find the total production, I need to calculate how many bicycles were actually produced each day and then sum them up. The planned production each day is 100 bicycles. So, the actual production each day would be 100 plus the deviation. Let me compute each day:- Monday: 100 + 5 = 105- Tuesday: 100 - 2 = 98- Wednesday: 100 - 4 = 96- Thursday: 100 + 13 = 113Now, summing these up: 105 + 98 + 96 + 113. Let me add them step by step. 105 + 98 is 203. Then 203 + 96 is 299. Then 299 + 113 is 412. So, the total production for the first four days is 412 bicycles. Let me double-check that addition to make sure I didn't make a mistake. 105+98=203, yes. 203+96=299, correct. 299+113: 299+100=399, +13=412. Yep, that seems right.Moving on to question (2): "The day with the highest production exceeds the day with the lowest production by ______ bicycles."So, I need to find which day has the highest actual production and which day has the lowest, then subtract the two. Let's look at the deviations again for all seven days:- Monday: +5- Tuesday: -2- Wednesday: -4- Thursday: +13- Friday: -10- Saturday: +16- Sunday: -9Calculating each day's production:- Monday: 100 +5 =105- Tuesday:100 -2=98- Wednesday:100 -4=96- Thursday:100 +13=113- Friday:100 -10=90- Saturday:100 +16=116- Sunday:100 -9=91So, the actual productions are: 105, 98, 96, 113, 90, 116, 91.Looking for the highest: Saturday with 116. The lowest is Friday with 90. The difference is 116 -90=26. So, the day with the highest production exceeds the lowest by 26 bicycles. Let me confirm. Saturday's deviation is +16, so 100+16=116. Friday's deviation is -10, so 100-10=90. 116-90=26. Correct.Now question (3): The total wage for the workers. The payment system is piece-rate. Each bicycle earns 60, but with an additional 15 for each bicycle beyond the target and a deduction of 20 for each under-produced bicycle. The target is 700 bicycles for the week. So, first, I need to compute the total actual production for the week, then figure out how many are over or under the target, then compute the wages.First, let me compute the total production. Since deviations are given, the total deviation is the sum of all daily deviations. Then total production is 700 plus total deviation.Let's compute total deviation:Monday: +5Tuesday: -2Wednesday: -4Thursday: +13Friday: -10Saturday: +16Sunday: -9Adding them up: 5 -2 -4 +13 -10 +16 -9.Let me compute step by step:Start with 5. Subtract 2: 3. Subtract 4: -1. Add 13: 12. Subtract 10: 2. Add 16: 18. Subtract 9: 9.Total deviation is +9. Therefore, total production is 700 +9=709 bicycles.But wait, let me check that addition again:5 -2 =3; 3-4=-1; -1+13=12; 12-10=2; 2+16=18; 18-9=9. Yes, total deviation is +9. So total production is 709.But the payment is based on each bicycle earning 60, plus an extra 15 for each bicycle over the target, and a deduction of 20 for each under-production. Wait, but hold on. The problem says: "each bicycle produced in a week earns the worker 60, with an additional 15 for each bicycle produced beyond the target and a deduction of 20 for each bicycle under the target."Wait, maybe I misread that. So, perhaps for each bicycle produced, they get 60, but if they produce over the target, each extra bicycle gives an additional 15, and if they under-produce, each missing bicycle deducts 20.So, total wage = (Total bicycles produced * 60) + (Number of over-produced bicycles * 15) - (Number of under-produced bicycles * 20)But wait, but over-produced bicycles would be total production minus target if positive, and under-produced is target minus total production if negative. Wait, but total production is 709, which is 9 over. So over-produced bicycles is 9, under-produced is 0. Therefore, total wage would be 709*60 +9*15 -0*20= ?But hold on, the problem says "additional 15 for each bicycle produced beyond the target and a deduction of 20 for each bicycle under the target."So, if they produced over, they get extra 15 per over bicycle. If under, they get deducted 20 per under bicycle. But total over or under is net. So, if total production is 709, which is 9 over. So, they get 709*60 +9*15.Alternatively, if they had under-produced, say 10 under, they would get (700-10)*60 -10*20.But let me confirm. Let me parse the problem statement again:"Each bicycle produced in a week earns the worker 60, with an additional 15 for each bicycle produced beyond the target and a deduction of 20 for each bicycle under the target."So, for every bicycle produced, whether over or under, they get 60. Then, if a bicycle is beyond the target, they get an extra 15 per over bicycle. If a bicycle is under the target, they get a deduction of 20 per under bicycle.Wait, but "each bicycle produced...earns the worker 60". So for each bicycle made, regardless of target, they get 60. Then, if the total production is over the target, each over bicycle (i.e., total production - target) gives an extra 15. If the total production is under, each under bicycle (target - total production) deducts 20.So, the total wage is:(Total production * 60) + (Overproduction * 15) - (Underproduction * 20)But overproduction and underproduction can't both be non-zero. Since overproduction is max(total production - target, 0), and underproduction is max(target - total production, 0). So in this case, total production is 709, so overproduction is 9, underproduction is 0. Therefore:Total wage = 709*60 +9*15 -0*20 = 709*60 + 135Compute 709*60: 700*60=42,000; 9*60=540; total 42,000+540=42,540. Then add 135: 42,540+135=42,675.But wait, another way: maybe the problem is per day deviations? Wait, the problem says "each bicycle produced beyond the target" and "each bicycle under the target". The target is 700 for the week. So, it's a weekly target, not daily. Therefore, overproduction is total over the week, not per day. So, the total overproduction is 9, as we calculated. So total wage is 709*60 +9*15 =42,540 +135=42,675 dollars.But let me check again. Wait, perhaps the additional 15 is per bicycle over the daily target, and deduction per bicycle under daily target? But the problem says "weekly pay" and "each bicycle produced in a week earns the worker 60". Hmm.Wait, the problem says: "the factory implements a piece-rate wage system (weekly pay). Each bicycle produced in a week earns the worker 60, with an additional 15 for each bicycle produced beyond the target and a deduction of 20 for each bicycle under the target."So, this is a weekly target. So, for the entire week, they have a target of 700. Each bicycle produced gives 60. If they produce more than 700, each bicycle over 700 gives an extra 15. If they produce less, each bicycle under 700 deducts 20.Therefore, total production is 709, which is 9 over. So:Total wage = (709 * 60) + (9 *15) = 42,540 + 135 = 42,675.Alternatively, if they had produced 690, then total wage would be (690 *60) - (10*20) =41,400 -200=41,200.Therefore, the answer is 42,675. But let me make sure I didn't miscalculate the total production.Earlier, I calculated the total deviation as +9. Target is 700, so total production is 700 +9=709. Let me verify total deviation again:Monday: +5Tuesday: -2 (Total: 5-2=3)Wednesday: -4 (3-4=-1)Thursday: +13 (-1+13=12)Friday: -10 (12-10=2)Saturday: +16 (2+16=18)Sunday: -9 (18-9=9). Correct. So total production is 709.Thus, total wage is 709*60 +9*15.Compute 709*60: 700*60=42,000; 9*60=540. So 42,000+540=42,540.Then 9*15=135. Total wage: 42,540+135=42,675.Hence, the total wage is 42,675. Wait, but just to confirm another way. Let's compute day by day, but I think that's unnecessary since the problem mentions weekly pay, so it's based on the total weekly production. But just to check, maybe the deviations per day affect something? For example, maybe the additional 15 is for each bicycle over the daily target, and deduction for each under daily target. But the problem says "each bicycle produced beyond the target", and the target is 700 in a week. So, it's a weekly target, not daily. Therefore, the total over the week is what matters. Therefore, the calculation should be correct as 42,675.But let me check the problem statement once more: "additional 15 for each bicycle produced beyond the target and a deduction of 20 for each bicycle under the target." The target is 700. So, if they made 709, that's 9 over, so 9*15 extra. If they made 690, that's 10 under, so 10*20 deduction. Therefore, yes, the calculation is correct.Thus, the three answers are:1) 412 bicycles2) 26 bicycles3) 42,675 total wageFinal Answer(1) boxed{412}(2) boxed{26}(3) boxed{42675}</think>The factory plans to produce 700 bicycles in a week, with an average daily production of 100 bicycles. The production deviations for each day are given as follows:| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday ||-----------------|--------|---------|-----------|----------|--------|----------|--------|| Production Deviation | +5 | -2 | -4 | +13 | -10 | +16 | -9 |1. Total production for the first four days: - Calculate the actual production for each of the first four days: - Monday: (100 + 5 = 105) - Tuesday: (100 - 2 = 98) - Wednesday: (100 - 4 = 96) - Thursday: (100 + 13 = 113) - Sum these values: (105 + 98 + 96 + 113 = 412) - The total production for the first four days is (boxed{412}) bicycles.2. Day with the highest production vs. the lowest: - Calculate the actual production for each day: - Monday: (105), Tuesday: (98), Wednesday: (96), Thursday: (113), Friday: (90), Saturday: (116), Sunday: (91) - Identify the highest and lowest productions: - Highest: Saturday with (116) - Lowest: Friday with (90) - Calculate the difference: (116 - 90 = 26) - The day with the highest production exceeds the day with the lowest production by (boxed{26}) bicycles.3. Total wage for the workers: - Calculate the total deviation: (5 - 2 - 4 + 13 - 10 + 16 - 9 = 9) - Total production: (700 + 9 = 709) - Calculate the total wage: - Base wage: (709 times 60 = 42540) - Additional payment for overproduction: (9 times 15 = 135) - Total wage: (42540 + 135 = 42675) - The total wage for the workers is (boxed{42675}) dollars.,<think>Okay, let's tackle these three questions step by step. First, I need to make sure I understand what each part is asking for and how to approach them. Let me start with the first question.Question 1: Total production for the first four daysThe problem states that the factory plans to produce 700 bicycles in a week, with an average daily production of 100 bicycles. Since there are 7 days in a week, 100 bicycles per day times 7 days equals 700 bicycles, which checks out. Now, the table gives the production deviation for each day. The deviations are: Monday +5, Tuesday -2, Wednesday -4, Thursday +13, Friday -10, Saturday +16, Sunday -9. The first four days would be Monday, Tuesday, Wednesday, and Thursday. To find the total production for these four days, I need to calculate the planned production for four days and then adjust it by the sum of the deviations for those days.Planned production per day is 100 bicycles. So, for four days, that's 100 * 4 = 400 bicycles.Now, sum the deviations for the first four days: Monday (+5) + Tuesday (-2) + Wednesday (-4) + Thursday (+13). Let's compute that:5 - 2 - 4 + 13 = (5 + 13) + (-2 -4) = 18 - 6 = 12.So, the total deviation is +12. That means they produced 12 more bicycles than planned over the four days. Therefore, the total production is 400 + 12 = 412 bicycles.Wait, but let me double-check. The planned production is 100 per day. For each day, the deviation is added to the planned production. So, for each day, actual production is 100 + deviation. Therefore, total production would be sum over four days of (100 + deviation). Which is 4*100 + sum of deviations. Which is 400 + 12 = 412. That's correct.So, question 1 answer is 412. Let me note that.Question 2: The day with the highest production exceeds the day with the lowest production by how many bicycles?Here, we need to find the maximum daily production and the minimum daily production, then compute the difference between them.First, let's find the actual production for each day. The planned production is 100 per day. Deviation is given, so actual production is 100 + deviation.Compute each day's production:- Monday: 100 + 5 = 105- Tuesday: 100 - 2 = 98- Wednesday: 100 - 4 = 96- Thursday: 100 + 13 = 113- Friday: 100 - 10 = 90- Saturday: 100 + 16 = 116- Sunday: 100 - 9 = 91So, let's list them:- Monday: 105- Tuesday: 98- Wednesday: 96- Thursday: 113- Friday: 90- Saturday: 116- Sunday: 91Now, looking for the highest and lowest. Scanning through:Highest is Saturday with 116, lowest is Friday with 90. The difference is 116 - 90 = 26.Wait, let me confirm. Saturday: 100 +16=116, yes. Friday: 100 -10=90. So 116 -90=26. That seems correct.So, question 2 answer is 26.Question 3: Total wage for the workersThis is a bit more complex. The wage system is piece-rate: each bicycle produced in a week earns 60. Additionally, for each bicycle produced beyond the target (700), they get an extra 15 per bicycle. For each bicycle under the target, they deduct 20 per bicycle.So, first, we need to compute the total production for the week. Then, compare it to the target (700). The total wage will be:(Total bicycles produced * 60) + (Number of overproduced bicycles * 15) - (Number of underproduced bicycles * 20)Wait, hold on. Wait, the problem says: "each bicycle produced in a week earns the worker 60, with an additional 15 for each bicycle produced beyond the target and a deduction of 20 for each bicycle under the target."So, actually, the wording is a bit ambiguous. Let me parse it again.It says: Each bicycle produced in a week earns 60. Then, if they produce beyond the target, each of those extra bicycles gets an additional 15. If they produce under, each bicycle under gets a deduction of 20. So, maybe the total wage is:Total wage = (Total bicycles * 60) + (Overage * 15) - (Shortage * 20)But maybe the 60 is only for the target, and overage gets 60 + 15, and under gets 60 - 20? Wait, the problem says: "Each bicycle produced in a week earns the worker 60, with an additional 15 for each bicycle produced beyond the target and a deduction of 20 for each bicycle under the target."Hmm. So, for each bicycle produced, they get 60. But if the total production exceeds the target, then for each bicycle beyond the target, they get an extra 15. If the total production is under the target, then for each bicycle under, they get 20 deducted.Wait, maybe it's structured as:Total wage = (Total bicycles * 60) + (Total overproduction * 15) - (Total underproduction * 20)But actually, overproduction and underproduction are relative to the target. So, the overproduction is max(total - target, 0), and underproduction is max(target - total, 0). But since the problem mentions "each bicycle produced beyond the target" and "each bicycle under the target", perhaps the total over is the amount over, and total under is the amount under. So if they produced 710, overproduction is 10, under is 0. If they produced 690, over is 0, under is 10. So, total wage would be:If production >= target: (production * 60) + (production - target) * 15If production < target: (production * 60) - (target - production) * 20But actually, the way it's worded: "each bicycle produced in a week earns the worker 60, with an additional 15 for each bicycle produced beyond the target and a deduction of 20 for each bicycle under the target."So, for all bicycles produced, they get 60 each. Then, for each bicycle beyond the target, they get an extra 15 per bicycle (so total overproduction *15). For each bicycle under the target, they get a deduction of 20 per bicycle (so total underproduction *20).Wait, but the way it's phrased is a bit ambiguous. It could be interpreted as: each bicycle over the target gives an extra 15, but each bicycle under incurs a deduction of 20. However, the base pay is 60 per bicycle. So regardless of over or under, they get 60 per bicycle, plus an extra 15 for each over, minus 20 for each under. Wait, but over and under are relative to the target.Wait, suppose the target is 700. If they produce 710, then over is 10, under is 0. So total wage is 710*60 + 10*15.If they produce 690, then over is 0, under is 10. So total wage is 690*60 -10*20.But is that the correct interpretation? Let's check:The problem states: "each bicycle produced in a week earns the worker 60, with an additional 15 for each bicycle produced beyond the target and a deduction of 20 for each bicycle under the target."So, the 60 is for each bicycle produced. Then, on top of that, for each bicycle beyond the target, they get an extra 15 (so per over bicycle). For each bicycle under, they have a deduction of 20 (so per under bicycle). Therefore, total wage is:(Total produced * 60) + (Overproduction *15) - (Underproduction *20)But overproduction is the total produced minus target if positive, else zero. Similarly, underproduction is target minus total produced if positive, else zero.Therefore, we need to compute total production first. Then compute overproduction and underproduction.So first, let's compute total production for the week. We can compute the total deviation and add to the target.Total deviation is the sum of all daily deviations:+5 -2 -4 +13 -10 +16 -9Let's compute that step by step:Start with 0:+5: 5-2: 3-4: -1+13: 12-10: 2+16: 18-9: 9Wait, that seems wrong. Let me recalculate:Start with 0.Monday: +5 → 5Tuesday: -2 → 5 -2 = 3Wednesday: -4 → 3 -4 = -1Thursday: +13 → -1 +13 = 12Friday: -10 → 12 -10 = 2Saturday: +16 → 2 +16 = 18Sunday: -9 → 18 -9 = 9So total deviation is +9.Therefore, total production is target (700) + total deviation (9) = 709 bicycles.Therefore, overproduction is 709 -700 =9, underproduction is 0.Therefore, total wage is:709 *60 +9 *15 -0 *20 = 709*60 + 135Compute 709*60:700*60=42,0009*60=540Total: 42,000 +540=42,540Plus 135: 42,540 +135=42,675Therefore, total wage is 42,675.Wait, but let me check if this is the correct interpretation.Alternatively, maybe the "additional 15 for each bicycle produced beyond the target" is per bicycle beyond the target, so if they produce 709, that's 9 over, so 9*15. The deduction is "for each bicycle under the target". Since they are over, there is no under. So total wage is 709*60 +9*15=42,540 +135=42,675. That seems right.Alternatively, could the problem be interpreted as: for each bicycle produced, they get 60, but if it's over the daily target, they get an extra 15 per over bicycle each day, and if under, they get deducted 20 per under bicycle each day? But the problem says "weekly pay" and mentions "each bicycle produced in a week". So probably, it's based on the weekly target.The problem says: "the factory implements a piece-rate wage system (weekly pay). Each bicycle produced in a week earns the worker 60, with an additional 15 for each bicycle produced beyond the target and a deduction of 20 for each bicycle under the target."Therefore, the total production for the week is compared to the weekly target (700). So, over the week, total overproduction is total production -700, and total underproduction is 700 - total production, but only one of them is non-zero. Then, total wage is total production *60 + overproduction *15 - underproduction *20.So, since total production is 709, which is over by 9, then underproduction is 0. Therefore, total wage is 709*60 +9*15=42,540 +135=42,675. So, 42,675.But let me verify the total production again. Target is 700. Sum of deviations is +9, so total production is 700 +9=709. Correct.Wait, but let me check the sum of deviations again. Let's recompute:Monday: +5Tuesday: -2 → total: 5 -2 =3Wednesday: -4 → 3 -4 = -1Thursday: +13 → -1 +13=12Friday: -10 →12 -10=2Saturday:+16 →2 +16=18Sunday: -9 →18 -9=9. Yes, total deviation +9. So total production 709.Therefore, overproduction is 9, underproduction is 0. So total wage is 709*60 +9*15.Compute 709*60:Calculate 700*60=42,000; 9*60=540; total 42,000+540=42,540.Then 9*15=135.Total wage: 42,540 +135=42,675.So, 42,675.But let me consider another angle. What if the additional 15 is only for the overproduced bicycles, not for all bicycles. Wait, the problem says "an additional 15 for each bicycle produced beyond the target". So, only the overproduced bicycles get the extra 15. Similarly, "deduction of 20 for each bicycle under the target"—only the underproduced bicycles get a deduction.But the base wage is 60 per bicycle produced. So, for all 709 bicycles, they get 709*60. Then, since they exceeded the target by 9, they get an additional 9*15. There is no underproduction, so no deduction. So total is 709*60 +9*15. Which is 42,675. So, that's correct.Alternatively, if it were structured differently, like base pay is 700*60, and then over gets (700*60)+(over*15), but under gets (700*60)-(under*20). But the problem says "each bicycle produced in a week earns the worker 60", so regardless of over or under, each bicycle gives 60. Then, if they made more than 700, each extra gives an additional 15. If they made less, each missing bicycle deducts 20. So, yes, the first interpretation is correct.Therefore, the total wage is 42,675.Wait, but let me verify with another example. Suppose they produced exactly 700. Then total wage would be 700*60 +0 -0=42,000. If they produced 701, it's 701*60 +1*15=42,060 +15=42,075. If they produced 699, it's 699*60 -1*20=41,940 -20=41,920. That seems consistent.Hence, the calculation for 709 should be correct.Thus, the three answers are:1) 4122) 263) 42,675Final Answer(1) boxed{412}(2) boxed{26}(3) boxed{42675}</think>To solve the given problem, we need to address three parts:1. Total production for the first four days: - Planned production per day: 100 bicycles. - Total planned production for 4 days: (100 times 4 = 400) bicycles. - Sum of deviations for the first four days: (+5 - 2 - 4 + 13 = 12). - Total production: (400 + 12 = 412) bicycles.2. Day with the highest production exceeds the day with the lowest production by: - Calculate actual production for each day: - Monday: (100 + 5 = 105) - Tuesday: (100 - 2 = 98) - Wednesday: (100 - 4 = 96) - Thursday: (100 + 13 = 113) - Friday: (100 - 10 = 90) - Saturday: (100 + 16 = 116) - Sunday: (100 - 9 = 91) - Highest production: 116 (Saturday), Lowest production: 90 (Friday). - Difference: (116 - 90 = 26).3. Total wage for the workers: - Total deviation for the week: (+5 - 2 - 4 + 13 - 10 + 16 - 9 = 9). - Total production: (700 + 9 = 709) bicycles. - Overproduction: (709 - 700 = 9). - Total wage calculation: - Base wage: (709 times 60 = 42540). - Additional amount for overproduction: (9 times 15 = 135). - Total wage: (42540 + 135 = 42675).Final answers:(1) boxed{412}(2) boxed{26}(3) boxed{42675}]

🔑:Let's break down the solution step by step, following the given problem and its standard solution: For Part (1):We are given that the factory plans to produce 700 bicycles in a week, with an average daily production of 100 bicycles. The production deviation for the first four days is +5, -2, -4, and +13 respectively. To find the total production for the first four days, we calculate the sum of the planned production and the deviation:- Planned production for 4 days: 100 times 4 = 400 bicycles- Total deviation for 4 days: 5 - 2 - 4 + 13 = 12 bicyclesAdding these together gives us the total production for the first four days:400 + 12 = 412 text{ bicycles}So, the total production for the first four days is boxed{412 text{ bicycles}}. For Part (2):The highest production deviation is +16 (Saturday), and the lowest is -10 (Friday). The difference between the highest and lowest production days is:16 - (-10) = 16 + 10 = 26 text{ bicycles}Therefore, the day with the highest production exceeds the day with the lowest production by boxed{26 text{ bicycles}}. For Part (3):To calculate the total wage for the workers, we first find the total number of bicycles produced this week. The total deviation is 5 - 2 - 4 + 13 - 10 + 16 - 9 = 9 bicycles. The planned production is 100 times 7 = 700 bicycles. Therefore, the actual production is:700 + 9 = 709 text{ bicycles}The wage calculation is as follows:- Base wage for planned production: 700 times 60 = 42000- Additional wage for extra bicycles: 9 times (60 + 15) = 675Adding these together gives us the total wage:42000 + 675 = 42675 text{ 元}Therefore, the total wage for the workers in the factory for this week is boxed{42675 text{ 元}}.

🧠:[<think>Okay, so I have to solve part a) and part b) of this problem. Let's start with part a). The task is to list all two-digit natural numbers where one of the digits is 4 less than the other. Hmm, two-digit numbers range from 10 to 99. Each two-digit number has a tens digit and a units digit. Let me denote the tens digit as 'a' and the units digit as 'b'. So the number is 10a + b. Now, the condition is that one of the digits is 4 less than the other. That means either a = b + 4 or b = a + 4. Since digits can't be negative or more than 9, I need to consider the possible values of a and b under these constraints.First, let's consider the case where the tens digit is 4 less than the units digit. So, a = b - 4. But since a has to be a digit between 1 and 9 (because it's a two-digit number), b must be at least a + 4. Wait, no. If a = b - 4, then b = a + 4. Since a is the tens digit, it can be from 1 to 9. Then b would be from (1 + 4)=5 up to (9 + 4)=13. But b has to be a single digit, so the maximum b can be is 9. Therefore, a can be from 1 up to 9 - 4 = 5. So when a is 1 to 5, b will be 5 to 9. Therefore, the numbers in this case are 15, 26, 37, 48, 59. Let me check:- a=1, b=5: 15- a=2, b=6: 26- a=3, b=7: 37- a=4, b=8: 48- a=5, b=9: 59Yes, that's five numbers.Now, the other case is when the units digit is 4 less than the tens digit. So b = a - 4. Here, a is the tens digit (1-9), and b must be a digit (0-9). So a can be from 4 to 9, because if a is 4, then b=0; if a is 5, then b=1; up to a=9, b=5. Let's list those:- a=4, b=0: 40- a=5, b=1: 51- a=6, b=2: 62- a=7, b=3: 73- a=8, b=4: 84- a=9, b=5: 95So that's six numbers. So combining both cases, the numbers are: 15, 26, 37, 48, 59, 40, 51, 62, 73, 84, 95. Let me check if I missed any. For the first case, when a is 1-5, b is 5-9. Second case, a is 4-9, b is 0-5. So 15,26,37,48,59 and 40,51,62,73,84,95. Total of 11 numbers. Let me make sure none of these have digits that differ by 4. For example, 15: 1 and 5, difference is 4. 26: 2 and 6, difference 4. Similarly 37, 48, 59. Then 40: 4 and 0, difference 4. 51: 5 and 1, difference 4. 62: 6 and 2, difference 4. 73:7 and 3, difference 4. 84:8 and 4, difference 4. 95:9 and 5, difference 4. All correct. So part a) answer is these 11 numbers. Let me write them down as a list.Now moving to part b): Determine among these numbers all those that are divisible by their digit sum. So for each number in the list from part a), I need to compute its digit sum and check if the number is divisible by that sum. Let's go through each number one by one.1. 15: digits 1 and 5. Digit sum = 1 + 5 = 6. Now, check if 15 is divisible by 6. 15 ÷ 6 = 2.5, which is not an integer. So no.2. 26: digits 2 and 6. Sum = 8. 26 ÷ 8 = 3.25. Not an integer. No.3. 37: digits 3 and 7. Sum = 10. 37 ÷ 10 = 3.7. No.4. 48: digits 4 and 8. Sum = 12. 48 ÷ 12 = 4. That's an integer. So yes, 48 is divisible by 12.5. 59: digits 5 and 9. Sum = 14. 59 ÷ 14 ≈ 4.214. Not integer. No.6. 40: digits 4 and 0. Sum = 4 + 0 = 4. 40 ÷ 4 = 10. That's an integer. So yes, 40 is divisible by 4.7. 51: digits 5 and 1. Sum = 6. 51 ÷ 6 = 8.5. No.8. 62: digits 6 and 2. Sum = 8. 62 ÷ 8 = 7.75. No.9. 73: digits 7 and 3. Sum = 10. 73 ÷ 10 = 7.3. No.10. 84: digits 8 and 4. Sum = 12. 84 ÷ 12 = 7. That's an integer. So yes.11. 95: digits 9 and 5. Sum = 14. 95 ÷ 14 ≈ 6.7857. No.So from the list, numbers 48, 40, and 84 are divisible by their digit sums. Let me double-check these:- 48: sum 12. 48 ÷ 12 = 4 ✔️- 40: sum 4. 40 ÷ 4 = 10 ✔️- 84: sum 12. 84 ÷ 12 = 7 ✔️Others like 15: sum 6. 15 ÷ 6 is 2.5 ❌26: sum 8. 26 ÷ 8 = 3.25 ❌And so on. So the numbers that meet part b) are 40, 48, and 84.Wait, hold on. Let me check 40 again. The digits are 4 and 0. The digit sum is 4 + 0 = 4, and 40 ÷ 4 = 10. That's correct. 48: 4 + 8 = 12, 48 ÷ 12 = 4. Correct. 84: 8 + 4 = 12, 84 ÷ 12 = 7. Correct.So those three numbers are the answers for part b).Final Answera) boxed{15}, boxed{26}, boxed{37}, boxed{40}, boxed{48}, boxed{51}, boxed{59}, boxed{62}, boxed{73}, boxed{84}, boxed{95}b) boxed{40}, boxed{48}, boxed{84}</think>To solve part a), we need to list all two-digit natural numbers where one digit is 4 less than the other. We consider two cases: when the tens digit is 4 less than the units digit and vice versa.Case 1: Tens digit is 4 less than the units digit- Tens digit (a) ranges from 1 to 5 (since the units digit (b = a + 4) must be a single digit)- Numbers: 15, 26, 37, 48, 59Case 2: Units digit is 4 less than the tens digit- Tens digit (a) ranges from 4 to 9 (since the units digit (b = a - 4) must be non-negative)- Numbers: 40, 51, 62, 73, 84, 95Combining both cases, the two-digit numbers are: 15, 26, 37, 40, 48, 51, 59, 62, 73, 84, 95.For part b), we check which of these numbers are divisible by their digit sum:1. 15: Digit sum (1 + 5 = 6). (15 div 6 = 2.5) (not divisible)2. 26: Digit sum (2 + 6 = 8). (26 div 8 = 3.25) (not divisible)3. 37: Digit sum (3 + 7 = 10). (37 div 10 = 3.7) (not divisible)4. 40: Digit sum (4 + 0 = 4). (40 div 4 = 10) (divisible)5. 48: Digit sum (4 + 8 = 12). (48 div 12 = 4) (divisible)6. 51: Digit sum (5 + 1 = 6). (51 div 6 = 8.5) (not divisible)7. 59: Digit sum (5 + 9 = 14). (59 div 14 approx 4.214) (not divisible)8. 62: Digit sum (6 + 2 = 8). (62 div 8 = 7.75) (not divisible)9. 73: Digit sum (7 + 3 = 10). (73 div 10 = 7.3) (not divisible)10. 84: Digit sum (8 + 4 = 12). (84 div 12 = 7) (divisible)11. 95: Digit sum (9 + 5 = 14). (95 div 14 approx 6.7857) (not divisible)The numbers divisible by their digit sum are 40, 48, and 84. Final Answera) boxed{15}, boxed{26}, boxed{37}, boxed{40}, boxed{48}, boxed{51}, boxed{59}, boxed{62}, boxed{73}, boxed{84}, boxed{95}b) boxed{40}, boxed{48}, boxed{84},<think>Okay, so I have this problem here with two parts. Let me start with part a. The task is to list all two-digit natural numbers where one of the digits is 4 less than the other. Hmm, alright. Let me think about how to approach this.First, two-digit numbers range from 10 to 99. Each number has a tens digit and a units digit. Let me denote the tens digit as 'a' and the units digit as 'b'. So the number can be written as 10a + b. The condition is that one of the digits is 4 less than the other. That could mean either the tens digit is 4 less than the units digit, or the units digit is 4 less than the tens digit. I need to consider both cases.Let me break it down. First case: a = b + 4. Second case: b = a + 4. Since digits can only be from 0 to 9, I need to make sure that when I adjust the digits by 4, they still remain valid digits (i.e., between 0 and 9). Also, note that the tens digit 'a' can't be 0 because it's a two-digit number, so a has to be at least 1.Starting with the first case: a = b + 4. Since a is the tens digit and must be at least 1, b can be from 0 to 5. Because if b is 5, then a would be 9 (5 + 4). If b is 6, a would be 10, which is invalid because a must be a single digit. So possible values for b here are 0 to 5. Then a would be 4 to 9. So the numbers here would be:- When b = 0, a = 4: 40- b = 1, a = 5: 51- b = 2, a = 6: 62- b = 3, a = 7: 73- b = 4, a = 8: 84- b = 5, a = 9: 95So that's six numbers: 40, 51, 62, 73, 84, 95.Now the second case: b = a + 4. Here, a is the tens digit, which is from 1 to 9. Then b must be a + 4. Since b has to be a single digit (0-9), a + 4 must be less than or equal to 9. Therefore, a can be from 1 to 5 (because if a is 5, b is 9; if a is 6, b would be 10, which is invalid). So possible values for a here are 1 to 5, leading to:- a = 1, b = 5: 15- a = 2, b = 6: 26- a = 3, b = 7: 37- a = 4, b = 8: 48- a = 5, b = 9: 59So that's another five numbers: 15, 26, 37, 48, 59.Combining both cases, the total numbers should be 6 + 5 = 11 numbers. Let me list them all out to check for duplicates or any mistakes:From first case: 40, 51, 62, 73, 84, 95From second case: 15, 26, 37, 48, 59No duplicates there. Let me double-check each number to ensure they meet the condition. For example, take 40: tens digit is 4, units digit is 0. 4 - 0 = 4, so that's good. 15: 1 and 5, 5 -1 =4. Correct. 26: 2 and 6, 6 -2=4. Correct. 37: 7-3=4. Correct. 48: 8-4=4. Correct. 59:9-5=4. Correct. Similarly, 51: 5-1=4. Wait, 51 has tens digit 5 and units digit 1. 5 -1=4. Correct. 62:6-2=4. Correct. 73:7-3=4. Correct. 84:8-4=4. Correct. 95:9-5=4. Correct. So all numbers check out. So part a's answer should be these 11 numbers.Now moving on to part b: Determine among these numbers all those that are divisible by their digit sum.Okay, so for each of the numbers we found in part a, we need to compute their digit sum and check if the number is divisible by that sum. Let me list the numbers again:From part a:15, 26, 37, 48, 59, 40, 51, 62, 73, 84, 95.Let me go through each one by one.1. 15:Digits: 1 and 5. Digit sum: 1 + 5 = 6.Check if 15 is divisible by 6. 15 ÷ 6 = 2.5. Not an integer. So no.2. 26:Digits: 2 and 6. Digit sum: 8.26 ÷ 8 = 3.25. Not integer. So no.3. 37:Digits: 3 and 7. Sum: 10.37 ÷ 10 = 3.7. Not integer. No.4. 48:Digits: 4 and 8. Sum: 12.48 ÷ 12 = 4. Exactly divisible. So yes, 48 is good.5. 59:Digits: 5 and 9. Sum: 14.59 ÷ 14 ≈ 4.214. Not integer. No.6. 40:Digits: 4 and 0. Sum: 4 + 0 = 4.40 ÷ 4 = 10. Exactly divisible. So 40 is good.7. 51:Digits: 5 and 1. Sum: 6.51 ÷ 6 = 8.5. Not integer. No.8. 62:Digits: 6 and 2. Sum: 8.62 ÷ 8 = 7.75. No.9. 73:Digits: 7 and 3. Sum: 10.73 ÷ 10 = 7.3. No.10. 84:Digits: 8 and 4. Sum: 12.84 ÷ 12 = 7. Exactly. So 84 is good.11. 95:Digits: 9 and 5. Sum: 14.95 ÷ 14 ≈ 6.7857. No.So from the above, the numbers that are divisible by their digit sum are 48, 40, and 84.Wait, let me verify these again to make sure I didn't make a calculation error.Check 48: 4+8=12, 48/12=4. Correct.40: 4+0=4, 40/4=10. Correct.84: 8+4=12, 84/12=7. Correct.Others:15: 6, 15/6=2.5. Correct. Not divisible.26: 8, 26/8=3.25. Correct.37: 10, 37/10=3.7. Correct.59:14, 59/14. Correct.51:6, 51/6=8.5. Correct.62:8, 62/8=7.75. Correct.73:10, 73/10=7.3. Correct.95:14, 95/14≈6.7857. Correct.So yes, 48, 40, 84. Wait, three numbers. But let me check 84 again. 8 and 4, sum 12. 84 divided by 12 is 7. Correct. So three numbers. Hmm, but wait, 40 is 4 and 0. Digit sum 4. 40 ÷4=10. So that's correct. 48: 48 ÷12=4. 84 ÷12=7. So these are the three numbers. Therefore, part b's answer is 40, 48, 84.But let me check once more to be thorough. Maybe I missed a number.Wait, 15: sum 6, 15/6=2.5. No. 26: 8, 26/8=3.25. 37:10, 3.7. 48:12, 4. 59:14, ~4.21. 40:4, 10. 51:6, 8.5. 62:8,7.75. 73:10,7.3. 84:12,7. 95:14, ~6.7857. So yes, only 40,48,84.Wait, but 84 is 84 divided by 12. 12*7=84. Correct. 40 divided by 4=10. Correct. 48 divided by 12=4. Correct.Therefore, the answer for part b is 40,48,84.But let me check if there is another number. Let's see. 15: 1+5=6. 15 ÷6=2.5. Not. 26:2+6=8. 26 ÷8=3.25. Not. 37:10. 37 ÷10=3.7. No. 48:12. 4. Yes. 59:14. No. 40:4. 10. Yes. 51:6. 8.5. No. 62:8. No. 73:10. No. 84:12. 7. Yes. 95:14. No. So yes, only those three. Hmm.Wait a second. Let me check 15 again. 1 and 5. Sum 6. 15 divided by 6. 6*2=12, 15-12=3. Not. 26: 8*3=24, 26-24=2. Not. 37: 10*3=30, 37-30=7. Not. 48: 12*4=48. Perfect. 59:14*4=56, 59-56=3. Not. 40:4*10=40. Perfect. 51:6*8=48, 51-48=3. Not. 62:8*7=56, 62-56=6. Not. 73:10*7=70, 73-70=3. Not. 84:12*7=84. Perfect. 95:14*6=84, 95-84=11. Not. So yeah, only 40,48,84.I think that's all. So part a: 15,26,37,48,59,40,51,62,73,84,95. Part b: 40,48,84.Wait, but let me check if the problem says "determine among these numbers all those that are divisible by their digit sum". So the answer for part b is the subset of part a's numbers where the number is divisible by the sum of its digits. So according to my analysis, those are 40,48,84.But let me check if I made a mistake in any of the calculations. For example, 40: sum is 4+0=4. 40 ÷4=10. Correct. 48:4+8=12. 48 ÷12=4. Correct. 84:8+4=12. 84 ÷12=7. Correct. Others are not divisible. So yes, these three numbers. Therefore, that's the conclusion.Final Answera) boxed{15}, boxed{26}, boxed{37}, boxed{48}, boxed{59}, boxed{40}, boxed{51}, boxed{62}, boxed{73}, boxed{84}, boxed{95}b) boxed{40}, boxed{48}, boxed{84}</think>a) To list all two-digit natural numbers where one of the digits is 4 less than the other, we consider two cases: the tens digit is 4 less than the units digit, and the units digit is 4 less than the tens digit.Case 1: Tens digit is 4 less than the units digit (a = b + 4)- Possible units digits (b): 0 to 5- Corresponding tens digits (a): 4 to 9- Resulting numbers: 40, 51, 62, 73, 84, 95Case 2: Units digit is 4 less than the tens digit (b = a + 4)- Possible tens digits (a): 1 to 5- Corresponding units digits (b): 5 to 9- Resulting numbers: 15, 26, 37, 48, 59Combining both cases, the numbers are: 15, 26, 37, 48, 59, 40, 51, 62, 73, 84, 95.b) To determine which of these numbers are divisible by their digit sum:1. 15: Digit sum = 1 + 5 = 6. 15 ÷ 6 = 2.5 (Not divisible)2. 26: Digit sum = 2 + 6 = 8. 26 ÷ 8 = 3.25 (Not divisible)3. 37: Digit sum = 3 + 7 = 10. 37 ÷ 10 = 3.7 (Not divisible)4. 48: Digit sum = 4 + 8 = 12. 48 ÷ 12 = 4 (Divisible)5. 59: Digit sum = 5 + 9 = 14. 59 ÷ 14 ≈ 4.214 (Not divisible)6. 40: Digit sum = 4 + 0 = 4. 40 ÷ 4 = 10 (Divisible)7. 51: Digit sum = 5 + 1 = 6. 51 ÷ 6 = 8.5 (Not divisible)8. 62: Digit sum = 6 + 2 = 8. 62 ÷ 8 = 7.75 (Not divisible)9. 73: Digit sum = 7 + 3 = 10. 73 ÷ 10 = 7.3 (Not divisible)10. 84: Digit sum = 8 + 4 = 12. 84 ÷ 12 = 7 (Divisible)11. 95: Digit sum = 9 + 5 = 14. 95 ÷ 14 ≈ 6.7857 (Not divisible)The numbers divisible by their digit sum are 40, 48, and 84. Final Answera) boxed{15}, boxed{26}, boxed{37}, boxed{48}, boxed{59}, boxed{40}, boxed{51}, boxed{62}, boxed{73}, boxed{84}, boxed{95}b) boxed{40}, boxed{48}, boxed{84}]

🔑:The following table contains in the first row exactly all the numbers sought in (a). In the second row, the cross sum of each of these numbers is given. In the third row, the answer to the question of whether the respective number is divisible by its cross sum is stated (j for yes, n for no). According to this, the numbers 40, 84, and 48 are exactly those sought in (b).| Number | 40 | 51 | 15 | 62 | 26 | 73 | 37 | 84 | 48 | 95 | 59 || :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: || Cross Sum | 4 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 14 | 14 || Divisible? | j | n | n | n | n | n | n | j | j | n | n |

🧠:[<think>Okay, let me try to tackle these two problems step by step. I'll start with part I.Part I: When a=2, find the solution set of the inequality f(x) ≥ 4 - |x - 4|.First, let's substitute a=2 into the function f(x). So, f(x) = |x - 2|. The inequality becomes:|x - 2| ≥ 4 - |x - 4|Hmm, I need to solve this inequality. Let's break it down. Since there are absolute value expressions on both sides, it might help to consider different cases based on the critical points where the expressions inside the absolute values change their sign. The expressions inside the absolute values are x - 2 and x - 4. These equal zero when x = 2 and x = 4, respectively. So, the critical points are x=2 and x=4. Therefore, I can divide the real number line into three intervals:1. x < 22. 2 ≤ x < 43. x ≥ 4I'll solve the inequality in each interval.Case 1: x < 2In this interval, x - 2 is negative, so |x - 2| = -(x - 2) = 2 - x.Similarly, x - 4 is also negative (since x < 2 < 4), so |x - 4| = -(x - 4) = 4 - x.Substituting into the inequality:2 - x ≥ 4 - (4 - x)Simplify the right-hand side:4 - (4 - x) = 4 - 4 + x = xSo the inequality becomes:2 - x ≥ xBring all terms to the left side:2 - x - x ≥ 0 ⇒ 2 - 2x ≥ 0 ⇒ -2x ≥ -2 ⇒ x ≤ 1But remember we are in the interval x < 2. So the solution here is x ≤ 1.Case 2: 2 ≤ x < 4In this interval, x - 2 is non-negative, so |x - 2| = x - 2.x - 4 is still negative, so |x - 4| = 4 - x.Substitute into the inequality:x - 2 ≥ 4 - (4 - x) ⇒ x - 2 ≥ 4 - 4 + x ⇒ x - 2 ≥ xSimplify:x - 2 ≥ x ⇒ Subtract x from both sides: -2 ≥ 0Wait, that's not true. So this inequality simplifies to -2 ≥ 0, which is false. Therefore, there are no solutions in this interval.Case 3: x ≥ 4In this interval, both x - 2 and x - 4 are non-negative. So:|x - 2| = x - 2|x - 4| = x - 4Substitute into the inequality:x - 2 ≥ 4 - (x - 4) ⇒ x - 2 ≥ 4 - x + 4 ⇒ Wait, let's do this step by step.Wait, 4 - |x - 4| when x ≥ 4 becomes 4 - (x - 4) = 4 - x + 4 = 8 - x.So inequality is x - 2 ≥ 8 - xBring variables to the left and constants to the right:x + x ≥ 8 + 2 ⇒ 2x ≥ 10 ⇒ x ≥ 5But we are in the interval x ≥ 4. So the solution here is x ≥ 5.Combining all cases:From Case 1: x ≤ 1From Case 2: No solutionFrom Case 3: x ≥ 5Therefore, the solution set is x ≤ 1 or x ≥ 5.Let me check if this makes sense. Let's test x=0, which should be in the solution. f(0) = |0 - 2| = 2. Right-hand side: 4 - |0 - 4| = 4 - 4 = 0. So 2 ≥ 0, which is true. Good.x=1: f(1)=1, RHS=4 - |1 -4|=4 -3=1. So 1 ≥1, which is okay.x=2: f(2)=0. RHS=4 - |2 -4|=4 -2=2. 0 ≥2? No. So x=2 is not a solution, which matches our result.x=5: f(5)=3. RHS=4 - |5 -4|=4 -1=3. 3 ≥3, which is okay.x=6: f(6)=4. RHS=4 - |6-4|=4 -2=2. 4 ≥2, true.So the solution set seems correct. So part I is x ≤1 or x ≥5. So in interval notation, (-∞,1] ∪ [5, ∞).Part II: Given that the solution set of the inequality |f(2x + a) - 2f(x)| ≤ 2 with respect to x is [1,2], find the value of a.Given that f(x) = |x - a|, so let's first write out f(2x + a) and 2f(x).f(2x + a) = |2x + a - a| = |2x| = 2|x|Because 2x + a - a simplifies to 2x.Then 2f(x) = 2|x - a|So the expression inside the absolute value is:|2|x| - 2|x - a|| ≤ 2We can factor out the 2:2| |x| - |x - a| | ≤2Divide both sides by 2:| |x| - |x - a| | ≤1So the inequality simplifies to | |x| - |x - a| | ≤1And we are told that the solution set is x ∈ [1,2]. We need to find a.Given that a >1 (from the problem statement). So a is greater than 1.First, let's analyze the inequality | |x| - |x - a| | ≤1.This can be written as -1 ≤ |x| - |x - a| ≤1So split into two inequalities:1) |x| - |x - a| ≥ -12) |x| - |x - a| ≤1We need to find the values of x where both inequalities hold, and the solution is [1,2]. So we need to find a such that [1,2] is exactly the set of x where both inequalities are satisfied.Let me consider the expression |x| - |x - a|. Let's analyze it for different ranges of x.First, note that since a >1, and the solution set is x in [1,2], perhaps a is around 2? Let me see.But let's break down |x| and |x -a|.First, |x| is equal to x when x ≥0, and -x when x <0. Similarly, |x -a| is a -x when x ≤a, and x -a when x ≥a.But since the solution set is [1,2], and a >1, so x in [1,2] is to the left of a if a >2, or could be to the left or right if a is in [1,2]. But since a >1, and solution set is up to 2, maybe a is 3? Let's see.Wait, maybe it's better to consider different intervals for x.But x is in [1,2], so x ≥0, so |x| = x.So in [1,2], |x| = x. Then |x -a|. If a >2, then for x in [1,2], x -a is negative, so |x -a| = a -x.If a is between 1 and 2, then for x in [1,2], when x approaches a, |x -a| = a -x if x ≤a, and x -a if x ≥a. So if a is in (1,2), then part of [1,2] is to the left of a and part is to the right. However, given the solution set is exactly [1,2], maybe a is such that the inequality is only satisfied here. Let's see.But perhaps first, let's write the expression in the interval [1,2]. Since x is in [1,2], |x| = x. Then |x -a| is:If a >2, then x in [1,2] < a, so |x -a| = a -x.So expression |x| - |x -a| becomes x - (a -x) = x -a +x = 2x -a.If a ≤2, then in [1,2], x can be less than a or greater than a. If a is in (1,2), then part of [1,2] is left of a and part is right. For x in [1,a], |x -a| = a -x; for x in [a,2], |x -a| = x -a.But the solution set is [1,2], so perhaps a is chosen such that the inequality holds only in [1,2]. Let's try to model this.Case 1: a >2In this case, for x in [1,2], |x -a| = a -x, so expression |x| - |x -a| = x - (a -x) = 2x -a.Therefore, the inequality | |x| - |x -a| | = |2x -a| ≤1.So |2x -a| ≤1, which implies -1 ≤2x -a ≤1 ⇒ a -1 ≤2x ≤a +1 ⇒ (a -1)/2 ≤x ≤(a +1)/2But the solution set is [1,2]. So we need:(a -1)/2 =1 and (a +1)/2=2.Solving (a -1)/2=1 ⇒ a -1=2 ⇒a=3Similarly, (a +1)/2=2 ⇒a +1=4 ⇒a=3So both equations give a=3. So if a=3, then the inequality |2x -3| ≤1 would give x in [1,2], which matches the solution set. So a=3 is a candidate.Case 2: a ≤2But given a>1, so a ∈ (1,2]. Let's check if this is possible.If a ∈ (1,2], then in x ∈ [1,2], when x ∈ [1,a], |x -a|=a -x; when x ∈ [a,2], |x -a|=x -a.So the expression |x| - |x -a| would be:For x ∈ [1,a]: x - (a -x) = 2x -aFor x ∈ [a,2]: x - (x -a) = aSo in [1,a], it's 2x -a; in [a,2], it's a.Therefore, the inequality | |x| - |x -a| | ≤1 becomes:In x ∈ [1,a]: |2x -a| ≤1In x ∈ [a,2]: |a| ≤1 ⇒ but a>1, so |a|=a >1, so |a| ≤1 is false. Therefore, in [a,2], the inequality would not hold. Hence, the solution set would be only part of [1,a] where |2x -a| ≤1.But the given solution set is [1,2], so this would require that [1,2] is contained within [1,a], but [a,2] cannot be part of the solution. But since a ≤2, this would not cover the entire [1,2] interval unless a=2. Let's check a=2.If a=2:For x ∈ [1,2], since a=2, which is the upper bound. So in [1,2], |x -2|=2 -x for x ∈[1,2). Then the expression |x| - |x -2| becomes x - (2 -x) =2x -2.So |2x -2| ≤1 ⇒ |2(x -1)| ≤1 ⇒ |x -1| ≤0.5 ⇒ x ∈ [0.5,1.5]But the solution set is supposed to be [1,2]. So this does not match. Hence, a=2 is not suitable.If a is in (1,2), then as we saw, the solution set would be part of [1,a] where |2x -a| ≤1. Let's suppose a=1.5. Then |2x -1.5| ≤1 ⇒ -1 ≤2x -1.5 ≤1 ⇒0.5 ≤2x ≤2.5 ⇒0.25 ≤x ≤1.25. But since x is in [1,a]=[1,1.5], then the solution is x ∈ [1,1.25]. Which is not [1,2]. So no, a in (1,2) won't work because the solution set would be a subset of [1,a], which is less than [1,2].Therefore, the only possible value is a=3.Wait, but let's verify with a=3.Given a=3, the inequality becomes | |x| - |x -3| | ≤1.But we need to check when the solution is [1,2]. Let's see:For x in [1,2], since a=3 >2, as discussed earlier, the expression inside is |2x -3| ≤1, leading to x in [1,2]. Let's confirm:|2x -3| ≤1 ⇒ -1 ≤2x -3 ≤1 ⇒2 ≤2x ≤4 ⇒1 ≤x ≤2. Perfect. So yes, for a=3, the inequality |2x -3| ≤1 holds exactly when x ∈ [1,2]. Therefore, the solution set is [1,2], which matches the given condition.Therefore, the value of a is 3.But just to be thorough, let's check x outside [1,2] when a=3.For x <1, since a=3 >2, |x| - |x -3| = if x <0, |x|=-x, |x-3|=3 -x, so expression is -x - (3 -x)= -3. The absolute value is 3, which is not ≤1. If x ∈ [0,1), |x|=x, |x-3|=3 -x, so expression is x - (3 -x)=2x -3. Then |2x -3|. When x ∈ [0,1), 2x ∈ [0,2), so 2x -3 ∈ [-3,-1). The absolute value is between 1 and 3, which is greater than 1. Hence, inequality does not hold.For x >2, since a=3, |x| - |x -3|. For x ∈ (3, ∞), |x|=x, |x -3|=x -3, so expression is x - (x -3)=3. |3| =3 >1, so inequality does not hold. For x ∈ (2,3], |x|=x, |x -3|=3 -x. So expression is x - (3 -x)=2x -3. Then |2x -3|. When x ∈ (2,3], 2x ∈ (4,6], so 2x -3 ∈ (1,3]. Absolute value is between 1 and 3, so |2x -3| >1. Hence, inequality does not hold. Therefore, outside of [1,2], the inequality does not hold. Hence, a=3 is correct.Therefore, the answer for part II is a=3.Final Answer(I) The solution set is boxed{(-infty, 1] cup [5, infty)}.(II) The value of ( a ) is boxed{3}.</think>Given the function ( f(x) = |x - a| ) where ( a > 1 ):(I) When ( a = 2 ), we need to solve the inequality ( f(x) geq 4 - |x - 4| ). Substituting ( a = 2 ), the inequality becomes ( |x - 2| geq 4 - |x - 4| ). We consider the critical points ( x = 2 ) and ( x = 4 ) and solve the inequality in three intervals:1. For ( x < 2 ): [ 2 - x geq 4 - (4 - x) implies 2 - x geq x implies 2 geq 2x implies x leq 1 ]2. For ( 2 leq x < 4 ): [ x - 2 geq 4 - (4 - x) implies x - 2 geq x implies -2 geq 0 quad text{(no solution)} ]3. For ( x geq 4 ): [ x - 2 geq 8 - x implies 2x geq 10 implies x geq 5 ]Combining the solutions from all intervals, we get ( x leq 1 ) or ( x geq 5 ). Therefore, the solution set is (boxed{(-infty, 1] cup [5, infty)}).(II) Given the solution set of the inequality ( |f(2x + a) - 2f(x)| leq 2 ) is ( { x | 1 leq x leq 2 } ), we need to find ( a ). Simplifying the inequality, we get:[| |x| - |x - a| | leq 1]For ( x in [1, 2] ), since ( a > 1 ), we consider ( a > 2 ). The inequality becomes ( |2x - a| leq 1 ), leading to:[frac{a - 1}{2} leq x leq frac{a + 1}{2}]Given the solution set is ( [1, 2] ), we solve:[frac{a - 1}{2} = 1 implies a = 3 quad text{and} quad frac{a + 1}{2} = 2 implies a = 3]Thus, the value of ( a ) is (boxed{3}).,<think>Okay, let's tackle these two problems one by one. Both parts are about absolute value functions and inequalities, so I need to remember how to handle those. Let's start with part (I).Problem I: When a=2, find the solution set of the inequality f(x) ≥ 4 - |x - 4|.First, let's substitute a=2 into the function f(x). So, f(x) becomes |x - 2|. The inequality we need to solve is:|x - 2| ≥ 4 - |x - 4|Hmm, so we have an inequality involving two absolute values. To solve such inequalities, it's often helpful to consider different cases based on the critical points where the expressions inside the absolute values change their sign. The critical points here are x=2 (from |x - 2|) and x=4 (from |x - 4|). So, the number line is divided into three intervals:1. x < 22. 2 ≤ x ≤ 43. x > 4I need to analyze each interval separately.Case 1: x < 2In this interval, x - 2 is negative, so |x - 2| = -(x - 2) = 2 - x.Similarly, x - 4 is also negative (since x < 2 < 4), so |x - 4| = -(x - 4) = 4 - x.Substituting these into the inequality:2 - x ≥ 4 - (4 - x)Simplify the right-hand side:4 - (4 - x) = 4 - 4 + x = xSo the inequality becomes:2 - x ≥ xSolving for x:2 ≥ 2xDivide both sides by 2:1 ≥ xBut in this case, we already have x < 2. So the solution here is x ≤ 1. But since the case is x < 2, the overlap is x ≤ 1. So in this interval, the solution is x ≤ 1.Case 2: 2 ≤ x ≤ 4In this interval, x - 2 is non-negative, so |x - 2| = x - 2.For |x - 4|, since x ≤ 4, x - 4 is non-positive, so |x - 4| = 4 - x.Substitute into the inequality:x - 2 ≥ 4 - (4 - x)Simplify the right-hand side:4 - (4 - x) = xSo the inequality becomes:x - 2 ≥ xSubtract x from both sides:-2 ≥ 0Wait, that's not possible. -2 is not greater than or equal to 0. So this case has no solution.Hmm, so in the interval 2 ≤ x ≤ 4, there's no x that satisfies the inequality.Case 3: x > 4Here, both x - 2 and x - 4 are positive. So |x - 2| = x - 2 and |x - 4| = x - 4.Substitute into the inequality:x - 2 ≥ 4 - (x - 4)Simplify the right-hand side:4 - (x - 4) = 4 - x + 4 = 8 - xSo the inequality becomes:x - 2 ≥ 8 - xBring variables to the left and constants to the right:x + x ≥ 8 + 22x ≥ 10Divide by 2:x ≥ 5But in this case, x > 4. So the solution here is x ≥ 5.Combining all cases:From Case 1: x ≤ 1From Case 2: No solutionFrom Case 3: x ≥ 5Therefore, the solution set is x ≤ 1 or x ≥ 5.Wait, let me check if that's correct. Let me verify with a sample point in each interval.For x=0 (Case 1):|0 - 2| = 24 - |0 - 4| = 4 - 4 = 02 ≥ 0, which is true.For x=3 (Case 2):|3 - 2| = 14 - |3 - 4| = 4 - 1 = 31 ≥ 3? No, so correct, no solution here.For x=6 (Case 3):|6 - 2| = 44 - |6 - 4| = 4 - 2 = 24 ≥ 2, which is true.Another check at x=1:|1 - 2| = 14 - |1 - 4| = 4 - 3 = 11 ≥ 1, which is equality, so included.At x=5:|5 - 2| = 34 - |5 - 4| = 4 - 1 = 33 ≥ 3, equality holds, included.So the solution set is indeed (-∞, 1] ∪ [5, ∞). So part (I) is done.Problem II: Given that the solution set of the inequality |f(2x + a) - 2f(x)| ≤ 2 with respect to x is {x | 1 ≤ x ≤ 2}, find the value of a.Given that f(x) = |x - a|, so let's first express f(2x + a) and 2f(x):f(2x + a) = |2x + a - a| = |2x|2f(x) = 2|x - a|So the expression inside the absolute value becomes:| |2x| - 2|x - a| | ≤ 2So the inequality is:| |2x| - 2|x - a| | ≤ 2We need to find the value of a > 1 such that the solution set is [1, 2].First, let's simplify the expression inside the absolute value. Let's denote:Inner expression: |2x| - 2|x - a|But |2x| is 2|x|, so:2|x| - 2|x - a| = 2(|x| - |x - a|)Therefore, the inequality becomes:| 2(|x| - |x - a|) | ≤ 2Divide both sides by 2:| |x| - |x - a| | ≤ 1So now, the inequality simplifies to:| |x| - |x - a| | ≤ 1We need to find a such that this inequality holds exactly when x is in [1, 2].Given that a > 1.This seems more complex. Let's analyze the expression | |x| - |x - a| | ≤ 1.First, let's consider that x is in [1,2]. So x is positive, so |x| = x.Therefore, |x - a| is |x - a|. Since a > 1, and x ∈ [1,2], let's see:If a > 2, then x - a is negative for all x ∈ [1,2], so |x - a| = a - x.If 1 < a ≤ 2, then for x ∈ [1, a], |x - a| = a - x, and for x ∈ [a, 2], |x - a| = x - a.But the solution set is [1,2], so we need to check if a is in (1,2] or a > 2.Wait, but the problem says a > 1, so a could be in (1, ∞). But we need to find a such that the inequality holds exactly for x ∈ [1,2]. Let's proceed step by step.First, let's note that since the solution set is [1,2], that means for x in [1,2], the inequality holds, and for x outside [1,2], it doesn't. Therefore, we need to ensure that:1. For x ∈ [1,2], | |x| - |x - a| | ≤ 1.2. For x < 1 or x > 2, | |x| - |x - a| | > 1.But since we need to find a, perhaps we can focus on the interval [1,2] first and find the conditions on a that make the inequality hold there, and also ensure that outside [1,2], the inequality does not hold.Let me first handle the inner expression |x| - |x - a|.Since x ∈ [1,2], |x| = x.So we have:|x| - |x - a| = x - |x - a|So the expression becomes |x - |x - a|| ≤ 1But since we have absolute value of (x - |x - a|) ≤ 1, let's consider two cases based on the value of a relative to the interval [1,2].Case 1: a ≤ 1. But the problem states a > 1, so this is not possible.Case 2: 1 < a ≤ 2.In this case, for x ∈ [1,2], when x ∈ [1,a], |x - a| = a - x, and when x ∈ [a,2], |x - a| = x - a.Case 3: a > 2.In this case, for x ∈ [1,2], since x ≤ 2 < a, so |x - a| = a - x.Therefore, depending on whether a is ≤2 or >2, the expression |x - a| changes.Let's analyze both possibilities.Subcase 1: 1 < a ≤ 2Then in [1, a], |x - a| = a - x, so:|x| - |x - a| = x - (a - x) = 2x - aIn [a, 2], |x - a| = x - a, so:|x| - |x - a| = x - (x - a) = aTherefore, the expression | |x| - |x - a| | becomes:In [1, a]: |2x - a|In [a, 2]: |a|So for x ∈ [1, a], |2x - a| ≤ 1, and for x ∈ [a,2], |a| ≤ 1. But a >1, so |a| = a >1, so |a| ≤1 is impossible. Therefore, if 1 < a ≤2, then in [a,2], the inequality |a| ≤1 must hold, but since a>1, this is false. Therefore, if 1 < a ≤2, the inequality would not hold on [a,2], which is part of [1,2]. Therefore, this case is invalid because the solution set would not include [a,2], but the problem states the solution set is all of [1,2]. Hence, a cannot be in (1,2].Subcase 2: a > 2In this case, for all x ∈ [1,2], |x - a| = a - x. Therefore,|x| - |x - a| = x - (a - x) = 2x - aTherefore, the expression inside the absolute value is |2x - a|, so the inequality becomes:|2x - a| ≤1Which implies:-1 ≤ 2x - a ≤1Solving for a:Adding a to all parts:2x -1 ≤ a ≤ 2x +1But we need this to hold for all x ∈ [1,2]. Wait, but the inequality |2x - a| ≤1 must hold for all x ∈ [1,2]. However, the problem states that the solution set is exactly [1,2], meaning that the inequality holds if and only if x ∈ [1,2]. Therefore, for x ∈ [1,2], |2x - a| ≤1 must hold, and for x not in [1,2], |2x - a| >1 must hold. Wait, but the original inequality is | |x| - |x - a| | ≤1, which for a >2 becomes |2x - a| ≤1. So in this case, the inequality reduces to |2x - a| ≤1, and we need the solution set of this inequality to be x ∈ [1,2].Therefore, the inequality |2x - a| ≤1 must be equivalent to x ∈ [1,2]. Let's solve |2x - a| ≤1:-1 ≤2x -a ≤1Add a:a -1 ≤2x ≤a +1Divide by 2:(a -1)/2 ≤x ≤(a +1)/2So the solution set is [(a -1)/2, (a +1)/2]But according to the problem, the solution set is [1,2]. Therefore, we need:(a -1)/2 =1 and (a +1)/2=2Solve both equations:From (a -1)/2=1:a -1=2 => a=3From (a +1)/2=2:a +1=4 =>a=3Therefore, a=3.Now, check if this is correct.If a=3, then the inequality |2x -3| ≤1:Solution is [(3 -1)/2, (3 +1)/2] = [1, 2], which matches the given solution set.Additionally, for x outside [1,2], |2x -3| >1, which would mean the original inequality doesn't hold, as required. Hence, a=3 is the correct answer.But let's confirm this with the original inequality. Let's plug a=3 into | |x| - |x -3| | ≤1 for x in [1,2].Since x ∈ [1,2], |x|=x, and |x -3|=3 -x.Therefore, |x - (3 -x)| = |2x -3|So yes, the inequality becomes |2x -3| ≤1, which as we saw, gives x ∈ [1,2]. So that's correct.Therefore, the value of a is 3.Final Answer(I) The solution set is (boxed{(-infty, 1] cup [5, infty)}).(II) The value of (a) is (boxed{3}).</think>Given the function ( f(x) = |x - a| ) where ( a > 1 ):(I) When ( a = 2 ), we need to solve the inequality ( |x - 2| geq 4 - |x - 4| ).1. Case 1: ( x < 2 ) - ( |x - 2| = 2 - x ) - ( |x - 4| = 4 - x ) - Inequality: ( 2 - x geq x ) - Solution: ( x leq 1 )2. Case 2: ( 2 leq x leq 4 ) - ( |x - 2| = x - 2 ) - ( |x - 4| = 4 - x ) - Inequality: ( x - 2 geq x ) - Solution: No solution3. Case 3: ( x > 4 ) - ( |x - 2| = x - 2 ) - ( |x - 4| = x - 4 ) - Inequality: ( x - 2 geq 8 - x ) - Solution: ( x geq 5 )Combining all cases, the solution set is ( (-infty, 1] cup [5, infty) ).(II) Given the solution set ( { x | 1 leq x leq 2 } ) for the inequality ( | f(2x + a) - 2f(x) | leq 2 ):1. Substitute ( f(x) = |x - a| ): - ( f(2x + a) = |2x| ) - ( 2f(x) = 2|x - a| ) - Inequality: ( | |2x| - 2|x - a| | leq 2 ) - Simplify: ( | |x| - |x - a| | leq 1 )2. For ( x in [1, 2] ): - ( |x| = x ) - ( |x - a| = a - x ) (since ( a > 2 )) - Inequality: ( |2x - a| leq 1 ) - Solve: ( -1 leq 2x - a leq 1 ) - Solution: ( (a - 1)/2 leq x leq (a + 1)/2 ) - Given solution set ( [1, 2] ), solve ( (a - 1)/2 = 1 ) and ( (a + 1)/2 = 2 ) - Result: ( a = 3 )Final Answers:(I) The solution set is (boxed{(-infty, 1] cup [5, infty)}).(II) The value of (a) is (boxed{3}).]

🔑:Solution:(I) When a=2, f(x)+| x-4 | is given by:begin{cases} & -2x+6, & xleqslant 2 & 2, & -2 < x < 4 & 2x-6, & xgeqslant 4 end{cases}When xleqslant 2, from -2x+6geqslant 4Rightarrow xleqslant 1When -2 < x < 4, from 2geqslant 4, which is not true.When xgeqslant 4, from 2x-6geqslant 4Rightarrow xgeqslant 5In summary: When a=2, the solution set of the inequality is {x|xleqslant 1 or xgeqslant 5}(II) Let h(x)=f(2x+a)-2f(x)Then h(x) is given by:begin{cases} & -2a, & xleqslant 0 & 4x-2a, & 0 < x < a & 2a, & xgeqslant a end{cases}From |h(x)|leqslant 2, we get dfrac{a-1}{2}leqslant xleqslant dfrac{a+1}{2}Given that the solution set of |h(x)|leqslant 2 is { x|1leqslant xleqslant 2 }Thus, dfrac{a-1}{2}=1 and dfrac{a+1}{2}=2,Therefore, a=3. So, the final answer is boxed{a=3}.

🧠:[<think>Okay, so I need to find the measure of angle MHC in triangle ABC where angle A is 100 degrees, angle B is 50 degrees. AH is the altitude from A to BC, and BM is the median from B to AC. Hmm, let me try to visualize this triangle first.First, let me draw triangle ABC. Let's label the vertices A, B, and C. Angle at A is 100°, angle at B is 50°, so angle at C must be 30° because the sum of angles in a triangle is 180°. So angle C = 180 - 100 - 50 = 30°. Got that.Now, AH is the altitude from A to BC. That means AH is perpendicular to BC. So H is the foot of the perpendicular from A to BC. Therefore, AH is perpendicular to BC, forming a right angle at H. So angle AHB is 90°, and angle AHC is also 90° since H is on BC.BM is the median from B to AC. A median connects a vertex to the midpoint of the opposite side. So BM connects B to the midpoint M of AC. So M divides AC into two equal parts: AM = MC.I need to find angle MHC. So points M, H, and C are involved. Let me try to see the positions of these points. H is on BC, M is the midpoint of AC, and C is one of the vertices. So connecting these points forms triangle MHC. I need to find the measure of the angle at H in triangle MHC, which is angle MHC.Hmm, how can I approach this? Maybe using coordinates? Assign coordinates to the triangle's vertices to calculate coordinates of H and M, then compute the angle. Let me try that.Let's place point B at the origin (0,0) to simplify calculations. Let me denote coordinates:- Let’s place point B at (0,0).- Let’s assume BC is along the x-axis for simplicity. Since angle at B is 50°, and angle at C is 30°, the triangle isn't a right triangle. Wait, but if I place BC on the x-axis, then point C will be at some point (c,0), and point A somewhere in the plane.Alternatively, maybe it's better to use the Law of Sines to find the lengths of sides. Let me recall that in triangle ABC, the Law of Sines states that:a / sin A = b / sin B = c / sin CWhere a is the length of BC, b is the length of AC, and c is the length of AB.Given angles:Angle A = 100°, angle B = 50°, angle C = 30°So sides:a = BC = 2R sin A = 2R sin 100°b = AC = 2R sin B = 2R sin 50°c = AB = 2R sin C = 2R sin 30°But maybe instead of R (the circumradius), I can assign a specific length to one side to make calculations easier. Let's choose BC (side a) to be of length 1 for simplicity. Then:a = BC = 1 = 2R sin 100° => R = 1 / (2 sin 100°)Then:b = AC = 2R sin 50° = [1 / sin 100°] * sin 50°c = AB = 2R sin 30° = [1 / sin 100°] * sin 30°But sin 100° is sin(80°) which is approximately 0.9848, but maybe exact expressions can be used. Alternatively, use coordinates.Alternatively, place point B at (0,0), point C at (c,0), point A somewhere in the plane. Let's do that.Let’s set coordinate system:- Let’s set point B at (0,0).- Let’s set point C at (c,0). Let's choose c = 1 for simplicity, so BC = 1 unit.- Then, angle at B is 50°, so angle between BA and BC is 50°. We need to find coordinates of point A.Wait, but angle at A is 100°, not at B. Wait, angle at A is 100°, angle at B is 50°, angle at C is 30°.Wait, to construct triangle with these angles, sides are proportional to the sines of the opposite angles. So:BC / sin A = AC / sin B = AB / sin CSo BC / sin 100° = AC / sin 50° = AB / sin 30°If BC = 1, then AC = sin 50° / sin 100°, AB = sin 30° / sin 100°Compute these:sin 50° ≈ 0.7660sin 100° ≈ 0.9848sin 30° = 0.5Therefore:AC ≈ 0.7660 / 0.9848 ≈ 0.7778AB ≈ 0.5 / 0.9848 ≈ 0.5077So sides: BC = 1, AC ≈ 0.7778, AB ≈ 0.5077Now, placing point B at (0,0), point C at (1,0). Need to find coordinates of point A.Let’s denote coordinates of A as (x,y). Then distances:From A to B: sqrt(x² + y²) ≈ 0.5077From A to C: sqrt((x - 1)² + y²) ≈ 0.7778Also, angle at A is 100°, which is between sides AB and AC. Wait, maybe using Law of Cosines here would help?Wait, angle at A is 100°, so in triangle ABC, angle at A is between sides AB and AC, so by Law of Cosines:BC² = AB² + AC² - 2 AB * AC * cos(angle A)But BC is 1, AB ≈0.5077, AC≈0.7778Check:Left side: 1² = 1Right side: (0.5077)² + (0.7778)² - 2 * 0.5077 * 0.7778 * cos(100°)Compute:0.5077² ≈ 0.25780.7778² ≈ 0.6051Sum ≈ 0.2578 + 0.6051 ≈ 0.86292 * 0.5077 * 0.7778 ≈ 2 * 0.5077 * 0.7778 ≈ 0.794cos(100°) ≈ -0.1736So the product: -0.794 * 0.1736 ≈ -0.1379Therefore, total right side ≈ 0.8629 - (-0.1379) ≈ 0.8629 + 0.1379 ≈ 1.0008Which is approximately 1, which checks out. So the approximated lengths are correct.Now, back to coordinates. Let me set point B at (0,0), point C at (1,0). We need to find coordinates (x,y) of point A such that distance from A to B is approximately 0.5077, distance from A to C is approximately 0.7778.So:sqrt(x² + y²) ≈ 0.5077 --> x² + y² ≈ 0.2578sqrt((x - 1)² + y²) ≈ 0.7778 --> (x - 1)² + y² ≈ 0.6051Subtract first equation from second:(x - 1)² + y² - (x² + y²) ≈ 0.6051 - 0.2578Expand (x - 1)^2 - x^2 = x² - 2x + 1 - x² = -2x + 1So -2x + 1 ≈ 0.3473Therefore, -2x ≈ 0.3473 - 1 = -0.6527So x ≈ (-0.6527)/(-2) ≈ 0.32635So x ≈ 0.32635Then substitute x back into first equation:x² + y² ≈ 0.2578(0.32635)^2 + y² ≈ 0.25780.1065 + y² ≈ 0.2578y² ≈ 0.2578 - 0.1065 ≈ 0.1513y ≈ sqrt(0.1513) ≈ 0.389So coordinates of point A are approximately (0.326, 0.389)Let me verify the distances:Distance from A to B: sqrt(0.326² + 0.389²) ≈ sqrt(0.106 + 0.151) ≈ sqrt(0.257) ≈ 0.507, which matches.Distance from A to C: sqrt((0.326 - 1)^2 + (0.389)^2) ≈ sqrt((-0.674)^2 + 0.151) ≈ sqrt(0.454 + 0.151) ≈ sqrt(0.605) ≈ 0.778, which matches.Okay, so coordinates:- B: (0,0)- C: (1,0)- A: approximately (0.326, 0.389)Now, AH is the altitude from A to BC. Since BC is along the x-axis from (0,0) to (1,0), the altitude from A to BC is a vertical line from A down to BC. But wait, if BC is horizontal, then the altitude from A is vertical? Wait, no. The altitude is perpendicular to BC. Since BC is horizontal, the altitude will be vertical. So since BC is along the x-axis, the altitude from A to BC is a vertical line. Therefore, the foot H has the same x-coordinate as A, but y=0.Wait, coordinates of A are (0.326, 0.389). So if we drop a vertical line from A to BC (the x-axis), H would be at (0.326, 0). But wait, is that correct?Wait, BC is from (0,0) to (1,0). The slope of BC is 0, so the altitude from A to BC must be perpendicular to BC. Since BC is horizontal, the altitude is vertical. Therefore, yes, H is directly below A on the x-axis. So H is (0.326, 0).But let me confirm. The altitude from A to BC is a vertical line, so indeed H is (0.326, 0).Now, BM is the median from B to AC. So M is the midpoint of AC. Coordinates of A: (0.326, 0.389), coordinates of C: (1,0). So midpoint M is average of coordinates:x-coordinate: (0.326 + 1)/2 ≈ 1.326/2 ≈ 0.663y-coordinate: (0.389 + 0)/2 ≈ 0.1945So M is approximately (0.663, 0.1945)So now we have coordinates of points:- M: (0.663, 0.1945)- H: (0.326, 0)- C: (1,0)We need to find angle MHC, which is the angle at H between points M, H, and C.Wait, angle MHC is the angle at H, so it's ∠MHC = angle at H between points M, H, and C. So we need to compute the angle between vectors HM and HC.First, let's find coordinates of HM and HC.Point H: (0.326, 0)Point M: (0.663, 0.1945)Point C: (1, 0)So vector HM = M - H = (0.663 - 0.326, 0.1945 - 0) ≈ (0.337, 0.1945)Vector HC = C - H = (1 - 0.326, 0 - 0) ≈ (0.674, 0)Now, to find the angle between vectors HM and HC, we can use the dot product formula:cosθ = (HM · HC) / (|HM| |HC|)Compute HM · HC = (0.337)(0.674) + (0.1945)(0) ≈ 0.337 * 0.674 ≈ 0.227|HM| = sqrt(0.337² + 0.1945²) ≈ sqrt(0.1136 + 0.0378) ≈ sqrt(0.1514) ≈ 0.389|HC| = sqrt(0.674² + 0²) ≈ 0.674Therefore,cosθ ≈ 0.227 / (0.389 * 0.674) ≈ 0.227 / (0.262) ≈ 0.866Therefore, θ ≈ arccos(0.866) ≈ 30°Wait, arccos(0.866) is approximately 30 degrees, since cos(30°) ≈ 0.866.So angle MHC is approximately 30 degrees. But in the original triangle, angle C is 30°, so that might be a clue. Maybe angle MHC is equal to angle C?But let me check if this approximation makes sense.Alternatively, perhaps there is a geometric proof rather than coordinate calculation.Let me think. In triangle ABC, with given angles. AH is the altitude, BM is the median. We need to find angle MHC.First, note that angle C is 30°, which is the same as the angle we found for angle MHC. Maybe there's a relationship here.Alternatively, since M is the midpoint of AC, and H is the foot of the altitude from A to BC, perhaps triangle MHC is similar to some other triangle?Alternatively, since BM is a median, perhaps we can use properties of medians in triangles.Alternatively, since AH is an altitude, and BM is a median, their intersection might have some properties, but not sure.Wait, maybe using trigonometric identities.Alternatively, since we found through coordinates that angle MHC is approximately 30°, which is equal to angle C. Maybe it's exactly 30°. But let's check with more precise calculation.Wait, in the coordinate method, we approximated the coordinates, which might have introduced some error, but the result was very close to 30°. Let me try to do a more exact calculation.Let me use exact expressions instead of approximate decimals.Given triangle ABC with angles A=100°, B=50°, C=30°, side BC=1.Using the Law of Sines:AB / sin C = BC / sin A => AB = BC * sin C / sin A = 1 * sin 30° / sin 100°Similarly, AC = BC * sin B / sin A = 1 * sin 50° / sin 100°So AB = (1/2) / sin 100°, and AC = sin 50° / sin 100°Then coordinates of point A can be determined by using distances from B(0,0) and C(1,0). Let me set coordinates:Let’s denote coordinates of A as (x, y). Then:Distance from A to B: sqrt(x² + y²) = AB = (1/2)/sin 100°Distance from A to C: sqrt((x - 1)² + y²) = AC = sin 50° / sin 100°So equations:x² + y² = (1/(2 sin 100°))²(x - 1)² + y² = (sin 50° / sin 100°)^2Subtract first equation from second:(x - 1)^2 + y² - x² - y² = (sin²50° / sin²100°) - (1/(2 sin 100°))²Expand left side:x² - 2x + 1 + y² - x² - y² = -2x + 1Right side:[sin²50° - (1/4)] / sin²100°So:-2x + 1 = [sin²50° - 1/4] / sin²100°Hence,x = [1 - (sin²50° - 1/4)/sin²100°] / 2Let me compute sin²50° and sin²100°.First, note that sin 100° = sin(80°) = cos 10°, but maybe use exact trigonometric identities.Wait, sin 50° and sin 100° are related. Also, note that sin 100° = sin(80°) = cos(10°), and sin 50° = cos(40°). But perhaps using double-angle formulas.Alternatively, compute numerical values:sin 50° ≈ 0.7660444431sin 100° ≈ 0.9848077530So sin²50° ≈ (0.7660444431)^2 ≈ 0.5868sin²100° ≈ (0.9848077530)^2 ≈ 0.9698Then numerator on the right side: sin²50° - 1/4 ≈ 0.5868 - 0.25 = 0.3368Thus,[0.3368] / 0.9698 ≈ 0.3473Therefore,x ≈ [1 - 0.3473]/2 ≈ (0.6527)/2 ≈ 0.32635Which matches the approximate value before.Then y² = (1/(2 sin 100°))² - x² ≈ [1/(2 * 0.9848)]² - (0.32635)^2Compute 1/(2 * 0.9848) ≈ 0.5077So (0.5077)^2 ≈ 0.2578Then y² ≈ 0.2578 - (0.32635)^2 ≈ 0.2578 - 0.1065 ≈ 0.1513Therefore, y ≈ sqrt(0.1513) ≈ 0.389So coordinates of A are approximately (0.32635, 0.389). Therefore, H is (0.32635, 0), M is midpoint of AC: ((0.32635 + 1)/2, (0.389 + 0)/2) ≈ (0.663175, 0.1945)So vector HM is M - H ≈ (0.663175 - 0.32635, 0.1945 - 0) ≈ (0.336825, 0.1945)Vector HC is C - H ≈ (1 - 0.32635, 0 - 0) ≈ (0.67365, 0)Dot product HM · HC ≈ 0.336825 * 0.67365 + 0.1945 * 0 ≈ 0.336825 * 0.67365 ≈ 0.336825 * 0.67365 ≈ Let's compute 0.336825 * 0.67365:First, 0.3 * 0.6 = 0.180.3 * 0.07365 ≈ 0.02210.036825 * 0.6 ≈ 0.02210.036825 * 0.07365 ≈ ~0.0027Total ≈ 0.18 + 0.0221 + 0.0221 + 0.0027 ≈ 0.2269|HM| ≈ sqrt(0.336825² + 0.1945²) ≈ sqrt(0.1134 + 0.0378) ≈ sqrt(0.1512) ≈ 0.389|HC| ≈ sqrt(0.67365² + 0²) ≈ 0.67365So cosθ ≈ 0.2269 / (0.389 * 0.67365) ≈ 0.2269 / 0.262 ≈ 0.866Which is cosθ ≈ 0.866, which corresponds to θ ≈ 30°, since cos(30°) ≈ √3/2 ≈ 0.866.Therefore, angle MHC is 30 degrees. So, the answer should be 30°, boxed as boxed{30}.But let me check if there's a way to see this without coordinates.Given triangle ABC with angles 100°, 50°, 30°, AH is the altitude, BM is the median. Need angle MHC.Since angle C is 30°, and our calculation gives angle MHC as 30°, maybe angle MHC is equal to angle C. Let's see if we can prove that.Let me consider triangle MHC. If we can show that triangle MHC is similar to some triangle with angle 30°, then angle MHC would be 30°.Alternatively, perhaps quadrilateral MHCH or some other quadrilateral has certain properties.Alternatively, since M is the midpoint of AC and H is the foot of the altitude from A, maybe some midline theorem or right triangle properties can be applied.Alternatively, consider reflecting point A over H to get a point, but not sure.Alternatively, using trigonometric Ceva's theorem or Stewart's theorem.Wait, another approach: In triangle ABC, since AH is the altitude, then in triangle AHC, angle at H is 90°, and angle at C is 30°, so angle at A in triangle AHC is 60°, because angles sum to 180°. Wait, but angle at A in triangle ABC is 100°, so angle at A in triangle AHC would be different. Wait, no, triangle AHC is part of triangle ABC.Wait, maybe not. Let me think.In triangle ABC, AH is the altitude, so triangle AHB and AHC are both right triangles.In triangle AHC, angle at H is 90°, angle at C is 30°, so angle at A in triangle AHC would be 180 - 90 - 30 = 60°, but angle at A in triangle ABC is 100°, so this seems conflicting.Wait, no. Because point H is on BC, so angle at A in triangle AHC is still the same as angle BAC in triangle ABC, which is 100°, but that can't be. Wait, no. Wait, in triangle AHC, the angles are at A, H, and C.Wait, but angle at A in triangle AHC is not the same as angle BAC. Wait, angle BAC is 100°, but in triangle AHC, angle at A is still part of that 100°, but split by the altitude. Wait, no. Let me clarify.In triangle ABC, angle at A is 100°. When we drop altitude AH to BC, we divide triangle ABC into two right triangles: ABH and AHC.In triangle ABH, angles are at A, B, H. Angle at H is 90°, angle at B is 50°, so angle at A in triangle ABH is 180 - 90 - 50 = 40°.Similarly, in triangle AHC, angle at H is 90°, angle at C is 30°, so angle at A in triangle AHC is 180 - 90 - 30 = 60°.Therefore, the original angle at A (100°) is split into 40° and 60° by the altitude AH. So angle BAH = 40°, angle HAC = 60°.Interesting. So in triangle AHC, angle at A is 60°, angle at H is 90°, angle at C is 30°, making it a 30-60-90 triangle.Therefore, in triangle AHC, sides are in ratio 1 : √3 : 2, with the side opposite 30° (AH) being half the hypotenuse AC. Wait, but AC is the side opposite angle B in triangle ABC. Wait, maybe not. Wait, in triangle AHC:- AH is the altitude, which we can compute as AH = AB * sin(angle at B). Wait, no, in triangle ABH, AH = AB * sin(50°). Wait, but AB we've already calculated.Alternatively, since triangle AHC is a 30-60-90 triangle, then sides opposite 30°, 60°, 90° are HC, AH, AC respectively.Therefore, HC = (1/2) AC, AH = (√3/2) AC, but wait, in a 30-60-90 triangle, sides are opposite 30°, 60°, 90°, so the side opposite 30° is half the hypotenuse. But in triangle AHC, the hypotenuse is AC, so the side opposite 30° (which is AH) should be half of AC. Wait, but angle at C is 30°, so side opposite angle C (which is 30°) is AH. Wait, no.Wait, in triangle AHC:- angle at H: 90°- angle at C: 30°- angle at A: 60°Therefore, side opposite angle H (90°) is AC.Side opposite angle C (30°) is AH.Side opposite angle A (60°) is HC.In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2, with the side opposite 30° being the shortest.Therefore, in triangle AHC:- AH (opposite 30°) = (1/2) * AC- HC (opposite 60°) = (√3/2) * AC- AC (hypotenuse) = ACBut in reality, we know that angle at C is 30°, so side AH is opposite angle C, so AH = (1/2) ACBut we can check with previous values:From earlier, AC ≈ 0.7778, so AH should be ≈ 0.7778 / 2 ≈ 0.3889. But from coordinates, AH is the y-coordinate of point A, which was approximately 0.389. That matches. So AH = 0.389 ≈ 0.3889. Correct.Therefore, HC = (√3/2) * AC ≈ 0.866 * 0.7778 ≈ 0.673, which matches the coordinate calculation where HC was approximately 0.674.Therefore, triangle AHC is indeed a 30-60-90 triangle.Now, M is the midpoint of AC. So AM = MC = AC / 2 ≈ 0.7778 / 2 ≈ 0.3889. Wait, but in triangle AHC, HC ≈ 0.673, which is longer than MC ≈ 0.3889. Wait, but HC is part of BC, which is length 1. Wait, HC is from H to C, which in coordinates was approximately 0.674.But M is the midpoint of AC, so coordinates of M are ((Ax + Cx)/2, (Ay + Cy)/2) ≈ ((0.326 + 1)/2, (0.389 + 0)/2) ≈ (0.663, 0.1945)Now, in triangle MHC, we have points M, H, C. From coordinates, we found angle at H is approx 30°, which matches angle C. But angle C is 30°, which is the angle at vertex C in triangle ABC. Is there a relationship here?Alternatively, perhaps triangle MHC is similar to triangle ABC, but scaled down.But in triangle MHC, we have point M as midpoint of AC, H as foot of altitude from A, and point C.Alternatively, since M is the midpoint, and in triangle AHC, which is 30-60-90, then connecting M to H and C might form another triangle with certain angles.Alternatively, since in triangle AHC, M is the midpoint of hypotenuse AC, then in a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices. Wait, but triangle AHC is a right triangle at H. Therefore, the midpoint M of hypotenuse AC is the circumcircle center, so HM = CM = AM = radius. Therefore, HM = CM.Wait, in right triangle AHC, with hypotenuse AC, midpoint M of AC, then indeed HM = CM = AM. Because in a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices.Therefore, HM = CM. So triangle HMC is isoceles with HM = CM. Therefore, angle MHC (at H) is equal to angle MCH (at C). Wait, but angle MCH is part of angle C in triangle ABC, which is 30°. Wait, but angle MCH is angle between MC and HC. Since M is the midpoint of AC, and C is a vertex, angle MCH is the same as angle ACB, which is 30°, but is that true?Wait, point M is the midpoint of AC, so line MC is half of AC. But angle at C in triangle ABC is 30°, but angle MCH is angle between MC and HC. Since HC is along BC, which is the base.Wait, perhaps triangle HMC is isoceles with HM = CM, and angle at C is 30°, so angles at H and M are equal. Wait, but angle at H is what we need, angle MHC. If triangle HMC is isoceles with HM = CM, then angle MHC = angle MCH. But angle MCH is angle between MC and HC. Since HC is along BC, and MC connects to midpoint of AC. Hmm, maybe angle MCH is equal to angle ACB = 30°, but need to verify.Alternatively, since HM = CM, and if angle MCH = 30°, then angle MHC = 30°, hence angle at H is 30°. Therefore, angle MHC = 30°.Therefore, through properties of the right triangle and midpoint of hypotenuse, we can conclude angle MHC is 30°, which matches our coordinate calculation.Therefore, the answer is 30°, boxed as boxed{30}.Final Answerboxed{30}</think>Given triangle ( triangle ABC ) with ( angle A = 100^circ ), ( angle B = 50^circ ), and ( angle C = 30^circ ). ( AH ) is the altitude from ( A ) to ( BC ), and ( BM ) is the median from ( B ) to ( AC ). We need to find ( angle MHC ).1. Triangle Properties: - ( angle A = 100^circ ), ( angle B = 50^circ ), so ( angle C = 30^circ ). - ( AH ) is the altitude from ( A ) to ( BC ), making ( triangle AHB ) and ( triangle AHC ) right triangles. - ( BM ) is the median from ( B ) to ( AC ), so ( M ) is the midpoint of ( AC ).2. Coordinate System: - Place ( B ) at ( (0, 0) ) and ( C ) at ( (1, 0) ). - Calculate coordinates of ( A ) using the Law of Sines and distances, approximating ( A ) at ( (0.326, 0.389) ).3. Finding Points ( H ) and ( M ): - ( H ) is the foot of the altitude from ( A ) to ( BC ), so ( H ) is ( (0.326, 0) ). - ( M ) is the midpoint of ( AC ), so coordinates of ( M ) are ( left( frac{0.326 + 1}{2}, frac{0.389 + 0}{2} right) approx (0.663, 0.1945) ).4. Vectors and Dot Product: - Vectors ( overrightarrow{HM} ) and ( overrightarrow{HC} ) are calculated. - Using the dot product, the angle between vectors ( overrightarrow{HM} ) and ( overrightarrow{HC} ) is found to be approximately ( 30^circ ).5. Geometric Proof: - In right triangle ( triangle AHC ), ( M ) is the midpoint of hypotenuse ( AC ), making ( HM = CM ). - ( triangle HMC ) is isosceles with ( HM = CM ), leading to ( angle MHC = angle MCH ). - Since ( angle MCH = 30^circ ), it follows that ( angle MHC = 30^circ ).Thus, the measure of ( angle MHC ) is (boxed{30}).,<think>Okay, so I need to find the measure of angle MHC in triangle ABC. Let me start by understanding the given information and then figure out how to approach this problem step by step.First, the triangle ABC has angles at A and B given as 100 degrees and 50 degrees, respectively. Since the sum of angles in any triangle is 180 degrees, angle C must be 180 - 100 - 50 = 30 degrees. So, angles are: ∠A = 100°, ∠B = 50°, ∠C = 30°. That's a start.Now, the problem mentions AH is the altitude from A to side BC. An altitude is a perpendicular segment from a vertex to the opposite side (or its extension). So, AH is perpendicular to BC. That means ∠AHB and ∠AHC are both right angles (90°). Wait, no. Actually, since AH is the altitude from A to BC, it meets BC at point H, making AH perpendicular to BC. So, H is the foot of the altitude from A onto BC. Therefore, triangle AHB and triangle AHC are both right-angled triangles.Next, BM is the median from B to side AC. A median connects a vertex to the midpoint of the opposite side. So, point M is the midpoint of AC. Therefore, AM = MC.The goal is to find angle MHC. So, points M, H, and C form triangle MHC, and we need to find the measure of the angle at H, which is ∠MHC.Let me visualize the triangle. Let me sketch this in my mind. Triangle ABC with angle A = 100°, angle B = 50°, angle C = 30°. Let me label the triangle: vertex A at the top, B at the bottom left, and C at the bottom right. AH is the altitude from A to BC, so H is somewhere on BC. BM is the median from B to AC, so M is the midpoint of AC.Hmm. To find angle MHC, I need to consider triangle MHC. Maybe I can use coordinate geometry? Assign coordinates to the points and compute the angle using vectors or slopes. Alternatively, use trigonometric laws like the Law of Sines or Cosines. Let me think.Alternatively, maybe some properties of triangles, medians, and altitudes can help here. Let me recall that in a triangle, the median divides the side into two equal parts, and the altitude creates right angles. Since angle C is 30°, and H is on BC, perhaps some relationships there can help.First, maybe assign coordinates to make this concrete. Let me place point B at the origin (0,0). Then, let me assume BC is along the x-axis for simplicity. Wait, but AH is the altitude from A to BC. If BC is along the x-axis, then AH would be vertical? Hmm. Let's see.But maybe it's better to assign coordinates such that BC is horizontal. Let's let point B be at (0,0), point C at (c,0), so BC is along the x-axis. Then, point A is somewhere in the plane. Since angle at B is 50°, angle at C is 30°, angle at A is 100°. Let me see if I can assign coordinates accordingly.Alternatively, maybe use the Law of Sines to find the lengths of sides. Let me denote the sides as follows: side opposite angle A is BC, denoted as a; side opposite angle B is AC, denoted as b; side opposite angle C is AB, denoted as c. Wait, actually, in standard notation, side a is opposite angle A, side b opposite angle B, side c opposite angle C. Wait, but in triangle ABC, side BC is opposite angle A, so that would be side a. Similarly, side AC is opposite angle B (50°), so that's side b, and side AB is opposite angle C (30°), so that's side c.Given that, using Law of Sines: a/sin A = b/sin B = c/sin C.So, a / sin 100° = b / sin 50° = c / sin 30°.Let me compute ratios. Let me compute the common ratio, say k, such that a = k sin 100°, b = k sin 50°, c = k sin 30°. Then, if I can assign lengths based on this ratio, maybe I can assign specific lengths by choosing k. Let's set k = 1 for simplicity, but maybe later we can adjust if needed.But perhaps to make calculations easier, let's assign BC = a = 1. Then, using Law of Sines, we can compute the other sides.Wait, if BC = a = 1, then 1 / sin 100° = b / sin 50°, so b = (sin 50° / sin 100°). Similarly, c = (sin 30° / sin 100°).Compute sin 100°, sin 50°, sin 30°. Let's recall that sin 100° is sin(80°) ≈ 0.9848, sin 50° ≈ 0.7660, sin 30° = 0.5. So, approximate values:b ≈ (0.7660 / 0.9848) ≈ 0.7778c ≈ (0.5 / 0.9848) ≈ 0.5077So, sides are approximately BC = 1, AC ≈ 0.7778, AB ≈ 0.5077.Alternatively, maybe assign BC = 1 unit for simplicity, then coordinates can be assigned accordingly.Let me set coordinate system with point B at (0,0), point C at (1,0). Then, point A is somewhere in the plane. We need to find coordinates of A such that angles at A, B, C are 100°, 50°, 30°, respectively.But perhaps using Law of Sines with BC = 1, so a = 1. Then, coordinates of A can be found using Law of Sines.Alternatively, since we know all angles and one side (BC = 1), we can compute coordinates.But perhaps another approach. Let me consider triangle ABC with BC = 1, angle at B is 50°, angle at C is 30°, so angle at A is 100°. Then, coordinates:Let me place B at (0,0), C at (1,0). Then, point A is somewhere above the x-axis. Let's find coordinates of A.Using Law of Sines, sides:AB = c = (sin 30°) / sin 100° ≈ 0.5 / 0.9848 ≈ 0.5077AC = b = (sin 50°) / sin 100° ≈ 0.7660 / 0.9848 ≈ 0.7778So, coordinates of A can be found by solving for the position such that distances from A to B is 0.5077, from A to C is 0.7778, and BC is 1.Let me denote coordinates of A as (x,y). Then, distance from A to B: sqrt(x² + y²) ≈ 0.5077Distance from A to C: sqrt((x - 1)² + y²) ≈ 0.7778So, two equations:1) x² + y² ≈ (0.5077)² ≈ 0.25782) (x - 1)² + y² ≈ (0.7778)² ≈ 0.6050Subtracting equation 1 from equation 2:(x - 1)² + y² - x² - y² ≈ 0.6050 - 0.2578Expanding (x - 1)²: x² - 2x + 1 - x² = -2x + 1Therefore, -2x + 1 ≈ 0.3472So, -2x ≈ 0.3472 - 1 = -0.6528Hence, x ≈ (-0.6528)/(-2) ≈ 0.3264Then, substitute x ≈ 0.3264 into equation 1:(0.3264)² + y² ≈ 0.25780.1065 + y² ≈ 0.2578Therefore, y² ≈ 0.2578 - 0.1065 ≈ 0.1513Thus, y ≈ sqrt(0.1513) ≈ 0.389Therefore, coordinates of A are approximately (0.3264, 0.389)So, in this coordinate system:B: (0,0)C: (1,0)A: (0.3264, 0.389)Now, AH is the altitude from A to BC. Since BC is on the x-axis, the altitude from A to BC is vertical, right? Because BC is horizontal. Wait, no. The altitude from A is perpendicular to BC. Since BC is along the x-axis, the altitude from A will be vertical. So, the foot H is directly below A on the x-axis. Therefore, H has the same x-coordinate as A, y-coordinate 0. So, H is (0.3264, 0)Wait, but in our coordinate system, BC is from (0,0) to (1,0). If point A is at (0.3264, 0.389), then the altitude from A to BC is indeed vertical, so H is (0.3264, 0). Correct.So, coordinates:H: (0.3264, 0)Then, BM is the median from B to AC. So, M is the midpoint of AC. Coordinates of A: (0.3264, 0.389), coordinates of C: (1,0). So, midpoint M is ((0.3264 + 1)/2, (0.389 + 0)/2) = (1.3264/2, 0.389/2) ≈ (0.6632, 0.1945)So, coordinates of M: approximately (0.6632, 0.1945)Now, we need to find angle MHC. Points M, H, C:M: (0.6632, 0.1945)H: (0.3264, 0)C: (1,0)So, angle at H: angle MHC, which is the angle at point H between points M, H, and C.To find angle MHC, which is the angle at H between M and C.To compute this angle, we can find the vectors HM and HC and then find the angle between them.First, compute vectors HM and HC.Coordinates:H: (0.3264, 0)M: (0.6632, 0.1945)C: (1,0)Vector HM = M - H = (0.6632 - 0.3264, 0.1945 - 0) = (0.3368, 0.1945)Vector HC = C - H = (1 - 0.3264, 0 - 0) = (0.6736, 0)Now, the angle between vectors HM and HC can be found using the dot product formula:cosθ = (HM · HC) / (|HM| |HC|)First, compute HM · HC:(0.3368)(0.6736) + (0.1945)(0) = 0.3368 * 0.6736 ≈ 0.2269Compute |HM|:sqrt(0.3368² + 0.1945²) ≈ sqrt(0.1134 + 0.0378) ≈ sqrt(0.1512) ≈ 0.389Compute |HC|:sqrt(0.6736² + 0²) = 0.6736Therefore, cosθ ≈ 0.2269 / (0.389 * 0.6736) ≈ 0.2269 / 0.262 ≈ 0.866Then, θ ≈ arccos(0.866) ≈ 30°Wait, arccos(0.866) is approximately 30 degrees, since cos(30°) ≈ 0.866.Therefore, angle MHC is approximately 30 degrees. Hmm, interesting. Given that angle C is 30°, is this a coincidence?But let me check my calculations again to ensure there's no mistake.First, coordinates of A were found by solving the distances from B and C. Using approximate values, which might have introduced some errors, but the result seems to align with angle C.Wait, angle C is 30°, and angle MHC is also 30°, that's interesting. Maybe there's a geometric reason for this, which my coordinate approach approximates.Alternatively, maybe the answer is exactly 30°, and the approximation just confirms that.But let's see if we can do this without coordinates, perhaps using triangle properties.Given triangle ABC with angles 100°, 50°, 30°. AH is the altitude from A to BC, so H is on BC. BM is the median from B to AC, so M is the midpoint of AC.We need to find angle MHC.Let me recall that in a triangle, the median, altitude, and other lines can create various triangle centers or other properties. However, angle MHC seems specific.Alternatively, perhaps triangle MHC has some properties. Let me consider triangle MHC. Points M, H, C.Given that M is the midpoint of AC, and H is the foot of the altitude from A.Alternatively, maybe triangle MHC is a 30-60-90 triangle? But in my coordinate calculation, angle at H was 30°, but let me verify.Alternatively, let's use exact trigonometric values instead of approximate decimal values. Let me try that.Let me denote BC = a, AC = b, AB = c.Using Law of Sines:a / sin A = b / sin B = c / sin CLet me write a = BC, angle A = 100°, so a / sin 100° = b / sin 50° = c / sin 30°Let’s let’s take BC = a = 1 (as before), then:b = (sin 50° / sin 100°)c = (sin 30° / sin 100°) = (0.5 / sin 100°)Expressed exactly in terms of sine functions.Now, coordinates:Let’s place B at (0,0), C at (1,0). Let’s find coordinates of A.Coordinates of A (x, y):Distance from A to B: sqrt(x² + y²) = c = 0.5 / sin 100°Distance from A to C: sqrt((x - 1)^2 + y^2) = b = sin 50° / sin 100°Let me compute these exactly.First, note that sin 100° = sin(80°) = cos(10°), since sin(90° + 10°) = cos(10°). Wait, actually sin(100°) = sin(80°) because sin(90° + 10°) = cos(10°), but sin(80°) is sin(90° - 10°) = cos(10°). Wait, sin(100°) = sin(80°) only in terms of numerical value, but actually sin(100°) = sin(80°) ≈ 0.9848, yes.But maybe using trigonometric identities can help.Alternatively, proceed with exact ratios.So, c = 0.5 / sin 100°, and b = sin 50° / sin 100°.Then, coordinates of A (x,y):x² + y² = c² = (0.5 / sin 100°)^2(x - 1)^2 + y² = b² = (sin 50° / sin 100°)^2Subtract the first equation from the second:(x - 1)^2 + y² - x² - y² = (sin²50° / sin²100°) - (0.25 / sin²100°)Simplify left side:x² - 2x + 1 + y² - x² - y² = -2x + 1Right side: [sin²50° - 0.25] / sin²100°Therefore:-2x + 1 = [sin²50° - 0.25] / sin²100°Solve for x:x = [1 - (sin²50° - 0.25)/sin²100° ] / 2Hmm, this seems complicated, but maybe we can compute sin50° and sin100° numerically.Compute sin50° ≈ 0.7660, sin100° ≈ 0.9848.Compute sin²50° ≈ 0.7660² ≈ 0.58680.5868 - 0.25 = 0.3368sin²100° ≈ (0.9848)^2 ≈ 0.9698Therefore, [0.3368] / 0.9698 ≈ 0.3472Therefore, x ≈ [1 - 0.3472]/2 ≈ (0.6528)/2 ≈ 0.3264, which matches the previous value. Then, y ≈ sqrt(c² - x²) ≈ sqrt(0.25 / sin²100° - x²). Wait, c = 0.5 / sin100° ≈ 0.5 / 0.9848 ≈ 0.5077, so c² ≈ 0.2578. Then x² ≈ 0.3264² ≈ 0.1065, so y² ≈ 0.2578 - 0.1065 ≈ 0.1513, so y ≈ 0.389. So, coordinates are consistent.Therefore, angle MHC is approximately 30°, which matches angle C. But why?Wait, angle C is 30°, which is at point C. But angle MHC is at point H. Is there a reason these angles are equal?Alternatively, perhaps triangle MHC is similar to another triangle in the figure.Alternatively, since H is the foot of the altitude, and M is the midpoint, maybe there's a property here.Alternatively, since angle at C is 30°, and in the coordinate system, HC is along the x-axis from H to C, and HM is going up to M. The calculation gave angle at H as 30°, same as angle at C. Maybe there is a reflection or something.Alternatively, perhaps using trigonometric identities.But let's see. In the coordinate system, vector HM is (0.3368, 0.1945), vector HC is (0.6736, 0). The angle between them is calculated using the dot product:cosθ = (0.3368 * 0.6736 + 0.1945 * 0) / (|HM| |HC|)= (0.3368 * 0.6736) / (sqrt(0.3368² + 0.1945²) * 0.6736)Simplify:= 0.3368 / sqrt(0.3368² + 0.1945²)Compute denominator: sqrt(0.1134 + 0.0378) = sqrt(0.1512) ≈ 0.389So, cosθ ≈ 0.3368 / 0.389 ≈ 0.866, which is approximately cos30°, as before.So, exact value would be 30°, so the answer is 30 degrees.But let's see if we can confirm this without coordinates.Alternative approach:In triangle ABC, with angles 100°, 50°, 30°, AH is the altitude, so AH is perpendicular to BC. BM is the median to AC, so M is the midpoint of AC.We need to find angle MHC.Let me consider triangle MHC. Let's see if we can find lengths or other angles.First, let's find coordinates using symbolic expressions instead of numbers.Let’s denote BC = a, so BC = a, angle at B is 50°, angle at C is 30°, angle at A is 100°.Using Law of Sines:AB / sin30° = BC / sin100° => AB = (BC * sin30°)/sin100°Similarly, AC = (BC * sin50°)/sin100°Let’s let BC = 1, so AB = sin30° / sin100°, AC = sin50° / sin100°Now, coordinates:Let’s place B at (0,0), C at (1,0). Let’s find coordinates of A.Let’s denote coordinates of A as (x, y). Then:Distance from A to B: sqrt(x² + y²) = AB = sin30° / sin100° = (0.5)/sin100°Distance from A to C: sqrt((x - 1)^2 + y²) = AC = sin50° / sin100°So, we can set up equations:x² + y² = (0.5 / sin100°)^2 ...(1)(x - 1)^2 + y² = (sin50° / sin100°)^2 ...(2)Subtract equation (1) from equation (2):(x - 1)^2 - x² = (sin²50° - 0.25)/sin²100°Expand left side:x² - 2x + 1 - x² = -2x + 1Right side: [sin²50° - 0.25]/sin²100°So:-2x + 1 = [sin²50° - 0.25]/sin²100°Solving for x:x = [1 - (sin²50° - 0.25)/sin²100°]/2Now, compute sin²50° and sin²100°.Using the identity sin²θ = (1 - cos2θ)/2Compute sin²50° = (1 - cos100°)/2Similarly, sin²100° = (1 - cos200°)/2, but cos200° = cos(180° + 20°) = -cos20°, so sin²100° = (1 + cos20°)/2Therefore,[sin²50° - 0.25] = [(1 - cos100°)/2 - 0.25] = (1 - cos100° - 0.5)/2 = (0.5 - cos100°)/2Thus,x = [1 - (0.5 - cos100°)/2 / ((1 + cos20°)/2)] / 2Simplify denominator:[(0.5 - cos100°)/2] divided by [(1 + cos20°)/2] = (0.5 - cos100°)/ (1 + cos20°)Therefore,x = [1 - (0.5 - cos100°)/(1 + cos20°)] / 2This seems complicated, but maybe we can relate angles 100° and 20°. Note that 100° = 90° + 10°, and 20° is just 20°.Alternatively, use trigonometric identities to relate cos100° and cos20°.Note that cos100° = cos(90° + 10°) = -sin10°, and cos20° is cos20°.So,x = [1 - (0.5 - (-sin10°))/ (1 + cos20°)] / 2 = [1 - (0.5 + sin10°)/(1 + cos20°)] / 2Hmm, not sure if this helps. Let me compute numerically:cos100° ≈ -0.1736cos20° ≈ 0.9397Therefore,(0.5 - cos100°) / (1 + cos20°) ≈ (0.5 - (-0.1736)) / (1 + 0.9397) ≈ (0.6736) / 1.9397 ≈ 0.3472Then, x ≈ [1 - 0.3472]/2 ≈ 0.6528 / 2 ≈ 0.3264, which matches previous result.So, x ≈ 0.3264, then y ≈ sqrt( (0.5 / sin100°)^2 - x² ) ≈ sqrt( (0.5 / 0.9848)^2 - 0.3264² ) ≈ sqrt(0.2578 - 0.1065) ≈ sqrt(0.1513) ≈ 0.389.Therefore, coordinates are as before.Therefore, angle MHC calculated as 30°, which is the same as angle C.This suggests that angle MHC is equal to angle C. Is this always the case? Or is it specific to this triangle?Wait, in this triangle, angle C is 30°, and angle MHC is also 30°. Perhaps there's a theorem or property here.Alternatively, since H is the foot of the altitude from A, and M is the midpoint of AC, then in triangle AHC, M is the midpoint of AC, and H is the foot of the altitude. Then, connecting M to H, forming triangle MHC, angle at H is equal to angle C. Maybe this is a general property?Alternatively, in this specific case, with angles 100°, 50°, 30°, this happens.Alternatively, perhaps using the Law of Sines in triangle MHC.In triangle MHC, we can compute sides MH, HC, and MC, then use the Law of Sines to find angle at H.First, compute HC: from H to C.Coordinates:H: (0.3264, 0)C: (1,0)So, HC = 1 - 0.3264 = 0.6736Compute MH: from M(0.6632, 0.1945) to H(0.3264,0)Distance MH: sqrt( (0.6632 - 0.3264)^2 + (0.1945 - 0)^2 ) ≈ sqrt(0.3368² + 0.1945²) ≈ sqrt(0.1134 + 0.0378) ≈ sqrt(0.1512) ≈ 0.389Compute MC: from M(0.6632, 0.1945) to C(1,0)Distance MC: sqrt( (1 - 0.6632)^2 + (0 - 0.1945)^2 ) ≈ sqrt(0.3368² + 0.1945²) ≈ same as MH, which is 0.389Wait, that's interesting. So, in triangle MHC, sides MH and MC are equal (both ≈ 0.389), making it an isosceles triangle with MH = MC.Therefore, angles at H and C are equal. But wait, angle at C in triangle MHC is not the same as angle C in triangle ABC. Wait, point C is the same, but in triangle MHC, angle at C is different. Wait, no. Wait, triangle MHC has vertices M, H, C. So, angle at C is angle MCH, which is different from angle BCA (which is 30°). However, in our calculation, we found that angle at H (angle MHC) is 30°, which is the same as angle C in triangle ABC. But in triangle MHC, angle at H is 30°, and sides MH ≈ MC ≈ 0.389, HC ≈ 0.6736.Wait, if triangle MHC has two sides equal (MH = MC), then it's isosceles with base HC and equal sides MH and MC. Therefore, the base angles at H and C should be equal. Wait, but according to our calculation, angle at H is 30°, and angle at C (in triangle MHC) would also be 30°, making the vertex angle at M 120°. But in reality, angle at C in triangle MHC is angle MCH, which is a different point. Wait, perhaps my previous statement is incorrect.Wait, in triangle MHC:- Vertex H: angle MHC is the angle at H between M and C.- Vertex M: angle HMC.- Vertex C: angle MCH.If MH = MC, then angles at H and C are equal. But according to our calculation, angle at H is 30°, which would make angle at C also 30°, and angle at M would be 120°. But let's verify.In our coordinate system, point C is at (1,0), point M is at (0.6632, 0.1945), point H is at (0.3264,0).So, to compute angle at C (angle MCH), we need to consider triangle MCH.Wait, no. Triangle MHC: points M, H, C. So, angle at C is angle HCM, which is different from the original angle at C (which was 30°).Wait, maybe I need to compute angle at C in triangle MHC. Let's do that.Coordinates:Point M: (0.6632, 0.1945)Point H: (0.3264, 0)Point C: (1, 0)Vectors for angle at C:Vector CM = M - C = (0.6632 - 1, 0.1945 - 0) = (-0.3368, 0.1945)Vector CH = H - C = (0.3264 - 1, 0 - 0) = (-0.6736, 0)Angle at C is between vectors CM and CH.Compute the angle between vectors CM and CH.Dot product: (-0.3368)(-0.6736) + (0.1945)(0) ≈ 0.2269|CM| = sqrt((-0.3368)^2 + (0.1945)^2) ≈ sqrt(0.1134 + 0.0378) ≈ 0.389|CH| = sqrt((-0.6736)^2 + 0^2) = 0.6736cosθ = 0.2269 / (0.389 * 0.6736) ≈ 0.2269 / 0.262 ≈ 0.866Thus, θ ≈ 30°, same as angle at H. Therefore, in triangle MHC, angles at H and C are both 30°, making it an isosceles triangle with MH = MC and base angles 30°. Therefore, the vertex angle at M is 120°.But how does this relate to the original triangle?Given that angle at C in triangle ABC is 30°, and angle at C in triangle MHC is also 30°, perhaps there's a reflection or some cyclic quadrilateral property.Alternatively, since M is the midpoint of AC, and AH is the altitude, perhaps there is some midpoint theorem or property at play.Alternatively, consider triangle AHC. Since AH is the altitude, triangle AHC is right-angled at H. So, AH ⊥ HC.M is the midpoint of AC. In a right-angled triangle, the midpoint of the hypotenuse is equidistant from all three vertices. Wait, yes! In triangle AHC, right-angled at H, the midpoint M of the hypotenuse AC is equidistant from A, H, and C. Therefore, AM = MC = MH.Therefore, in triangle MHC, since M is the midpoint of the hypotenuse AC in the right triangle AHC, then MH = MC. Therefore, triangle MHC is isosceles with MH = MC.Therefore, angles at H and C are equal. But angle at C in triangle MHC is angle MCH, which we calculated as 30°, so angle at H (angle MHC) is also 30°. Therefore, angle MHC = 30°.Ah! This is the key insight. Since triangle AHC is right-angled at H, and M is the midpoint of the hypotenuse AC, then by the property of right-angled triangles, M is the circumcircle center, and thus MA = MC = MH. Therefore, triangle MHC is isosceles with MH = MC, hence base angles are equal. But angle MCH is equal to angle ACB (original angle at C) which is 30°, wait, is that true?Wait, angle MCH is the angle at point C between points M, C, H. Is this angle equal to angle ACB?Point M is the midpoint of AC, so line CM connects C to the midpoint of AC. Line CH connects C to H, the foot of the altitude from A. So, angle MCH is different from angle ACB, unless H coincides with the midpoint, which it doesn't in this case.But wait, in our coordinate calculation, angle MCH was 30°, which is the same as angle ACB. Maybe that's a coincidence or a specific property of this triangle.Alternatively, since triangle AHC is right-angled at H, and M is the midpoint of AC, then HM = MC = MA. Therefore, triangle MHC is isosceles, so angles at H and C are equal. But angle at C in triangle MHC is equal to angle ACB?Wait, angle ACB is 30°, which is the angle at point C in triangle ABC. However, angle MCH is part of triangle MHC. How are these related?Wait, perhaps using the fact that line HC is common to both triangles ABC and MHC. In triangle ABC, angle at C is 30°, formed by sides BC and AC. In triangle MHC, angle at C is formed by sides HC and MC. Since M is the midpoint of AC, MC = AC/2. Therefore, triangle MHC is related to triangle ABC.Alternatively, since in triangle AHC, right-angled at H, with M as the midpoint of hypotenuse AC, then angle MHC is equal to angle MAH, which might relate to angle BAC.But I need to think differently.Since triangle AHC is right-angled at H, and M is the midpoint of AC, then HM = AM = CM. Therefore, triangle HMC is isosceles with HM = CM. Therefore, angles at H and C are equal. So, angle MHC = angle MCH.But angle MCH is the angle between CM and CH. CM is the median to AC, and CH is part of BC.But angle ACB in triangle ABC is 30°, which is the angle between BC and AC. However, angle MCH is between CM and CH (which is part of BC). Since M is the midpoint of AC, line CM is a median. The angle between CM and BC (at point C) is angle MCH. How is this related to angle ACB?Wait, angle ACB is the angle between AC and BC. Since M is the midpoint of AC, line CM is going from C to M, which is along the median. So, angle MCH is the angle between the median CM and side BC at point C. In this specific triangle, due to the ratios of the sides and angles, this angle ends up being equal to angle MHC, which is 30°, same as angle ACB.But this seems specific to the given angles. However, using the property that in a right-angled triangle, the midpoint of the hypotenuse is equidistant from all three vertices, we established that HM = CM, making triangle MHC isosceles, hence angles at H and C are equal. Then, if angle MCH is equal to angle ACB (30°), which might not be generally true, but in this case, due to the specific configuration, angle MCH equals 30°, leading angle MHC to also be 30°.Alternatively, maybe there's a way to show that angle MCH is equal to angle ACB.Wait, angle ACB is 30°, and angle MCH is part of triangle MCH. However, angle MCH is not necessarily equal to angle ACB. But in our coordinate calculation, they both turned out to be 30°, so maybe in this case, it's true.Alternatively, since M is the midpoint of AC and AH is the altitude, there could be some reflection symmetry or other properties.But regardless of the path, the coordinate geometry approach and the properties of the right-angled triangle AHC lead us to the conclusion that angle MHC is 30°. Therefore, the answer is 30 degrees.Final Answerboxed{30^circ}</think>Given triangle ( triangle ABC ) with ( angle A = 100^circ ), ( angle B = 50^circ ), and ( angle C = 30^circ ). We need to find the measure of ( angle MHC ) where ( AH ) is the altitude from ( A ) to ( BC ) and ( BM ) is the median from ( B ) to ( AC ).1. Determine the coordinates of points: - Place ( B ) at ( (0, 0) ) and ( C ) at ( (1, 0) ). - Using the Law of Sines, calculate the coordinates of ( A ) as approximately ( (0.3264, 0.389) ).2. Find the foot of the altitude ( H ): - Since ( AH ) is perpendicular to ( BC ), ( H ) has the same x-coordinate as ( A ) and y-coordinate 0, giving ( H approx (0.3264, 0) ).3. Determine the midpoint ( M ) of ( AC ): - The coordinates of ( M ) are the midpoint of ( A ) and ( C ), giving ( M approx (0.6632, 0.1945) ).4. Calculate vectors ( HM ) and ( HC ): - Vector ( HM = (0.3368, 0.1945) ) - Vector ( HC = (0.6736, 0) )5. Use the dot product to find the angle between vectors ( HM ) and ( HC ): - Compute the dot product: ( 0.3368 times 0.6736 approx 0.2269 ) - Compute magnitudes: ( |HM| approx 0.389 ) and ( |HC| = 0.6736 ) - Calculate ( cos theta approx 0.2269 / (0.389 times 0.6736) approx 0.866 ) - Determine ( theta approx 30^circ )6. Conclusion: - Using properties of right-angled triangles and midpoints, we confirm that ( angle MHC = 30^circ ).Thus, the measure of ( angle MHC ) is ( boxed{30^circ} ).]

🔑:begin{array}{l}text { 3. } angle M H C=30^{circ} text {. } left.text { Solution } begin{array}{l}M text { is the midpoint of } A C text {, } angle A H C=90^{circ}end{array}right} Rightarrow A M=M C=M H text {. } Rightarrow angle M A H=angle A H M=60^{circ} text {. } text { And } angle C=180^{circ}-150^{circ}=30^{circ} text {, then } angle M H C=30^{circ} text {. } end{array}

🧠:[<think>Okay, so I have this problem where I need to substitute four numbers—12, 14, 16, and 18—into the variables A, B, C, D in the equation A×B + B×C + B×D + C×D. Then, I need to find the maximum value possible by arranging these numbers in different orders. Let me try to figure this out step by step.First, let me make sure I understand the problem correctly. The equation given is A multiplied by B, plus B multiplied by C, plus B multiplied by D, plus C multiplied by D. So, written out, it's AB + BC + BD + CD. My task is to assign each of the numbers 12, 14, 16, 18 to A, B, C, D such that when I compute this expression, the result is as large as possible. And then find that maximum value.Alright. So the variables A, B, C, D each take one of those four numbers, no repeats. So we need to consider all possible permutations of these numbers into the four variables and compute the value for each permutation, then pick the maximum. But since there are 4 variables and 4 numbers, the number of permutations is 4 factorial, which is 24. That seems manageable, but maybe there's a smarter way to do this without checking all 24 possibilities.Let me see. Maybe I can analyze the expression to see which variables are multiplied together and how they contribute to the total. The expression is AB + BC + BD + CD. Let's break this down term by term:1. AB: Product of A and B.2. BC: Product of B and C.3. BD: Product of B and D.4. CD: Product of C and D.Looking at this, B is involved in three terms: AB, BC, BD. C is involved in two terms: BC and CD. D is involved in two terms: BD and CD. A is only involved in the first term, AB. So A is only multiplied by B. Therefore, maybe the placement of A is less critical than the placement of B, C, D.Since B is involved in three products, and C and D each in two, and A only in one, perhaps assigning the largest number to B would maximize the products where B is a factor. Similarly, since C and D are each in two products, maybe assigning the next largest numbers to C and D would help. But I need to verify this.Alternatively, perhaps factoring the expression might help. Let me see if I can factor any part of this expression.Looking at AB + BC + BD + CD. Let's factor B out of the first three terms:B(A + C + D) + CDHmm, so the expression becomes B*(A + C + D) + CD. Interesting. So if I can make B*(A + C + D) as large as possible, and CD as large as possible, that would be good. But since A, B, C, D are all distinct numbers from 12, 14, 16, 18, the sum A + C + D would be (12 + 14 + 16 + 18) minus B. Because all four numbers add up to 12+14+16+18=60. Therefore, A + C + D = 60 - B. So substituting back into the expression:B*(60 - B) + CDSo the total expression simplifies to 60B - B² + CDTherefore, our goal is to maximize 60B - B² + CD. Since we have four numbers: 12,14,16,18, and B must be one of them. So let's consider each possible value of B and then see how CD can be maximized given that B is fixed. Then compute 60B - B² + CD for each case, and see which one is the largest.That seems like a good approach. Let me proceed accordingly.First, list all possible values of B (12,14,16,18) and for each, assign the remaining three numbers to A, C, D. Then compute CD for the maximum possible CD (since in the expression, after choosing B, CD is the product of the remaining two numbers assigned to C and D. Wait, but hold on: after choosing B, the remaining numbers are assigned to A, C, D. But in the expression, CD is the product of C and D, so we need to assign the two largest remaining numbers to C and D to maximize CD. But the problem is, after choosing B, the remaining numbers are assigned to A, C, D. However, in the expression, A is only present in the term AB, which we've already accounted for in the 60B - B² term. Wait, no, because A is part of the 60 - B sum. Wait, let me clarify.Wait, when we rewrote the expression as 60B - B² + CD, that's correct because A + C + D = 60 - B. So in that term, B*(60 - B) already includes A, C, D. Then CD is just the product of C and D. But in reality, C and D are two of the remaining three numbers (since B is fixed and A is another). Therefore, after choosing B, the remaining numbers are A, C, D. However, CD is just the product of two of those three numbers, so to maximize CD, we need to choose the two largest numbers among A, C, D for C and D. But wait, C and D are variables; the numbers assigned to them are part of the permutation. So if I fix B, then the remaining three numbers can be assigned to A, C, D in any way, but to maximize CD, we need to assign the two largest remaining numbers to C and D, and the smallest remaining number to A. Because A is only in the term AB, which is already included in the 60B - B² term. Wait, but AB is part of the original expression, but in our rewritten expression, we have 60B - B² + CD. Wait, that seems conflicting.Wait, let me check the algebra again. The original expression is AB + BC + BD + CD. Let's rearrange:AB + BC + BD + CD = B(A + C + D) + CDBut since A + C + D = 60 - B (because A + B + C + D = 60), so substituting:B*(60 - B) + CDTherefore, the entire expression is indeed 60B - B² + CD.So CD is just the product of C and D. But once B is fixed, the remaining numbers are A, C, D, which are three numbers. So we need to assign the remaining three numbers (12,14,16,18 excluding B) to A, C, D. But CD is the product of C and D. To maximize CD, given three numbers, we need to pick the two largest numbers among the remaining three and assign them to C and D. Then the smallest remaining number is assigned to A. Because A is only part of the term AB, which is already included in the 60B - B² term. Wait, but in the expression 60B - B² + CD, the term AB is included in the 60B - B² part? Wait, no. Wait, let me check again.Original expression: AB + BC + BD + CD.Factoring B out of the first three terms: B(A + C + D) + CD.But A + C + D = 60 - B, so replacing that, we get B*(60 - B) + CD.Therefore, the entire expression is 60B - B² + CD.But in this expression, the CD is separate. So in order to compute the total expression, once B is fixed, CD is the product of variables C and D, which are two of the remaining three numbers (since A is another variable). Therefore, given that B is fixed, the remaining three numbers must be assigned to A, C, D, and CD is the product of the numbers assigned to C and D. So, to maximize CD, regardless of A, we should assign the two largest remaining numbers to C and D. The remaining number (the smallest one) will be assigned to A. Since A is only present in the term AB, which is already part of the 60B - B² term. However, in reality, A is part of the sum A + C + D, which is 60 - B. Therefore, assigning a smaller number to A would leave larger numbers for C and D, which would then make CD larger. Therefore, yes, if we assign the smallest remaining number to A, then C and D get the two larger numbers, which would maximize CD.Therefore, the strategy is:1. For each possible B (12,14,16,18): a. Calculate 60B - B². b. Assign the smallest remaining number to A, and the two larger remaining numbers to C and D. c. Compute CD as the product of the two larger remaining numbers. d. Total expression = 60B - B² + CD.2. Compare the totals for each B and select the maximum.This seems efficient. Let's proceed with each value of B.Case 1: B = 12Remaining numbers: 14,16,18. Assign the smallest to A, which is 14. Then C and D get 16 and 18. So CD = 16*18 = 288.Compute 60*12 - 12² + 288.60*12 = 72012² = 144So 720 - 144 = 576576 + 288 = 864Total: 864Case 2: B = 14Remaining numbers: 12,16,18. Assign the smallest to A, which is 12. Then C and D get 16 and 18. CD = 16*18 = 288.Compute 60*14 - 14² + 288.60*14 = 84014² = 196840 - 196 = 644644 + 288 = 932Total: 932Case 3: B = 16Remaining numbers: 12,14,18. Assign the smallest to A, which is 12. Then C and D get 14 and 18. CD = 14*18 = 252.Compute 60*16 - 16² + 252.60*16 = 96016² = 256960 - 256 = 704704 + 252 = 956Total: 956Case 4: B = 18Remaining numbers: 12,14,16. Assign the smallest to A, which is 12. Then C and D get 14 and 16. CD = 14*16 = 224.Compute 60*18 - 18² + 224.60*18 = 108018² = 3241080 - 324 = 756756 + 224 = 980Total: 980So, summarizing:- B=12 gives 864- B=14 gives 932- B=16 gives 956- B=18 gives 980Therefore, the maximum value is 980 when B=18, C=14, D=16, A=12. Wait, hold on. Wait, if B is 18, then the remaining numbers are 12,14,16. Assign A=12, so C and D are 14 and 16. So CD is 14*16=224. Then the total is 980. But is this correct? Let me verify.Wait, when B=18, the expression AB + BC + BD + CD becomes 12*18 + 18*14 + 18*16 + 14*16.Compute each term:12*18 = 21618*14 = 25218*16 = 28814*16 = 224Sum: 216 + 252 = 468; 468 + 288 = 756; 756 + 224 = 980. Correct.Similarly, for B=16, the assignment was A=12, C=14, D=18. Then AB=12*16=192; BC=16*14=224; BD=16*18=288; CD=14*18=252. Sum: 192+224=416; 416+288=704; 704+252=956. Correct.Similarly, B=14: A=12, C=16, D=18. AB=12*14=168; BC=14*16=224; BD=14*18=252; CD=16*18=288. Sum: 168+224=392; 392+252=644; 644+288=932. Correct.And B=12: A=14, C=16, D=18. AB=14*12=168; BC=12*16=192; BD=12*18=216; CD=16*18=288. Sum: 168+192=360; 360+216=576; 576+288=864. Correct.So, the maximum is indeed 980 when B=18. But wait, hold on. Wait, when B=18, A is 12, and C and D are 14 and 16. But in the original problem statement, A, B, C, D are four numbers which are 12,14,16,18. So substituting into the equation. So, in this case, A=12, B=18, C=14, D=16. So the assignment is A=12, B=18, C=14, D=16. Then the calculation is as above. So, that seems okay.But wait, is there another way to assign the numbers when B is 18 to get a higher value? For example, if instead of assigning A=12, we assigned A=14, then C and D would be 12 and 16. Then CD would be 12*16=192, which is lower than 224. Similarly, assigning A=16 would leave C and D as 12 and 14, CD=12*14=168, which is even worse. So assigning the smallest remaining number to A is indeed the right strategy to maximize CD.Similarly, for other values of B, assigning the smallest remaining number to A gives the maximum CD. Therefore, the approach seems valid.Therefore, the maximum value is 980. So I think that's the answer.But just to be thorough, let me check all permutations for B=18. There are 3! = 6 permutations of A, C, D. Let's see if any of them gives a higher total.When B=18, the remaining numbers are 12,14,16. Assigning them to A, C, D.Possible permutations:1. A=12, C=14, D=16: CD=14*16=224; total=60*18 -18² +224=1080-324+224=980.2. A=12, C=16, D=14: CD=16*14=224; same total=980.3. A=14, C=12, D=16: CD=12*16=192; total=1080-324+192=948.4. A=14, C=16, D=12: CD=16*12=192; total=948.5. A=16, C=12, D=14: CD=12*14=168; total=1080-324+168=924.6. A=16, C=14, D=12: CD=14*12=168; total=924.So indeed, the maximum when B=18 is 980, achieved when C and D are assigned the two larger numbers (14 and 16). Therefore, even if we permute A, C, D, the maximum is 980. So our initial conclusion holds.Similarly, let's check for B=16. Remaining numbers:12,14,18. Assigning A=12, C=14, D=18 gives CD=14*18=252; total=60*16 -16² +252=960-256+252=956.Other permutations:A=14, C=12, D=18: CD=12*18=216; total=960-256+216=920.A=18, C=12, D=14: CD=12*14=168; total=960-256+168=872.So indeed, the maximum is 956 when CD=14*18. So correct.Same with B=14. Assigning A=12, C=16, D=18: CD=16*18=288; total=840-196+288=932.Other permutations would result in lower CD.For B=12, assigning A=14, C=16, D=18 gives CD=16*18=288; total=720-144+288=864. Other permutations give lower CD.Therefore, our analysis is correct. The maximum value is 980 when B=18, A=12, C=14, D=16.But wait, hold on. Let me just check one thing. The original equation is AB + BC + BD + CD. So when B=18, A=12, C=14, D=16, then:AB=12*18=216BC=18*14=252BD=18*16=288CD=14*16=224Adding them up: 216+252=468; 468+288=756; 756+224=980. Correct.Alternatively, if I assign A=12, B=18, C=16, D=14, then CD=16*14=224, same as before. So the total is same.But just to make sure, is there a way to get a higher value by perhaps not assigning the largest number to B? Let's see. For example, when B=16, we got 956. When B=18, we got 980. So B=18 is better. If we tried a different assignment where B is not the largest, but perhaps middle numbers, but given the analysis, 18 as B gives the highest total.Alternatively, is there a scenario where making B a middle number but arranging C and D differently could give a higher total? For example, if B=16, and instead of assigning A=12, C=14, D=18, which gives CD=14*18=252, what if we assigned A=14, C=12, D=18. Then CD=12*18=216. Which is lower. So no. So the maximum when B=16 is 252. So the initial approach is still correct.Therefore, the conclusion is 980.But just to make sure, let's see if there is a different way of assigning variables where B is not the largest, but the product CD is significantly larger. For instance, suppose B=14. Then CD=16*18=288, which is higher than when B=18, which is CD=14*16=224. But in the total expression, when B=14, the total is 932, which is lower than 980. So even though CD is higher when B=14, the term 60B - B² is lower. So there's a trade-off.So let's check that: When B=14, 60B - B² = 60*14 - 14² = 840 - 196 = 644. Then CD=288. Total=644+288=932.When B=18, 60*18 -18²=1080 - 324=756. CD=224. Total=756+224=980. So even though CD is lower, the 60B - B² term is higher enough to compensate. So the total is higher.Similarly, for B=16, 60*16 -16²=960-256=704. CD=252. Total=704+252=956. So again, 60B - B² term is lower than when B=18, but higher than when B=14.Therefore, the term 60B - B² is quadratic in B. Let's analyze it. 60B - B² = -B² +60B. This is a quadratic equation that opens downward, with vertex at B=60/(2*1)=30. Therefore, the maximum value of this quadratic is at B=30, which would be 60*30 -30²=1800 -900=900. However, since B can only be 12,14,16,18, the closer B is to 30, the higher the value of 60B - B². But in our case, B=18 is the closest to 30, so 60B - B² is maximized at B=18 among our options. Therefore, even though CD is lower when B=18, the gain in the quadratic term is more significant, leading to a higher total.Therefore, the conclusion that B=18 gives the maximum total is correct.Therefore, the maximum value is 980.Final Answerboxed{980}</think>To find the maximum value of the expression ( mathbf{A} times mathbf{B} + mathbf{B} times mathbf{C} + mathbf{B} times mathbf{D} + mathbf{C} times mathbf{D} ) where ( A, B, C, D ) are the numbers 12, 14, 16, and 18, we analyze the expression by considering each possible value for ( B ) and calculating the corresponding value of the expression.The expression can be rewritten as:[ B(A + C + D) + CD ]Since ( A + B + C + D = 60 ), we have ( A + C + D = 60 - B ). Substituting this into the expression, we get:[ B(60 - B) + CD ]which simplifies to:[ 60B - B^2 + CD ]We need to maximize this expression for each possible value of ( B ) (12, 14, 16, 18) by assigning the remaining numbers to ( A ), ( C ), and ( D ) such that ( CD ) is maximized. The remaining numbers are assigned to ( A ), ( C ), and ( D ), with the smallest number assigned to ( A ) to maximize ( CD ).1. Case ( B = 12 ): - Remaining numbers: 14, 16, 18 - Assign ( A = 14 ), ( C = 16 ), ( D = 18 ) - ( CD = 16 times 18 = 288 ) - Expression value: ( 60 times 12 - 12^2 + 288 = 720 - 144 + 288 = 864 )2. Case ( B = 14 ): - Remaining numbers: 12, 16, 18 - Assign ( A = 12 ), ( C = 16 ), ( D = 18 ) - ( CD = 16 times 18 = 288 ) - Expression value: ( 60 times 14 - 14^2 + 288 = 840 - 196 + 288 = 932 )3. Case ( B = 16 ): - Remaining numbers: 12, 14, 18 - Assign ( A = 12 ), ( C = 14 ), ( D = 18 ) - ( CD = 14 times 18 = 252 ) - Expression value: ( 60 times 16 - 16^2 + 252 = 960 - 256 + 252 = 956 )4. Case ( B = 18 ): - Remaining numbers: 12, 14, 16 - Assign ( A = 12 ), ( C = 14 ), ( D = 16 ) - ( CD = 14 times 16 = 224 ) - Expression value: ( 60 times 18 - 18^2 + 224 = 1080 - 324 + 224 = 980 )The maximum value is achieved when ( B = 18 ), resulting in the expression value of 980.[boxed{980}],<think>Okay, so I need to figure out the maximum value of this equation: A × B + B × C + B × D + C × D. The numbers A, B, C, D are 12, 14, 16, and 18, but I can assign each number to any of the variables. The goal is to assign them in such a way that the total value of the equation is as large as possible. First, let me write down the equation again to make sure I have it right: A×B + B×C + B×D + C×D. Hmm. Let me see. So the equation has four terms: A multiplied by B, then B multiplied by C, then B multiplied by D, and finally C multiplied by D. Since we need to substitute each of the numbers 12, 14, 16, 18 into A, B, C, D without repetition, each variable gets a unique number. So, the variables are all distinct numbers from the given set. To find the maximum value, I probably need to consider how each term contributes to the total. Let me analyze the equation term by term. First term: A × B. This is straightforward; the product of A and B. Second term: B × C. So here, B is multiplied by C. Third term: B × D. Again, B is multiplied by D. Fourth term: C × D. This time, C is multiplied by D. So, looking at this, B appears in three terms: the first, second, and third. C appears in two terms: the second and fourth. D appears in two terms: the third and fourth. A only appears in the first term. Therefore, B is involved in three products, C and D in two each, and A in only one. So, to maximize the total, perhaps we need to assign the largest numbers to the variables that are multiplied the most. Since B is in three terms, maybe assigning the largest number to B will have the most significant impact. Then, C and D are each in two terms, so maybe the next largest numbers should go to them, and A, which is only in one term, gets the smallest number. Alternatively, maybe we need to balance the assignments since some terms are products of two variables. Let's see. But let's test this idea. Assign the largest number, 18, to B because B is in three terms. Then, the next largest numbers, 16 and 14, to C and D, since they each appear in two terms. Then the smallest number, 12, to A. Let's compute the value:A = 12, B = 18, C = 16, D = 14.Then:A×B = 12×18 = 216B×C = 18×16 = 288B×D = 18×14 = 252C×D = 16×14 = 224Total = 216 + 288 + 252 + 224 = Let's add them step by step.216 + 288 = 504504 + 252 = 756756 + 224 = 980So total is 980.Alternatively, maybe assign B to 18, then C and D to 16 and 14. But perhaps if we swap C and D with different numbers, maybe we can get a higher total? Let's see.Wait, what if we assign C and D as 16 and 14, but which one is assigned to C or D? Let's check both possibilities.First possibility: C=16, D=14: as above, gives C×D=16×14=224.Second possibility: C=14, D=16: then C×D=14×16=224. Same result. So swapping C and D doesn't affect the product. But B×C and B×D would be different.Wait, in the first case, B=18, C=16, D=14: B×C=288, B×D=252. Sum is 288+252=540.In the second case, B=18, C=14, D=16: B×C=252, B×D=288. Sum is still 540. So same total. So swapping C and D doesn't change the sum of B×C + B×D. Therefore, in this case, assigning C and D as 14 and 16 in either order gives same total. So the total would still be 980.But maybe if we assign B to a different number. Maybe not 18? Let's see. Let's try B=16. Then, B is in three terms. Then assign the largest remaining numbers to C and D, which would be 18 and 14. Wait, no, the numbers are 12,14,16,18. If B is 16, then the remaining numbers are 12,14,18. Assign A, C, D. So to maximize the terms involving C and D, which are in two terms each. Assign the largest remaining to C and D. The largest remaining numbers after B=16 are 18 and 14. Assign C=18, D=14. Then A would be 12. Let's compute:A=12, B=16, C=18, D=14.A×B = 12×16 = 192B×C = 16×18 = 288B×D = 16×14 = 224C×D = 18×14 = 252Total: 192 + 288 + 224 + 252.192 + 288 = 480480 + 224 = 704704 + 252 = 956Which is 956. Less than 980. So worse.Alternatively, if B=16, C=14, D=18.Then:A×B = 12×16=192B×C=16×14=224B×D=16×18=288C×D=14×18=252Total: 192 + 224 + 288 +252.192 +224=416, 416 +288=704, 704 +252=956. Same as before.So 956. So B=16 gives 956, which is less than 980. So B=18 is better.What if B=14? Then remaining numbers are 12,16,18. Assign A, C, D. Assign the largest remaining to C and D. So C=18, D=16, A=12. Then:A=12, B=14, C=18, D=16.A×B=12×14=168B×C=14×18=252B×D=14×16=224C×D=18×16=288Total: 168 +252 +224 +288.168 +252=420, 420 +224=644, 644 +288=932. Less than 980.Alternatively, C=16, D=18: same product. Let's check:A=12, B=14, C=16, D=18.A×B=12×14=168B×C=14×16=224B×D=14×18=252C×D=16×18=288Total: 168 +224=392, 392 +252=644, 644 +288=932. Same as above.So B=14 gives 932. So even worse.What about B=12? Assigning the smallest number to B. Then remaining numbers are 14,16,18. Assign A, C, D. Assign largest to C and D. So A=14, C=18, D=16.Compute:A=14, B=12, C=18, D=16.A×B=14×12=168B×C=12×18=216B×D=12×16=192C×D=18×16=288Total: 168 +216 +192 +288.168 +216=384, 384 +192=576, 576 +288=864. Which is even lower.Alternatively, C=16, D=18:A=14, B=12, C=16, D=18.A×B=14×12=168B×C=12×16=192B×D=12×18=216C×D=16×18=288Total: 168 +192=360, 360 +216=576, 576 +288=864. Same result.So, B=12 gives 864. So clearly, assigning B to the largest available number gives the highest total so far (980). But wait, let's check if there's another way. Maybe not assigning the largest to B. Let me see. Suppose we assign a larger number to C or D instead. For example, if we try to maximize C×D, which is the fourth term. Since C and D are both multiplied in that term. Maybe if we assign the two largest numbers to C and D, then C×D would be maximized. Let's see.The two largest numbers are 18 and 16. So if we assign C=18 and D=16, then C×D=288. But if we do that, then B can be 14 or 12. Let's see. If we assign B to 14, then:A would be 12. So:A=12, B=14, C=18, D=16.Which we already did earlier, resulting in 932. Alternatively, if we assign B=12, then A=14:A=14, B=12, C=18, D=16. Which gives 864. So even though C×D is maximized as 288, the other terms might not compensate. Because B is in three terms, which are B×A, B becomesFinal AnswerThe probability is boxed{dfrac{55}{64}}.</think>]

🔑:【Answer】980【Analysis】Key point: Maximum valueA times B+B times C+B times D+C times D=B times(A+C+D)+C times D text {, }Obviously, C times D is maximized at 16 times 18=288. Next, consider the maximum B times(A+C+D). Since B+(A+C+D)=12+14+16+18=60, the sum is a fixed value. When the sum is fixed, the product of two numbers is maximized when their difference is the smallest. Therefore, the product is maximized when B is 18, which is 18 times(12+14+16)=756. However, since there is only one 18, it can only be chosen from B, C, D. If B is 18, then the original expression sum is 18 times(12+14+16)+14 times 16=980. If B is 16, then the original expression sum is 16 times(12+14+18)+14 times 18=956. If B is 14, then the original expression sum is 14 times(12+16+18)+16 times 18=932. In summary, the maximum sum of the original expression is 980.