🧠:[<think>Okay, so I need to convert the decimal number 2009 into an octal number using the "divide by k and take the remainder" method. Hmm, octal is base 8, right? So that means I need to divide the number by 8 repeatedly and keep track of the remainders each time. Let me try to recall how this works.First, I remember that when converting from decimal to another base, you divide the number by the base (which is 8 here) and the remainder becomes the least significant digit. Then, you take the quotient from that division and divide it by the base again, getting another remainder, which is the next digit. You keep doing this until the quotient is zero. Then, you read the remainders from last to first to get the final number.Let me start with 2009 divided by 8. Let me do the division. So 2009 divided by 8. Let's see, 8 times 250 is 2000. So 2000 is 8*250. Then, 2009 - 2000 is 9. So the remainder is 9. Wait, but 9 is larger than 8, isn't it? Wait, no, because when you divide 2009 by 8, you have 8*251=2008, right? Because 8*250=2000, plus 8*1=8, so 2000+8=2008. So 2009 divided by 8 is 251 with a remainder of 1. Wait, 8*251=2008, so 2009-2008=1. So the first remainder is 1. So that's the first digit (rightmost digit) of the octal number.Then, take the quotient 251 and divide by 8 again. 251 divided by 8. Let's see, 8*30=240. So 251-240=11. Wait, that's 11, but 11 divided by 8 is 1 with remainder 3. Wait, no, 8*31=248, so 251-248=3. So 251 divided by 8 is 31 with a remainder of 3. So the next remainder is 3. That's the next digit to the left.Now take 31 and divide by 8. 8*3=24, so 31-24=7. So 31 divided by 8 is 3 with a remainder of 7. So the next digit is 7.Then take the quotient 3 and divide by 8. 8*0=0, so the remainder is 3. So the next remainder is 3, but since the quotient is now 0, we stop here.So collecting the remainders from last to first, that's 3, 7, 3, 1. So the octal number should be 3731. Let me verify that.Wait, let me check my steps again to make sure.First step: 2009 ÷ 8.2009 divided by 8. 8*250=2000. 2009-2000=9. 9 ÷8 is 1 with remainder 1. Wait, but how exactly is this?Wait, maybe I confused something. Let me do the division properly.2009 divided by 8.How many times does 8 go into 2009?8*251=2008. So 251*8=2008. 2009-2008=1. So quotient 251, remainder 1. Correct. So first remainder is 1.Then 251 ÷8. 8*31=248. 251-248=3. So quotient 31, remainder 3.Then 31 ÷8. 8*3=24. 31-24=7. Quotient 3, remainder 7.Then 3 ÷8. 0*8=0, remainder 3. So quotient 0, remainder 3.So the remainders, from last to first, are 3,7,3,1. So 3731 octal. Let me check by converting back.3*8^3 +7*8^2 +3*8 +1.Compute 3*512=1536.7*64=448. So 1536 +448=1984.3*8=24. 1984+24=2008. Then +1=2009. Yes, that works. So the answer should be 3731.But wait, let me check if I did the division steps correctly.First division: 2009 ÷8.251*8=2008, so 2009-2008=1. So remainder 1, quotient 251. Correct.Second division: 251 ÷8.31*8=248. 251-248=3. Remainder 3, quotient 31.Third division: 31 ÷8. 3*8=24. 31-24=7. Remainder 7, quotient 3.Fourth division: 3 ÷8. 0*8=0, remainder 3. Quotient 0. So we stop.Yes, so writing the remainders from last to first: 3,7,3,1. So 3731 in octal. Converting back to decimal gives 3*8³ + 7*8² +3*8 +1 = 3*512 +7*64 +24 +1 = 1536 +448 +24 +1 = 1536+448=1984, 1984+24=2008, 2008+1=2009. Correct.So that seems right. Therefore, the octal equivalent of 2009 is 3731.Final AnswerThe octal representation of the decimal number 2009 is boxed{3731}.</think>To convert the decimal number 2009 into an octal number using the "divide by k and take the remainder" method, we follow these steps:1. Divide 2009 by 8: - ( 2009 div 8 = 251 ) with a remainder of 1. The least significant digit is 1.2. Divide the quotient 251 by 8: - ( 251 div 8 = 31 ) with a remainder of 3. The next digit is 3.3. Divide the quotient 31 by 8: - ( 31 div 8 = 3 ) with a remainder of 7. The next digit is 7.4. Divide the quotient 3 by 8: - ( 3 div 8 = 0 ) with a remainder of 3. The most significant digit is 3.Reading the remainders from last to first, the octal representation of 2009 is ( 3731 ).To verify, we convert ( 3731_8 ) back to decimal:[3 cdot 8^3 + 7 cdot 8^2 + 3 cdot 8 + 1 = 3 cdot 512 + 7 cdot 64 + 3 cdot 8 + 1 = 1536 + 448 + 24 + 1 = 2009]Thus, the octal representation of the decimal number 2009 is (boxed{3731}).,<think>Okay, let me try to convert the decimal number 2009 into an octal number using the "divide by 8 and take the remainder" method. Hmm, I remember that octal is base 8, so each digit represents a power of 8. The rightmost digit is 8⁰, then 8¹, 8², and so on. To convert a decimal number to octal, you divide the number by 8, record the remainder, then divide the quotient by 8 again, and keep doing this until the quotient is zero. The octal number is then the remainders read from bottom to top. Let me see if I can apply this step by step.First, let's start with 2009. Divide by 8. What's 2009 divided by 8? Let me calculate that. 8 times 250 is 2000, right? So 2000 divided by 8 is 250. Then 2009 minus 2000 is 9. So 2009 divided by 8 is 251 with a remainder of 1. Wait, let me check that again. If 8*250=2000, then 2000 +9=2009. So 2009 divided by 8: 250*8=2000, 2009-2000=9. So 9 divided by 8 is 1 with a remainder of 1. Wait, no, that's not the right way. Wait, when you divide 2009 by 8, you have to find how many times 8 fits into 2009. Let me do it step by step.Alternatively, maybe using calculator-like division. 8 into 2009. 8*300=2400, which is too big. So 8*200=1600. 2009-1600=409. Then 8*50=400. So 200+50=250, 409-400=9. Then 8*1=8. So 250+1=251, remainder 1. So 2009 divided by 8 is 251 with a remainder of 1. So the first remainder is 1. That will be the least significant digit (rightmost digit) in the octal number.Now, take the quotient 251 and divide that by 8. Let's do that. 8*30=240, so 251-240=11. 11 divided by 8 is 1 with a remainder of 3. Wait, no. Wait, 8*31=248. 251-248=3. So 251 divided by 8 is 31 with a remainder of 3. So the next remainder is 3. That's the next digit to the left.Now take the quotient 31 and divide by 8. 8*3=24, 31-24=7. So 31 divided by 8 is 3 with a remainder of 7. So the next remainder is 7.Take the quotient 3 and divide by 8. 8*0=0, so 3 divided by 8 is 0 with a remainder of 3. So the remainder here is 3. Since the quotient is now 0, we stop.Now, collecting the remainders from last to first, we have 3, 7, 3, 1. So the octal number should be 3731. Let me verify this to make sure.Wait, let's check if 3*8³ + 7*8² + 3*8 +1 equals 2009.Calculate 8³ is 512, so 3*512=1536.8² is 64, so 7*64=448.8^1 is 8, so 3*8=24.Plus 1.Adding them up: 1536 + 448 = 1984. Then 1984 +24=2008, plus 1 is 2009. Perfect, that adds up correctly. So the octal conversion is indeed 3731.Wait, just to recap the steps:1. 2009 divided by 8: quotient 251, remainder 1.2. 251 divided by 8: quotient 31, remainder 3.3. 31 divided by 8: quotient 3, remainder 7.4. 3 divided by 8: quotient 0, remainder 3.Then writing the remainders from last to first: 3, 7, 3, 1 → 3731₈.I think that's right. Let me cross-check with another method. Alternatively, I can use the expansion method. Let me see.Starting from the right, the digits in octal are 8^0, 8^1, 8^2, 8^3. So the rightmost digit is 1 (from 2009), then next is 3 (from 251), then 7 (from 31), then 3 (from 3). So yes, 3731. That seems correct. Also, if I convert 3731 octal back to decimal, it should give me 2009. Which we already verified.Therefore, I think the answer is correct. So the octal equivalent of decimal 2009 is 3731.Final AnswerThe octal representation of decimal number 2009 is boxed{3731}.</think>To convert the decimal number 2009 into an octal number using the "divide by 8 and take the remainder" method, we follow these steps:1. Divide 2009 by 8: - Quotient: 251 - Remainder: 12. Divide the quotient 251 by 8: - Quotient: 31 - Remainder: 33. Divide the quotient 31 by 8: - Quotient: 3 - Remainder: 74. Divide the quotient 3 by 8: - Quotient: 0 - Remainder: 3Reading the remainders from last to first, we get the octal number 3731. Verifying this by converting back to decimal:[3 times 8^3 + 7 times 8^2 + 3 times 8^1 + 1 times 8^0 = 3 times 512 + 7 times 64 + 3 times 8 + 1 times 1 = 1536 + 448 + 24 + 1 = 2009]Thus, the octal representation of the decimal number 2009 is (boxed{3731}).]

🔑:2009 ÷ 8 = 251 remainder 1 251 ÷ 8 = 31 remainder 3 31 ÷ 8 = 3 remainder 7 3 ÷ 8 = 0 remainder 3 Therefore, 2009_{(10)} = 3731_{(8)} Hence, the answer is: boxed{3731_{(8)}}

🧠:[<think>Okay, let me try to work through this problem step by step. First, let me make sure I understand the problem correctly.We have two superheroes, Superman and Flash, who are called to do construction work. Their productivity is different: Flash is twice as productive as Superman. The payment each receives is based on the time they spend working, following the formula w = 90 / t, where w is the amount of money and t is the time in minutes. Now, Flash was 1 minute late. During that 1 minute, Superman managed to complete 10% of the total work. The rest of the construction (90%) was done by both working together. We need to find out how much money each superhero received.Alright, let's break this down. Let me note down the key points:1. Flash's productivity is twice Superman's.2. Payment formula: w = 90 / t for each. Wait, that seems a bit odd. So the amount of money they receive is 90 divided by the time they spend working. So the longer they work, the less money they get? That might be. Maybe it's structured as a fixed amount divided by their time, so they get paid more if they finish faster? Hmm, maybe the total payment is 90, and it's split based on time? Wait, the problem says "the payment for each superhero depends only on the time they spend on the work, as follows: w = 90 / t". So for each individual superhero, their payment is 90 divided by their own individual time. So if Superman works for t minutes, he gets 90 / t dollars. Similarly, Flash works for (t - 1) minutes because he was 1 minute late, so he gets 90 / (t - 1) dollars. Wait, maybe not. Wait, the total work is done in two parts: first, Superman works alone for 1 minute, then both work together until the job is done.Wait, actually, let me parse the problem again:"Flash was 1 minute late to the start of the construction, and during this time, Superman managed to complete 10% of the total work. The rest of the construction was completed together by both superheroes."So the timeline is:1. Superman starts working alone at time t=0.2. Flash arrives 1 minute late, so he starts working at t=1 minute.3. During that first 1 minute (from t=0 to t=1), Superman completes 10% of the work.4. From t=1 until the work is finished, both work together and complete the remaining 90%.So the total work time for Superman is the entire duration until completion, and Flash's work time is the duration from t=1 to completion, which is 1 minute less than Superman's total time.Our goal is to calculate how much money each received. The payment for each is w = 90 / t, where t is the time they spent working. Wait, but that would mean each is paid 90 divided by their own time? So if Superman worked for T minutes, he gets 90 / T dollars. Flash worked for (T - 1) minutes, so he gets 90 / (T - 1) dollars. But that seems strange because if they work longer, they get paid less. Maybe the total amount to be paid is 90, and each gets a share based on the inverse of their time? Hmm, maybe I need to clarify.Wait, the problem says: "The payment for each superhero depends only on the time they spend on the work, as follows: w = 90 / t, where w is the amount of money the superhero will receive, and t is the time in minutes they spend on the work."So each superhero's payment is individually calculated as 90 divided by their own time spent. So regardless of how much work they did, their payment is based only on their time. That is, Superman's payment is 90 divided by the total time he worked, and Flash's payment is 90 divided by the total time he worked. Wait, but if they are being paid based on time, then perhaps the longer they work, the less they get paid? That seems counterintuitive. Maybe it's supposed to represent a penalty for taking longer? Or maybe it's a fixed reward divided by time? Hmm. Maybe the problem is structured so that each superhero is paid 90 dollars, but divided by their time. So if Superman takes 10 minutes, he gets 9 dollars. If he takes 1 minute, he gets 90 dollars. So the faster they work, the more they get paid. So it's like a reward for finishing quickly. That makes sense. So they want to maximize their payment by working as little time as possible. But in this case, they have to complete the work, so their time is determined by how long it takes to finish the job.But the problem says "the payment for each superhero depends only on the time they spend on the work, as follows: w=90/t". So each of them will get paid 90 divided by their own time. So if Superman works t_S minutes, he gets 90/t_S, and Flash works t_F minutes, he gets 90/t_F. The total payment from the company would be 90/t_S + 90/t_F.But the problem is asking "how much money did each superhero receive for the work they completed?" So we need to compute 90/t_S and 90/t_F where t_S is Superman's total time, and t_F is Flash's total time.Given that Superman started alone, worked for 1 minute, during which he completed 10% of the work. Then both worked together for the remaining time to finish the remaining 90%.So first, let's figure out how long the total work took. Let's denote:- Let’s denote Superman's work rate as S (portion of the work per minute). Then Flash's rate is 2S, since he's twice as productive.In the first minute, Superman works alone at rate S, completing 10% (0.1) of the work. So:S * 1 = 0.1 => S = 0.1 per minute.Therefore, Superman's productivity is 0.1 (10%) per minute, and Flash's productivity is 0.2 per minute.Then, after the first minute, both work together. Their combined rate is S + 2S = 3S = 0.3 per minute.The remaining work is 90% (0.9). So time needed to complete the remaining work is 0.9 / 0.3 = 3 minutes.Therefore, the total time Superman worked is 1 + 3 = 4 minutes. Flash worked only 3 minutes (since he started 1 minute late).Therefore, Superman's payment is 90 / 4 = 22.50, and Flash's payment is 90 / 3 = 30.00.Wait, but let me check that again.Wait, the problem says "the payment for each superhero depends only on the time they spend on the work, as follows: w=90/t". So Superman worked 4 minutes, so he gets 90 / 4 = 22.5. Flash worked 3 minutes, gets 90 / 3 = 30. So that's straightforward?But let me double-check the work rates and the time calculation.First, Superman works alone for 1 minute and completes 10% of the work. Therefore, his rate is 10% per minute, or 0.1 work units per minute. Then Flash's rate is 0.2 work units per minute.When they work together, their combined rate is 0.1 + 0.2 = 0.3 work units per minute. The remaining work is 90%, so 0.9 work units. Therefore, time required is 0.9 / 0.3 = 3 minutes. Therefore, total time from start to finish is 1 + 3 = 4 minutes. Therefore, Superman worked 4 minutes, Flash worked 3 minutes.Therefore, payments:Superman: 90 / 4 = 22.5Flash: 90 / 3 = 30So the answer is Superman received 22.50 and Flash received 30.00.Wait, but let me think again. The problem says "the payment for each superhero depends only on the time they spend on the work". So maybe the payment is calculated based on how long they worked, but the total payment from the company could be different. Wait, the problem doesn't specify a total amount of money, except for the formula w=90/t. So each superhero individually gets 90 divided by their own time. So the company pays each of them separately according to that formula. So regardless of how much work each did, their payment is based only on their time. So even if Superman did 10% in the first minute and then both did the rest together, his payment is 90 divided by his total time (4 minutes), and Flash's payment is 90 divided by his total time (3 minutes). That seems to be what the problem states. So the answer is Superman gets 22.5, Flash gets 30.But let me check again to make sure there's no misunderstanding.Alternatively, maybe the payment formula is different. Maybe the total amount of money is 90, and each gets a portion based on their time? But the problem says "the payment for each superhero depends only on the time they spend on the work, as follows: w=90/t". So the wording is a bit ambiguous. It could mean that each superhero's payment is calculated as 90 divided by their own time. So if they work t minutes, they get 90/t dollars. So even if both work different times, each is paid 90 over their own time. So the company would pay Superman 90/4 and Flash 90/3, totaling 22.5 + 30 = 52.5 dollars. That seems possible. But perhaps the problem is in another interpretation: that the total payment is 90, which is split between the two based on the inverse of their time? Wait, but the problem states "the payment for each superhero depends only on the time they spend on the work, as follows: w=90/t". So for each, individually, their payment is 90 divided by their time. So regardless of the other's time, each one's payment is 90/t. So company pays both 90/t_S + 90/t_F.But in that case, the total payment would depend on how long each worked, which is determined by the problem. The problem doesn't state that 90 is the total payment; it's just the formula for each. So each is paid 90 divided by their own time. So yes, the answer would be Superman: 22.5, Flash: 30.But let me check again the work rates.Let me formalize it with variables.Let’s denote:- Let’s let Superman’s work rate be S (work per minute). Then Flash’s rate is 2S.In the first minute, Superman works alone and completes 10% of the work. Therefore:S * 1 = 0.1 => S = 0.1 work per minute.Therefore, Flash’s rate is 0.2 work per minute.After the first minute, both work together. The remaining work is 0.9.Their combined rate is S + 2S = 3S = 0.3 work per minute.Time to finish remaining work: t = 0.9 / 0.3 = 3 minutes.Therefore, total time Superman worked: 1 + 3 = 4 minutes.Total time Flash worked: 3 minutes.Therefore, payments:Superman: 90 / 4 = 22.5Flash: 90 / 3 = 30Yes, that seems consistent.Alternatively, let's check if the total work is considered as 1 unit. Superman does 0.1 in the first minute, so his rate is 0.1 per minute. Then together, their combined rate is 0.3 per minute. The remaining work is 0.9, so time is 3 minutes. Total time 4 minutes. So calculations hold.Therefore, the answer should be Superman receives 22.5 and Flash receives 30.Final AnswerSuperman received boxed{22.50} dollars and Flash received boxed{30.00} dollars.</think>Given the problem, we need to determine how much money each superhero received based on the time they spent working. The key steps are as follows:1. Productivity Rates: - Superman's productivity is ( S ) (work per minute). - Flash's productivity is twice that of Superman, so ( 2S ).2. Initial Work by Superman: - Superman works alone for the first 1 minute and completes 10% of the total work. - Therefore, ( S times 1 = 0.1 ) implies ( S = 0.1 ) work per minute.3. Combined Work Rate: - After the first minute, both Superman and Flash work together. - Their combined work rate is ( S + 2S = 3S = 0.3 ) work per minute.4. Remaining Work and Time: - The remaining work is 90% (0.9 work units). - Time required to complete the remaining work is ( frac{0.9}{0.3} = 3 ) minutes.5. Total Time Spent: - Superman worked for the entire duration: ( 1 + 3 = 4 ) minutes. - Flash worked for the duration after being 1 minute late: ( 3 ) minutes.6. Payment Calculation: - Payment formula: ( w = frac{90}{t} ), where ( t ) is the time spent working. - Superman's payment: ( frac{90}{4} = 22.5 ) dollars. - Flash's payment: ( frac{90}{3} = 30 ) dollars. Final AnswerSuperman received boxed{22.50} dollars and Flash received boxed{30.00} dollars.,<think>Okay, so we have this problem where Superman and Flash are doing construction work for Superheroes, Inc. The problem is about figuring out how much money each of them gets paid based on the time they spend working. Let me try to break this down step by step.First, the key points:1. Productivity: Flash is twice as productive as Superman. So if Superman can do, say, x units of work per minute, Flash can do 2x units per minute. Productivity here probably means work rate, right? So work rate of Flash = 2 * work rate of Superman.2. Payment Formula: Each superhero is paid according to w = 90 / t, where w is the money received and t is the time in minutes they spend working. So the longer they take, the less they get paid. Interesting. So it's inversely proportional to the time they spend. But how does this apply? Is it that each gets 90 divided by their own time? Or is it 90 divided by the total time? Wait, the problem says "the payment for each superhero depends only on the time they spend on the work." So each has their own w, calculated as 90 divided by their own t. So if Superman works for t minutes, he gets 90/t dollars, and similarly for Flash.3. Late Arrival: Flash was 1 minute late. During that 1 minute, Superman managed to complete 10% of the total work. Then, after that, both worked together until the rest (90%) was done. So the total work is split into two parts: 10% done by Superman alone in 1 minute, and the remaining 90% done by both working together.So, we need to find the time each spent working and then compute their payments using the formula.Let me formalize this:Let’s denote:- Let’s suppose the total work is W.- Superman's work rate is S (work per minute), so Flash's work rate is 2S.- Superman works alone for the first 1 minute, completing 10% of W. So:In 1 minute, Superman does S * 1 = 0.1 * W => S = 0.1 * W. So Superman's work rate is 0.1W per minute. Then Flash's work rate is 0.2W per minute.Then, after the first minute, the remaining work is 0.9W. Both work together, so their combined work rate is S + 2S = 3S = 0.3W per minute. Therefore, time taken to complete 0.9W at 0.3W per minute is 0.9 / 0.3 = 3 minutes. So they worked together for 3 minutes.Therefore, Superman's total time is 1 minute alone + 3 minutes with Flash = 4 minutes. So his payment is 90 / 4 = 22.50.Flash arrived 1 minute late, so he started working at minute 1, and worked for 3 minutes. Therefore, his time is 3 minutes. His payment is 90 / 3 = 30.00.But wait, let me check again to make sure.Total work W. Superman works alone for 1 minute, completing 10% of W. So in 1 minute, he does 0.1W. So his rate S is 0.1W per minute. Then Flash's rate is 0.2W per minute. When working together, their combined rate is 0.1 + 0.2 = 0.3W per minute. The remaining work is 0.9W. So time taken is 0.9W / 0.3W per minute = 3 minutes. Therefore, Superman worked 1 + 3 = 4 minutes, and Flash worked 3 minutes. Therefore, their payments are 90/4 and 90/3. So Superman gets 22.50 and Flash gets 30.00.But let me check if this is correct. Let's confirm if 10% of the work was done by Superman in 1 minute. If Superman's rate is 0.1W per minute, then in 1 minute, he does 0.1W, which is indeed 10%. Then, together, they work at 0.3W per minute for 3 minutes, which is 0.9W, so total work is 0.1 + 0.9 = 1.0W. That checks out.So payments: Superman's time is 4 minutes, so 90 / 4 = 22.5. Flash's time is 3 minutes, so 90 / 3 = 30. Therefore, Superman gets 22.50, Flash gets 30.00.Wait, but the problem states "The payment for each superhero depends only on the time they spend on the work," so even though Flash was late, the payment is based on the actual time each spent working. So Superman worked 4 minutes, Flash worked 3 minutes. So the calculation seems correct. Therefore, the answer should be Superman: 22.50, Flash: 30.00.But let me check again with different variables to make sure.Let’s denote Superman’s work rate as S. Then Flash’s work rate is 2S. The total work is W.In the first minute, Superman does S * 1 = 0.1W, so S = 0.1W per minute. Hence, Flash is 0.2W per minute.Remaining work: W - 0.1W = 0.9W. Working together, their combined rate is S + 2S = 3S = 0.3W per minute. Time taken: 0.9W / 0.3W per minute = 3 minutes. So total time for Superman: 1 + 3 = 4 minutes. For Flash: 3 minutes. So payment for Superman: 90 / 4 = 22.5, Flash: 90 / 3 = 30.Yes, this seems consistent.Therefore, the answer should be Superman received 22.50 and Flash received 30.00.Final AnswerSuperman received boxed{22.50} dollars and Flash received boxed{30.00} dollars.</think>Given the problem, we need to determine how much money Superman and Flash each received for their work. Here are the key steps:1. Productivity Rates: Flash's productivity is twice that of Superman. Let Superman's work rate be ( S ) (work per minute). Therefore, Flash's work rate is ( 2S ).2. Superman's Initial Work: Superman worked alone for the first minute and completed 10% of the total work. If the total work is ( W ), then Superman's work rate ( S ) is calculated as: [ S times 1 = 0.1W implies S = 0.1W text{ per minute} ]3. Remaining Work: The remaining 90% of the work (( 0.9W )) was completed by both Superman and Flash working together. Their combined work rate is: [ S + 2S = 3S = 3 times 0.1W = 0.3W text{ per minute} ]4. Time to Complete Remaining Work: The time taken to complete the remaining 90% of the work is: [ frac{0.9W}{0.3W} = 3 text{ minutes} ]5. Total Time Spent: - Superman worked for 1 minute alone and 3 minutes with Flash, totaling ( 1 + 3 = 4 ) minutes. - Flash started 1 minute late and worked for 3 minutes.6. Payment Calculation: - Superman's payment: ( frac{90}{4} = 22.50 ) dollars. - Flash's payment: ( frac{90}{3} = 30.00 ) dollars. Final AnswerSuperman received boxed{22.50} dollars and Flash received boxed{30.00} dollars.]

🔑:# Solution:9/10 of the work was completed together by the superheroes, with 3/10 done by Superman and 6/10 by Flash, since his productivity is twice as high. Since 1/10 of the entire work was completed by Superman in 1 minute, 3/10 of the work will take him 3 minutes. At the same time, Flash will also work for 3 minutes, as they will finish the work simultaneously. Then Superman will receive 90 / (1+3) = 22.5, and Flash will receive 90 / 3 = 30.## Grading Scheme:Fully correct solution - 15 points.Correctly found the working time of each superhero, but did not find the amount of money - 10 points out of 15.Incorrectly found the working time of each superhero, but the logic of the search is correct - 5 points out of 15.

🧠:[<think>Okay, so I need to find the dimension of the vector space V consisting of balanced polynomials of degree at most 2009 in two variables x and y. These polynomials have real coefficients and satisfy the condition that their average value on every circle centered at the origin is zero. First, I should recall what it means for the average value on each circle to be zero. A circle centered at the origin with radius r is defined by all points (x, y) such that x² + y² = r². The average value of a polynomial P(x, y) on this circle would be the integral of P over the circle divided by the circumference of the circle. So, for the average to be zero for every r > 0, the integral of P(x, y) over each such circle must be zero.Hmm, integrating over a circle... Maybe polar coordinates would be helpful here. If I switch to polar coordinates, where x = r cos θ and y = r sin θ, then the circle of radius r is parameterized by θ from 0 to 2π. The integral over the circle of radius r becomes an integral from 0 to 2π of P(r cos θ, r sin θ) multiplied by the arc length element, which is r dθ (since ds = r dθ for a circle of radius r). But the average value is (1/(2πr)) times this integral, because the circumference is 2πr. So the average value is (1/(2πr)) ∫₀²π P(r cos θ, r sin θ) r dθ = (1/(2π)) ∫₀²π P(r cos θ, r sin θ) dθ. Therefore, the condition that the average is zero for every r is equivalent to ∫₀²π P(r cos θ, r sin θ) dθ = 0 for all r > 0. So, if I can write P(x, y) in polar coordinates as P(r cos θ, r sin θ), then integrating over θ from 0 to 2π gives zero for each r. Now, polynomials in x and y can be expressed in polar coordinates as functions of r and θ. For example, x = r cos θ, y = r sin θ, so x² + y² = r², etc. Maybe it's useful to decompose the polynomial into homogeneous components. Let me recall that any polynomial can be written as a sum of homogeneous polynomials of different degrees. So, if P is a polynomial of degree at most 2009, then P = P₀ + P₁ + ... + P₂₀₀₉, where each Pₖ is homogeneous of degree k.Since the integral over θ is linear, the integral of P over the circle of radius r is the sum of the integrals of each Pₖ(r cos θ, r sin θ) over θ. Therefore, the condition that the integral is zero for all r > 0 would require that each homogeneous component's integral is zero for all r. But since each Pₖ is homogeneous of degree k, Pₖ(r cos θ, r sin θ) = rᵏ Pₖ(cos θ, sin θ). Therefore, the integral over θ becomes rᵏ ∫₀²π Pₖ(cos θ, sin θ) dθ. For this to be zero for all r > 0, the integral itself must be zero. Because otherwise, if the integral is non-zero, then for r > 0, multiplying by rᵏ would just scale it, but since r can be any positive number, the only way the integral times rᵏ is zero for all r is if the integral itself is zero. Therefore, the condition that the average is zero on every circle is equivalent to each homogeneous component Pₖ satisfying ∫₀²π Pₖ(cos θ, sin θ) dθ = 0. So the balanced condition decomposes into each homogeneous part individually having zero average over the unit circle. Therefore, the space V of balanced polynomials of degree at most 2009 is the direct sum of the spaces Vₖ for k from 0 to 2009, where each Vₖ is the space of homogeneous polynomials of degree k which satisfy ∫₀²π Pₖ(cos θ, sin θ) dθ = 0. Therefore, the dimension of V is the sum over k from 0 to 2009 of the dimensions of each Vₖ. But now, I need to figure out the dimension of each Vₖ. First, let's recall that the space of homogeneous polynomials of degree k in two variables has dimension k + 1. For example, for degree k, the monomials xᵏ, xᵏ⁻¹ y, ..., yᵏ form a basis, so there are k + 1 monomials, hence dimension k + 1.But now, in each Vₖ, we have a condition: the integral over the unit circle of Pₖ(cos θ, sin θ) dθ = 0. So this is a linear condition on the coefficients of Pₖ. Therefore, if this condition is non-trivial, the dimension of Vₖ would be (k + 1) - 1 = k, provided that the integral is not identically zero on the space of homogeneous polynomials of degree k.But wait, is the integral ∫₀²π Pₖ(cos θ, sin θ) dθ always zero for some polynomials and not others? Let's check. For example, when k is even or odd?Wait, maybe we can use Fourier analysis here. Expressing Pₖ(cos θ, sin θ) in terms of trigonometric polynomials. Since Pₖ is a homogeneous polynomial of degree k, when expressed in polar coordinates, it becomes rᵏ times a trigonometric polynomial of degree k. For example, xᵏ = rᵏ cosᵏ θ, yᵏ = rᵏ sinᵏ θ, and mixed terms like xᵐ yⁿ (with m + n = k) become rᵏ cosᵐ θ sinⁿ θ. So when we evaluate on the unit circle (r = 1), Pₖ(cos θ, sin θ) is a linear combination of terms like cosᵐ θ sinⁿ θ with m + n = k.The integral ∫₀²π cosᵐ θ sinⁿ θ dθ is zero unless both m and n are even. Wait, is that true? Actually, more precisely, the integral is zero unless m and n are both even. Wait, no. Let me recall: for example, if m is even and n is odd, the integral over 0 to 2π would be zero because of periodicity. Similarly, if either m or n is odd, the integral over the full circle would be zero. Because cosᵐ θ sinⁿ θ would be an odd function in some sense. For example, if n is odd, then sinⁿ θ is an odd function about θ = π, so integrating over 0 to 2π would cancel out. Similarly for m odd. Therefore, ∫₀²π cosᵐ θ sinⁿ θ dθ = 0 if either m or n is odd. If both m and n are even, then the integral is non-zero. Therefore, for a homogeneous polynomial Pₖ of degree k, when we express it in terms of cos θ and sin θ, only the terms where all exponents are even will contribute to the integral. Wait, but if k is even, then m and n can both be even? For example, if k is even, say k = 2p, then m and n can be both even: m = 2a, n = 2b, such that 2a + 2b = 2p. Similarly, if k is odd, then even if we split into m and n, one would have to be odd and the other even. But if k is odd, then m + n = odd, so one is even and the other is odd. But then, in that case, when we integrate over θ from 0 to 2π, since either m is even and n is odd, or vice versa, but in either case, the integral would be zero. Therefore, for k odd, the integral ∫₀²π Pₖ(cos θ, sin θ) dθ = 0 for any homogeneous polynomial of odd degree k. But if k is even, then there can be terms in Pₖ where m and n are both even, and those would contribute to the integral. So in that case, the integral is not automatically zero. Therefore, for even k, the integral is a non-trivial linear functional on the space of homogeneous polynomials of degree k, hence the dimension of Vₖ is (k + 1) - 1 = k. For odd k, since the integral is always zero, Vₖ is the entire space of homogeneous polynomials of degree k, so dimension k + 1.Wait, let me verify this. Suppose k is odd. Then any monomial xᵐ yⁿ with m + n = k must have one of m or n odd. Therefore, when expressed in polar coordinates, each term would have an odd power of either cos θ or sin θ. When integrating over θ from 0 to 2π, each term would integrate to zero. Therefore, the integral ∫₀²π Pₖ(cos θ, sin θ) dθ = 0 automatically. Therefore, for k odd, the condition is automatically satisfied, so all homogeneous polynomials of odd degree are in Vₖ. Hence, for odd k, Vₖ is the entire space, with dimension k + 1. For even k, the integral is not automatically zero, so we have a condition that the integral must be zero, which is a linear constraint, so the dimension is (k + 1) - 1 = k.Therefore, putting this together, the space V of balanced polynomials up to degree 2009 is the direct sum over k = 0 to 2009 of Vₖ, where each Vₖ has dimension:- If k is even: dim Vₖ = k- If k is odd: dim Vₖ = k + 1Therefore, to compute the total dimension, we need to sum over k = 0 to 2009:Sum_{k=0}^{2009} [dim Vₖ] = Sum_{k even} k + Sum_{k odd} (k + 1)Let me separate the sum into even and odd k. Let's denote E = {0, 2, 4, ..., 2008} (since 2009 is odd, the last even k is 2008) and O = {1, 3, 5, ..., 2009}.So, Sum_{k even} k is the sum of even numbers from 0 to 2008.Sum_{k odd} (k + 1) is the sum of (k + 1) for k odd from 1 to 2009. Note that (k + 1) for k odd is even, since k is odd, so k + 1 is even. For example, when k = 1, we get 2; when k = 3, we get 4, etc., up to k = 2009, we get 2010.So, Sum_{k odd} (k + 1) = Sum_{j=1}^{1005} (2j) where j ranges from 1 to 1005, since the number of odd numbers from 1 to 2009 is (2009 - 1)/2 + 1 = 1005. Wait, let's check: numbers from 1 to 2009, step 2: 1, 3, ..., 2009. The number of terms is (2009 - 1)/2 + 1 = 1004 + 1 = 1005. So yes, 1005 terms. Then (k + 1) when k = 2j - 1 (for j from 1 to 1005) would be (2j - 1 + 1) = 2j. Therefore, Sum_{k odd} (k + 1) = Sum_{j=1}^{1005} 2j = 2 * Sum_{j=1}^{1005} j = 2 * (1005 * 1006)/2 = 1005 * 1006.Similarly, Sum_{k even} k is the sum of even numbers from 0 to 2008. Let's note that the even numbers from 0 to 2008 can be written as 2j where j ranges from 0 to 1004 (since 2008 = 2 * 1004). Therefore, Sum_{k even} k = 2 * Sum_{j=0}^{1004} j = 2 * (1004 * 1005)/2 = 1004 * 1005.Therefore, the total dimension is:1004 * 1005 + 1005 * 1006Factor out 1005:1005 * (1004 + 1006) = 1005 * 2010Now compute 1005 * 2010. Let's compute that.1005 * 2000 = 2,010,0001005 * 10 = 10,050So total is 2,010,000 + 10,050 = 2,020,050Wait, but wait a second. Let me check that again. 1005 * 2010 = 1005 * (2000 + 10) = 1005*2000 + 1005*10 = 2,010,000 + 10,050 = 2,020,050. Yes.But let me verify the previous step:Sum_{k even} k = 1004*1005Sum_{k odd} (k + 1) = 1005*1006Therefore, total sum is 1004*1005 + 1005*1006 = 1005*(1004 + 1006) = 1005*2010 = 2,020,050.But wait, but the problem says the polynomials have degree at most 2009, so k ranges from 0 to 2009. However, we need to confirm that when we separated the sums, we accounted for all the terms. Let me verify the counts:Number of even k from 0 to 2009: Since 2009 is odd, the last even number is 2008. The even numbers start at 0 and go up by 2 each time. The number of terms is (2008 - 0)/2 + 1 = 1004 + 1 = 1005. Wait, 2008 divided by 2 is 1004, so 0 to 2008 inclusive is 1005 terms. Therefore, Sum_{k even} k is 1005 terms, sum = 1004*1005. Wait, but when we computed Sum_{k even} k as 2*Sum_{j=0}^{1004} j, which is 2*(1004*1005)/2 = 1004*1005. That seems correct.Similarly, the number of odd k from 1 to 2009 is (2009 - 1)/2 + 1 = 1005 terms. Then Sum_{k odd} (k + 1) is Sum_{j=1}^{1005} 2j, which is 2*(1005*1006)/2 = 1005*1006. That seems correct.Therefore, total dimension is 1004*1005 + 1005*1006 = 1005*(1004 + 1006) = 1005*2010 = 2,020,050.But wait a second, the answer seems quite large, but maybe it's correct. Let me cross-check with a smaller case. Let's take degree at most 1. Then the balanced polynomials would be V₀ ⊕ V₁. For k=0, which is even, the dimension of V₀ is 0 (since k=0 is even, dim V₀ = 0). For k=1, which is odd, dim V₁ = 1 + 1 = 2. So total dimension is 0 + 2 = 2. But polynomials of degree at most 1 are linear polynomials: ax + by + c. The balanced condition would require that the average over every circle is zero. The average of a constant c over any circle is c, so to have average zero, c must be zero. For the linear terms ax + by, since they are homogeneous of degree 1, which is odd, the integral over any circle would be zero, as per our previous reasoning. So the balanced polynomials are exactly the linear polynomials ax + by. So dimension 2. Which matches our calculation: if we set 2009=1, then 1005 would be (1 + 1)/2 = 1? Wait, wait, no. Wait, if 2009 is replaced by 1, then the sum is from k=0 to 1. So even k: 0, sum k=0: 0. Then odd k=1: dim V₁ = 1 + 1 = 2. So total dimension 0 + 2 = 2, which is correct.Similarly, let's check for degree at most 2. Then the balanced polynomials would be V₀ ⊕ V₁ ⊕ V₂. For k=0: even, dim V₀ = 0. For k=1: odd, dim V₁ = 2. For k=2: even, dim V₂ = 2. So total dimension 0 + 2 + 2 = 4. What's the actual dimension? Polynomials of degree at most 2: 1 (constant) + 2 (linear) + 3 (quadratic) = 6. The balanced condition requires that the average over each circle is zero. The constant term must be zero (dimension reduction by 1). The linear terms are allowed (dimension 2). The quadratic terms must have average zero over each circle. Quadratic polynomials are ax² + bxy + cy² + dx + ey + f. But constants must be zero, so f=0. Linear terms dx + ey are allowed. Quadratic terms: average of ax² + bxy + cy² over a circle of radius r. In polar coordinates, that's a r² cos²θ + b r² cosθ sinθ + c r² sin²θ. The average over θ would be (a r² * (1/2) + c r² * (1/2)) + 0, since the cross term b r² cosθ sinθ averages to zero. So the average is (a + c)/2 * r². For this to be zero for all r, we need a + c = 0. Therefore, the quadratic polynomials must satisfy a + c = 0. So the quadratic part has dimension 3 - 1 = 2. So total balanced polynomials of degree at most 2: 0 (constants) + 2 (linear) + 2 (quadratic) = 4, which matches our previous calculation. So our formula works here. For degree at most 2, the dimension is 4. If we apply the formula for N=2: even though 2 is even, but since the maximum degree is 2, which is even. Wait, in our original problem, the maximum degree is 2009, which is odd. Wait, but in the formula, we have Sum_{k=0}^{2009} dim Vₖ. So in the case of maximum degree 2, the formula would be:Sum_{k=0}^{2} dim Vₖ = dim V₀ + dim V₁ + dim V₂ = 0 + 2 + 2 = 4.Which is correct. Similarly, for maximum degree 3, it would be:Sum_{k=0}^3 dim Vₖ = 0 + 2 + 2 + 4 (since k=3 is odd, dim V₃ = 4). Wait, hold on. Wait, when k is odd, dim Vₖ = k + 1. For k=3, that's 4. The total would be 0 + 2 + 2 + 4 = 8. Let's check. Polynomials of degree at most 3. The balanced ones would have constants zero (dimension 0), linear terms (dim 2), quadratic terms with a + c = 0 (dim 2), and cubic terms. For cubic terms, since k=3 is odd, the integral over the circle is automatically zero, so all cubic terms are allowed. The space of cubic polynomials is 4-dimensional (x³, x² y, x y², y³). So total dimension would be 0 + 2 + 2 + 4 = 8. That seems correct. So the formula holds here.Therefore, returning to the original problem, with maximum degree 2009 (which is odd), the total dimension is 1005 * 2010 = 2,020,050.But let me check once more. Let's note that for each even k, we subtract 1 dimension, and for odd k, we don't. So the total dimension of all polynomials of degree at most 2009 is Sum_{k=0}^{2009} (k + 1) = Sum_{k=0}^{2009} (k + 1) = Sum_{m=1}^{2010} m = (2010)(2011)/2. This is the total dimension without any constraints. Then, the balanced condition imposes that for each even k, we have one linear constraint, so we subtract 1 for each even k. The number of even k from 0 to 2009 is 1005 (since 2008 is the last even, and (2008 - 0)/2 + 1 = 1005). Therefore, the dimension of V is total dimension minus 1005. Let's compute that:Total dimension: (2010)(2011)/2Subtract 1005: (2010)(2011)/2 - 1005But according to our previous calculation, the dimension is 1005 * 2010. Let's check if these are equal:(2010)(2011)/2 - 1005 = (2010*2011 - 2010)/2 = (2010*(2011 - 1))/2 = (2010*2010)/2 = (2010²)/2But 2010² / 2 = (2010/2)*2010 = 1005*2010, which matches our previous result. Therefore, both methods confirm that the dimension is 1005*2010 = 2,020,050.Therefore, the answer should be boxed{1005 times 2010} or boxed{2020050}. Since the problem says "Find the dimension of V," and typically dimensions are written as numbers, so 2020050 is the numerical answer. Let me check the multiplication once more: 1005 * 2010. 1005 * 2000 = 2,010,0001005 * 10 = 10,050Adding them together: 2,010,000 + 10,050 = 2,020,050. Yes, correct.Therefore, the dimension of V is 2,020,050, which can be written as boxed{2020050}.Final Answerboxed{2020050}</think>To find the dimension of the vector space ( V ) of balanced polynomials of degree at most 2009, we need to consider the condition that the average value of the polynomial on each circle centered at the origin is zero. This condition translates to the integral of the polynomial over each circle being zero. A polynomial ( P(x, y) ) can be decomposed into homogeneous components ( P_k ) of degree ( k ). The condition that the average value is zero for each circle implies that the integral of each homogeneous component ( P_k ) over the unit circle must be zero. For a homogeneous polynomial ( P_k ) of degree ( k ):- If ( k ) is odd, the integral over the unit circle is automatically zero, so all such polynomials are balanced. The dimension of the space of homogeneous polynomials of degree ( k ) is ( k + 1 ).- If ( k ) is even, the integral over the unit circle imposes a linear constraint, reducing the dimension by 1. The dimension of the space of balanced homogeneous polynomials of degree ( k ) is ( k ).To find the dimension of ( V ), we sum the dimensions of these spaces for ( k ) from 0 to 2009:- For even ( k ), the dimension is ( k ).- For odd ( k ), the dimension is ( k + 1 ).The sum is calculated as follows:- The number of even ( k ) from 0 to 2008 is 1005, and their sum is ( 1004 times 1005 ).- The number of odd ( k ) from 1 to 2009 is 1005, and the sum of ( k + 1 ) for these values is ( 1005 times 1006 ).Adding these results:[1004 times 1005 + 1005 times 1006 = 1005 times (1004 + 1006) = 1005 times 2010]Calculating the product:[1005 times 2010 = 2020050]Thus, the dimension of the vector space ( V ) is (boxed{2020050}).,<think>Okay, let's see. The problem is about balanced polynomials in two variables x and y with real coefficients. A balanced polynomial has the property that its average value on every circle centered at the origin is zero. We need to find the dimension of the vector space V of such polynomials with degree at most 2009.First, let's parse the problem again. The key points are:1. Polynomials in x and y with real coefficients.2. Balanced means the average on each circle (i.e., each circle centered at the origin) is zero.3. We are considering polynomials of degree at most 2009.4. The set of such balanced polynomials forms a vector space V over R.5. Find the dimension of V.So, the main task is to figure out the dimension of the space of balanced polynomials with degree ≤ 2009.First, I need to recall that the space of all polynomials in two variables of degree at most n has dimension (n+1)(n+2)/2. So for degree at most 2009, the dimension would be (2009+1)(2009+2)/2 = 2010*2011/2. But we need the dimension of the subspace V of balanced polynomials. So V is a subspace of this larger space, and we need to find its dimension.To find the dimension of V, perhaps we can find the number of constraints that being balanced imposes on the polynomials, and subtract that from the total dimension? But it's not immediately clear how many constraints there are, since the condition is for every circle centered at the origin. That is, for each radius r > 0, the average of the polynomial over the circle of radius r is zero. So infinitely many constraints, but the polynomials have finite degree. Maybe these constraints are redundant, so that only finitely many are needed?Alternatively, maybe we can use some properties of polynomials and their averages on circles. Let me think. If we express the polynomial in polar coordinates (r, θ), then perhaps we can expand it as a Fourier series in θ for each fixed r. The average over the circle of radius r would be the zeroth Fourier coefficient. So for the average to be zero for each r, the zeroth Fourier coefficient must be zero for all r. But since the polynomial has degree at most 2009, its expression in polar coordinates must involve terms up to r^2009 multiplied by trigonometric polynomials in θ of degree at most 2009.But wait, polynomials in x and y can be written in polar coordinates as polynomials in r and trigonometric functions of θ. For example, x = r cos θ, y = r sin θ. So a monomial like x^k y^m would become r^{k+m} cos^k θ sin^m θ. Therefore, when expressed in polar coordinates, each term would have a certain power of r multiplied by a trigonometric polynomial in θ.The average over the circle of radius r is the integral over θ from 0 to 2π of the polynomial divided by 2π. So for the average to be zero for all r, the integral over θ of each term in the polynomial must be zero for all r. But since each term has a factor of r^{k+m}, which is non-zero for r > 0, the integral over θ of the trigonometric part must be zero. However, the average over θ of cos^k θ sin^m θ is zero unless k and m are both even? Wait, no. Let me think again.Actually, the average over θ of cos^k θ sin^m θ is zero unless both k and m are even? Wait, no. For example, the average of cos θ over [0, 2π] is zero, the average of cos^2 θ is 1/2, the average of cos θ sin θ is zero, etc. So in general, the average of cos^k θ sin^m θ is non-zero only if k and m are both even. Wait, actually, even that's not quite right. Let me recall that when you have an odd power, the integral over the full circle is zero. For example, if k is odd, then cos^k θ is an odd function around θ = π, so integrating over 0 to 2π would give zero. Similarly for sin^m θ. Wait, no, cos θ is even, sin θ is odd. So, cos^k θ is even if k is even, even if k is odd? Wait, cos(-θ) = cos θ, so cos^k (-θ) = cos^k θ. So cos^k θ is always even. Similarly, sin(-θ) = -sin θ, so sin^m (-θ) = (-1)^m sin^m θ. Therefore, the product cos^k θ sin^m θ is even if m is even, and odd if m is odd. So integrating over the circle (from 0 to 2π), the integral of an odd function would be zero. But since the circle is symmetric, any function that's odd in θ would integrate to zero. So the average of cos^k θ sin^m θ is zero if m is odd. If m is even, then sin^m θ is even, and then the product is even. Wait, but cos^k θ sin^m θ is even or odd depending on m. Wait, actually, if m is even, sin^m θ is even, and cos^k θ is even, so their product is even. If m is odd, sin^m θ is odd, and cos^k θ is even, so their product is odd. Therefore, integrating over the symmetric interval [0, 2π], the integral of an odd function is zero. So the average of cos^k θ sin^m θ over the circle is zero unless m is even. Wait, but even then, if k is even and m is even, then the integral might not be zero. For example, if both k and m are even, then the integral is non-zero? Let's take k=0, m=0: integral of 1 is 2π. If k=2, m=0: integral of cos^2 θ is π. If k=0, m=2: integral of sin^2 θ is π. If k=1, m=0: integral of cos θ is zero. If k=0, m=1: integral of sin θ is zero. If k=1, m=1: integral of cos θ sin θ over [0, 2π] is zero. If k=2, m=2: integral of cos^2 θ sin^2 θ is π/4. So actually, the integral is non-zero only if both k and m are even? Wait, no. For example, if k=1, m=1, integral is zero. If k=2, m=0, integral is π. If k=0, m=2, integral is π. So actually, the average over θ is non-zero if either k is even or m is even? Wait, no, that's not correct. Wait, let's see. The average of cos^k θ sin^m θ is zero if either k or m is odd. Wait, no. Let's take k even and m odd. Then, cos^k θ is even, sin^m θ is odd, so the product is odd. Therefore, integral over symmetric interval is zero. Similarly, if k is odd and m even: product is odd. If both odd: product is even? Wait, no. If k is odd and m is even: cos^k θ is even (since cos is even, any power is even), sin^m θ is even (since m even), so product is even. Wait, no. Wait, cos^k θ is even regardless of k, because cos(-θ) = cos θ. Similarly, sin^m θ is even if m even, odd if m odd. Therefore, the product cos^k θ sin^m θ is even if m even, and odd if m odd. Therefore, integrating over [0, 2π], the integral is zero if m is odd, regardless of k. If m is even, then the product is even, and the integral may not be zero. Wait, so the integral of cos^k θ sin^m θ over the circle is zero if m is odd, and non-zero if m is even. But even when m is even, depending on k, the integral could still be zero or not. For example, if k is odd and m even: cos^k θ is even, sin^m θ is even, so product is even. Then the integral over [0, 2π] is twice the integral over [0, π]. But does that integral necessarily be non-zero? For example, cos^1 θ sin^0 θ: integral over [0, 2π] is zero. Wait, cos θ has integral zero. Wait, but m is even here. Wait, m=0, which is even, but k=1, which is odd. So in this case, integral of cos θ over [0, 2π] is zero. Wait, but according to earlier reasoning, if m is even, the integral could be non-zero, but here it is zero. So maybe my previous conclusion is wrong. Maybe the integral is non-zero only when both k and m are even? Wait, let's check:Take k=2, m=0: integral of cos^2 θ over [0, 2π] is π*2 = 2π? Wait, no. Wait, integral of cos^2 θ over [0, 2π] is π. Similarly, integral of sin^2 θ is π. Integral of cos^4 θ is (3π)/4, etc. So if k is even and m is even, the integral is non-zero. If either k or m is odd, the integral is zero. Wait, but when k is even and m is even, the integral is non-zero. If either is odd, even if the other is even, the integral is zero? Wait, no. For example, k=1 (odd), m=2 (even): integral of cos θ sin^2 θ over [0, 2π]. Let's compute that. Let u = sin θ, du = cos θ dθ. Then integral becomes integral of u^2 du from u=0 to u=0 (since sin(0) = sin(2π) = 0), which is zero. So yes, even if m is even but k is odd, the integral is zero. Similarly, if k is even and m is odd, integral is zero. If both are odd, integral is zero as well. So in conclusion, the integral over θ of cos^k θ sin^m θ is non-zero only if both k and m are even. Wait, but when both are even, then the integral is non-zero. For example, k=0, m=0: integral is 2π. k=2, m=0: integral is π. k=0, m=2: integral is π. k=2, m=2: integral of cos^2 θ sin^2 θ is π/4. Etc. So the average over θ (i.e., divide by 2π) of cos^k θ sin^m θ is zero unless both k and m are even, in which case it's non-zero.Wait, but let me check another case: k=1, m=1. Then integral of cos θ sin θ over [0, 2π] is 0.5 integral of sin(2θ) dθ over [0, 2π], which is zero. So yes. So only when both k and m are even does the integral not vanish.Therefore, in the expression of the polynomial in polar coordinates, when we average over θ, only the terms where both exponents of cos θ and sin θ are even will contribute to the average. All other terms will integrate to zero. Therefore, the average of the polynomial over the circle of radius r is equal to the sum over all monomials where both exponents are even, multiplied by their respective r^{k+m} and divided by 2π.But the problem states that for a balanced polynomial, this average must be zero for every r. Since r can be any positive real number, the coefficients multiplying each power of r must individually be zero. That is, the average over θ for each monomial term must be zero, except that when decomposed into terms with different powers of r, each coefficient must vanish.Wait, perhaps a better approach is to note that a homogeneous polynomial of degree n can be written in polar coordinates as r^n times a trigonometric polynomial of degree n. Then, the average over the circle of radius r would be r^n times the average of the trigonometric polynomial over θ. For the average to be zero for all r, the average of the trigonometric polynomial must be zero. Therefore, each homogeneous component of the polynomial must have an average of zero over the circle.Therefore, if we decompose the polynomial into homogeneous components (i.e., grouping terms by degree), then each homogeneous component must individually have an average of zero over the circle. Therefore, the space of balanced polynomials is the direct sum of the balanced homogeneous polynomials of each degree from 0 to 2009.Therefore, to find the dimension of V, we can compute the dimension of each homogeneous balanced polynomial of degree d (for d from 0 to 2009) and sum them up.So first, let's recall that the space of homogeneous polynomials of degree d in two variables has dimension d+1. For example, degree 0: constants, dimension 1; degree 1: ax + by, dimension 2; degree 2: ax² + bxy + cy², dimension 3; etc.Now, for each homogeneous polynomial of degree d, the condition of being balanced is that its average over the circle of radius r is zero. As discussed earlier, since the polynomial is homogeneous of degree d, its average over the circle of radius r is r^d times the average over the unit circle. Therefore, the average over any circle is zero if and only if the average over the unit circle is zero. Therefore, the balanced condition for homogeneous polynomials is equivalent to the average over the unit circle being zero.Therefore, for each homogeneous polynomial of degree d, we have a condition: the integral over the unit circle (i.e., θ from 0 to 2π) of the polynomial divided by 2π is zero. Therefore, this is a single linear condition on the coefficients of the homogeneous polynomial. However, we need to check if this condition is non-trivial (i.e., if the integral is not automatically zero for all polynomials of degree d).But wait, earlier analysis suggests that the average over the unit circle of a homogeneous polynomial is zero unless the polynomial contains terms where all variables have even exponents. Wait, but a homogeneous polynomial of degree d must have exponents k and m such that k + m = d. If d is odd, then in each monomial, either k or m must be odd. Then, as per earlier reasoning, the integral over the unit circle would be zero. So if d is odd, then any homogeneous polynomial of degree d automatically has average zero over the unit circle. If d is even, then there can exist monomials where both exponents are even (since d even, k and m can both be even). For example, d=2: x², y², xy. The average of x² over the unit circle is 1/2, average of y² is 1/2, average of xy is zero. Therefore, the average of a homogeneous polynomial of degree 2 is (a + c)/2, where the polynomial is ax² + bxy + cy². So the average is non-zero unless a + c = 0. Therefore, for even degrees, the average can be non-zero, hence the balanced condition imposes a non-trivial linear constraint.Therefore, putting this together:- For each even degree d, the space of homogeneous polynomials of degree d has dimension d + 1. The balanced condition (average zero) imposes 1 linear constraint, so the dimension of balanced homogeneous polynomials of degree d is (d + 1) - 1 = d.- For each odd degree d, all homogeneous polynomials automatically satisfy the balanced condition (average zero), so the dimension is d + 1.Therefore, the total dimension of V is the sum over d from 0 to 2009 of dim V_d, where V_d is the space of balanced homogeneous polynomials of degree d.So we need to compute:Sum_{d=0}^{2009} dim V_d = Sum_{d even} d + Sum_{d odd} (d + 1)Wait, let's check:Wait, for d even:dim V_d = d (since (d + 1) - 1 = d)For d odd:dim V_d = d + 1 (since there are no constraints)But wait, d starts from 0. Let's check for d=0:d=0: homogeneous polynomials of degree 0 are constants. The average over the circle is the constant itself. Therefore, to be balanced, the constant must be zero. Therefore, dim V_0 = 0.But according to the previous reasoning, d=0 is even. So dim V_0 = d = 0. That's correct, because the only homogeneous polynomial of degree 0 is the constant, which must be zero to have average zero.Similarly, for d=1 (odd):dim V_1 = 1 + 1 = 2. Wait, homogeneous polynomials of degree 1 are linear combinations of x and y, which have average zero over the unit circle (since they integrate to zero). Therefore, all degree 1 homogeneous polynomials are balanced. So the dimension is 2, which matches (d + 1) = 2.For d=2 (even):Homogeneous polynomials of degree 2: ax² + bxy + cy². The average is (a + c)/2. So the balanced condition is a + c = 0. Therefore, the space is {ax² + bxy - a y² | a, b real}, which has dimension 2. But according to our formula, dim V_2 = d = 2. Correct.So the formula works for d=0,1,2. Therefore, our general formula is:For each d ≥ 0,- If d is even, dim V_d = d (except for d=0, which is 0)Wait, but d=0: even, dim V_d = 0, which is equal to d=0. So yes, the formula holds.Therefore, the total dimension is:Sum_{d=0}^{2009} dim V_d = Sum_{d=0, d even}^{2009} dim V_d + Sum_{d=1, d odd}^{2009} dim V_d= Sum_{d=0, d even}^{2009} d + Sum_{d=1, d odd}^{2009} (d + 1)But note that for d=0, even, dim V_d = 0, so we can start the even sum from d=0.But let's separate the sums:First, handle even d:Number of even d from 0 to 2009: Since 2009 is odd, the last even d is 2008. So even d: 0, 2, 4, ..., 2008. Number of terms: (2008 - 0)/2 + 1 = 1005 terms.Sum_{d even} d = Sum_{k=0}^{1004} 2k = 2 * Sum_{k=0}^{1004} k = 2 * (1004)(1005)/2 = 1004*1005Similarly, for odd d:Odd d from 1 to 2009: 1, 3, 5, ..., 2009. Number of terms: (2009 - 1)/2 + 1 = 1005 terms.Sum_{d odd} (d + 1) = Sum_{d odd} d + Sum_{d odd} 1 = Sum_{d odd} d + 1005Sum_{d odd} d: Sum_{k=0}^{1004} (2k + 1) = Sum_{k=0}^{1004} 2k + Sum_{k=0}^{1004} 1 = 2*(1004)(1005)/2 + 1005 = 1004*1005 + 1005 = 1005*(1004 + 1) = 1005^2Therefore, Sum_{d odd} (d + 1) = 1005^2 + 1005 = 1005*(1005 + 1) = 1005*1006Therefore, total dimension is:Sum even d + Sum odd (d + 1) = 1004*1005 + 1005*1006Factor out 1005:1005*(1004 + 1006) = 1005*(2010)Therefore, the total dimension is 1005*2010Compute 1005*2010:Note that 1005*2000 = 2,010,0001005*10 = 10,050Therefore, total is 2,010,000 + 10,050 = 2,020,050But let's check with another method:1005*2010 = (1000 + 5)*(2000 + 10) = 1000*2000 + 1000*10 + 5*2000 + 5*10 = 2,000,000 + 10,000 + 10,000 + 50 = 2,020,050Yes, that's correct.However, wait a minute. The total dimension of all polynomials of degree at most 2009 is (2009 + 1)(2009 + 2)/2 = 2010*2011/2 = Let's compute that.2010*2011/2: First compute 2010*2011.2010*2000 = 4,020,0002010*11 = 22,110Total: 4,020,000 + 22,110 = 4,042,110Divide by 2: 2,021,055But according to our previous calculation, the dimension of V is 2,020,050. So the difference is 2,021,055 - 2,020,050 = 1,005.But according to the problem statement, V is the space of balanced polynomials. The difference in dimensions is 1,005, which would be the number of constraints. But given that for each even degree d, we have one constraint, except for d=0. Wait, from d=0 to 2009, the number of even degrees is 1005 (including d=0). For each even degree d ≥ 0, there is one constraint (dim V_d = d instead of d+1). Except for d=0, which goes from 1 to 0. Wait, for d=0, the dimension drops by 1 (from 1 to 0). For each even d ≥ 2, the dimension drops by 1 (from d+1 to d). Therefore, the total number of constraints is 1 (from d=0) plus (number of even d ≥ 2). The even d from 2 to 2008 (since 2009 is odd) is 1004 terms. So total constraints: 1 + 1004 = 1005. Therefore, the dimension of V is total dimension minus 1005, which is 2010*2011/2 - 1005. Let's check:2010*2011/2 = 2,021,0552,021,055 - 1,005 = 2,020,050, which matches our previous result. Therefore, the dimension of V is 2,020,050.But let me verify once more to make sure.Alternative approach: The space of all polynomials of degree at most n is the direct sum of homogeneous polynomials from degree 0 to n. The dimension is Sum_{d=0}^n (d + 1) = (n + 1)(n + 2)/2. The space V is the direct sum of balanced homogeneous polynomials from degree 0 to n. For each d:- If d is even, dim V_d = d (if d ≥ 2) or 0 (if d=0)Wait, but earlier we saw for d even:- For d=0: dim V_d = 0- For d ≥ 2 even: dim V_d = dBut wait, for d=0, which is even, the dimension is 0. For d even ≥ 2, it's d. For d odd, it's d + 1.Therefore, Sum_{d=0}^n dim V_d = Sum_{d odd} (d + 1) + Sum_{d even} dim V_d= Sum_{d odd} (d + 1) + Sum_{d even ≥ 2} d + dim V_0= Sum_{d odd} (d + 1) + Sum_{d even ≥ 2} d + 0But since d even ≥ 2 can be written as 2k for k from 1 to floor(n/2). Similarly, d odd from 1 to n (if n is odd) or n-1 (if n is even). Since n=2009 is odd, d odd goes up to 2009.So Sum_{d odd=1}^{2009} (d + 1) + Sum_{k=1}^{1004} 2k (since d even=2k, k from 1 to 1004, since 2*1004=2008)Wait, but 2008 is the highest even d ≤ 2009.Therefore, Sum_{d even ≥2} d = Sum_{k=1}^{1004} 2k = 2*(1004)(1005)/2 = 1004*1005Sum_{d odd=1}^{2009} (d + 1) = Sum_{d odd=1}^{2009} d + Sum_{d odd=1}^{2009} 1Sum_{d odd=1}^{2009} d: There are 1005 odd numbers from 1 to 2009 (since 2009 = 2*1004 +1). The sum is (number of terms)*(first term + last term)/2 = 1005*(1 + 2009)/2 = 1005*2010/2 = 1005*1005Sum_{d odd=1}^{2009} 1 = 1005Therefore, Sum_{d odd=1}^{2009} (d +1) = 1005*1005 + 1005 = 1005*(1005 +1) = 1005*1006Therefore, total dimension:Sum_{d even ≥2} d + Sum_{d odd} (d +1) = 1004*1005 + 1005*1006Factor out 1005:1005*(1004 + 1006) = 1005*2010 = 2,020,050Same result as before. Therefore, the dimension of V is 2,020,050.Alternatively, note that 1005*2010 = (2010/2)*2010 = 1005*2010, but that's not particularly helpful.Alternatively, confirm 1005*2010:1005 multiplied by 2000 is 2,010,0001005 multiplied by 10 is 10,050Adding them gives 2,010,000 + 10,050 = 2,020,050. Yep.Therefore, the answer is 2,020,050.But the problem asks for the dimension of V, the balanced polynomials of degree at most 2009. So the answer is boxed{2020050}Final Answerboxed{2020050}</think>To determine the dimension of the vector space ( V ) of balanced polynomials of degree at most 2009, we need to consider the properties of such polynomials. A balanced polynomial has an average value of zero on every circle centered at the origin. 1. Homogeneous Polynomials Decomposition: - A polynomial can be decomposed into homogeneous polynomials of degree ( d ) for ( 0 leq d leq 2009 ). - For each homogeneous polynomial of degree ( d ), the average over the unit circle must be zero.2. Dimension Analysis: - For even ( d ): - The average of a homogeneous polynomial of even degree ( d ) over the unit circle is non-zero unless specific coefficients cancel out. This imposes one linear constraint, reducing the dimension from ( d + 1 ) to ( d ). - Exception: For ( d = 0 ) (constant term), the dimension is 0 because the only constant polynomial with average zero is the zero polynomial. - For odd ( d ): - All homogeneous polynomials of odd degree automatically have an average of zero over the unit circle, so the dimension remains ( d + 1 ).3. Summing Dimensions: - Even degrees (0, 2, 4, ..., 2008): - Sum of dimensions: ( sum_{k=0}^{1004} 2k = 1004 times 1005 ) - Odd degrees (1, 3, 5, ..., 2009): - Sum of dimensions: ( sum_{k=0}^{1004} (2k + 1 + 1) = 1005 times 1006 )4. Total Dimension Calculation: - Sum the contributions from even and odd degrees: [ 1004 times 1005 + 1005 times 1006 = 1005 times (1004 + 1006) = 1005 times 2010 = 2020050 ]Thus, the dimension of the vector space ( V ) of balanced polynomials of degree at most 2009 is (boxed{2020050}).]

🔑:1. Understanding the Problem: We need to find the dimension of the vector space ( V ) of balanced polynomials of degree at most 2009. A polynomial is balanced if its average value on each circle centered at the origin is 0.2. Homogeneous Polynomials: Split the polynomial into its homogeneous parts. For example, ( x^2 + xy + 3x - y + 2 ) becomes ( (x^2 + xy) + (3x - y) + 2 ). The average value of a degree-( n ) homogeneous polynomial on a circle of radius ( r ) is a constant multiple of ( r^n ).3. Average Value Calculation: The average value of the whole polynomial is a polynomial of degree at most 2009. This average value is a linear map from ( V ) to the space ( P_{2009}[r] ) of polynomials of degree at most 2009 in ( r ).4. Image of the Map: We seek the nullity of this map, which is the dimension of ( V ) minus the dimension of its image. The circle of radius ( -r ) is also the circle of radius ( r ), so everything in the image is even. Since ( (x^2 + y^2)^n ) has average value ( r^{2n} ), the image contains every even power of ( r ) up to ( r^{2008} ). Including ( r^0 ), there are 1005 of these.5. Dimension Calculation: The nullity is ( text{dim } V - 1005 ). There are ( d+1 ) monomials ( x^d, x^{d-1}y, ldots, y^d ) of each degree ( d ). Therefore, [ text{dim } V = 1 + 2 + 3 + cdots + 2010 = frac{2010 cdot 2011}{2} = 2021055 ] Subtract 1005 to get the final answer: [ text{dim } V - 1005 = 2021055 - 1005 = 2020050 ]The final answer is (boxed{2020050}).

🧠:[<think>Okay, so I need to find the length of the side of an equilateral triangle when a point inside it is 5, 7, and 8 units away from the three sides. Hmm, let me think. I remember something about the distances from a point inside an equilateral triangle to its sides adding up to the height of the triangle. Is that right?Let me recall. For an equilateral triangle, the height (h) can be calculated using the formula h = (√3/2) * side length (a). So h = (√3/2) * a. And if a point is inside the triangle, the sum of its distances from the three sides should equal the height. So if the distances are 5, 7, and 8, their sum should be 5 + 7 + 8 = 20. Therefore, h = 20. Then the side length would be (20 * 2)/√3 = 40/√3. Rationalizing the denominator, that's (40√3)/3, which can be written as (40/3)√3. Let me check the answer choices. Option E is (40/3)√3, so that seems to be the answer. But wait, let me verify this again because sometimes these problems can have tricks.Is there a theorem that states that the sum of the distances from any interior point to the sides of an equilateral triangle equals the height? I think that's Viviani's theorem. Wait, Viviani's theorem says that in an equilateral triangle, the sum of the perpendicular distances from any interior point to the sides is equal to the height of the triangle. Yes, that's correct. So if that's the case, then indeed, 5 + 7 + 8 = 20 should be equal to the height. Then the side length is (20 * 2)/√3 = 40/√3, which is equivalent to (40√3)/3. So that's 40/3 multiplied by √3, which matches option E. Therefore, the answer should be E.But wait, let me check if there's any possibility that the configuration doesn't exist. For example, could the distances given not satisfy some triangle inequality or something? But since the point is inside the triangle, the distances just need to be positive and their sum equal to the height. Since 5, 7, and 8 are all positive and sum to 20, which would be the height, then as long as each distance is less than the height, which they are (since 5, 7, 8 are all less than 20), then such a point should exist. Therefore, the configuration is possible. So the data is sufficient, and the answer is E. So options C, D, E are possible. Let me check the answer choices again.Option C says the side length is 20, but if the height is 20, then the side length is (20 * 2)/√3 ≈ 40 / 1.732 ≈ 23.09, which is not 20. So C is incorrect. Option D is 14√3 ≈ 24.25, which is also different. Option E is (40/3)√3 ≈ 23.09, which matches the calculation. So E is correct. Therefore, the answer is E.Final Answerboxed{E}</think>To find the length of the side of an equilateral triangle when a point inside it is 5, 7, and 8 units away from the three sides, we use Viviani's theorem. This theorem states that the sum of the perpendicular distances from any interior point to the sides of an equilateral triangle is equal to the height of the triangle.Given distances: 5, 7, and 8 units. Their sum is:[5 + 7 + 8 = 20]This sum equals the height ( h ) of the equilateral triangle. The height ( h ) of an equilateral triangle with side length ( a ) is given by:[h = frac{sqrt{3}}{2} cdot a]Setting ( h = 20 ), we solve for ( a ):[20 = frac{sqrt{3}}{2} cdot a implies a = frac{20 times 2}{sqrt{3}} = frac{40}{sqrt{3}} = frac{40 sqrt{3}}{3}]Thus, the side length of the triangle is:[frac{40}{3} sqrt{3}]The final answer is (boxed{E}).,<think>Okay, let's see. The problem says there's a point inside an equilateral triangle, and the distances from this point to the three sides are 5, 7, and 8 units. We need to find the length of the side of the triangle. The options are given, so maybe one of them will fit. Hmm, let me think.First, I remember that in an equilateral triangle, the distance from any interior point to the three sides adds up to the height of the triangle. Wait, is that right? Let me recall. If you have a point inside a triangle, the sum of the distances from that point to each of the sides is equal to the triangle's height. But is this only true for equilateral triangles, or any triangle? I think it's for any triangle. The formula is like the sum of the distances multiplied by the respective base lengths over 2 equals the area, but in an equilateral triangle, all sides are the same. So maybe in this case, since all sides are equal, the sum of the distances should equal the height. Let me check that.For an equilateral triangle with side length 'a', the height h is (√3/2)a. If a point inside the triangle has distances d1, d2, d3 from the three sides, then maybe d1 + d2 + d3 = h. Is that the case?Wait, actually, I think that's true. Because in any triangle, the sum of the distances from an interior point to the three sides is equal to the triangle's height. But let me confirm that.Suppose we have a triangle with area A, and base length b. The height h is 2A/b. If a point is inside the triangle, the distances to the three sides, say d1, d2, d3, then the areas formed by connecting the point to the vertices would sum up to A. Each of these smaller triangles would have areas (d1 * a)/2, (d2 * a)/2, (d3 * a)/2, where a is the side length. But wait, in a general triangle, the sides can be different. Wait, maybe for an equilateral triangle, since all sides are equal, the sum of the distances would be equal to the height.Yes, that makes sense. Let me try to write it out.If the triangle is equilateral with side length 'a', then the area is (√3/4)a². The height h is (√3/2)a. If a point is inside, then the sum of the distances to the three sides should be equal to the height. Because each distance times the corresponding side length divided by 2 gives the area of the small triangle, and adding those up gives the total area. Since all sides are the same length, the sum of the distances times (a/2) equals the area. So:(d1 + d2 + d3) * (a/2) = (√3/4)a²Divide both sides by a/2:d1 + d2 + d3 = (√3/4)a² / (a/2) = (√3/4)a² * (2/a) = (√3/2)aWhich is exactly the height. So yes, the sum of the distances from any interior point to the three sides of an equilateral triangle is equal to the height. Therefore, d1 + d2 + d3 = h = (√3/2)a.So in this problem, the distances given are 5, 7, and 8. Therefore, 5 + 7 + 8 = 20 = (√3/2)a. Then, solving for 'a' gives a = (20 * 2)/√3 = 40/√3. Rationalizing the denominator, that's (40√3)/3, which is the same as (40/3)√3. Let me check the answer options.Option E is "The side length is (40/3)√3", which matches. So that should be the answer. Let me double-check if that makes sense.Wait, but hold on. Is there a possibility that the given configuration cannot exist? The problem says a point inside the triangle. So, if the sum of the distances is 20, and the side length is (40/3)√3, which is approximately 23.09 units. But the distances given are 5,7,8. Each of these distances has to be less than the height. The height h = (√3/2)a = (√3/2)*(40/3)√3 = (√3/2)*(40√3/3) = (40*3)/(2*3) )= 40/2 = 20. Wait, the height is 20? But the distances are 5,7,8. Each distance is less than the height, which is 20. So that's fine. So all distances are positive and less than the height, so such a point can exist. Therefore, the answer should be E.Wait, but let me check the options again. Option C is 20. If I miscalculated somewhere. Let me recheck.Sum of distances: 5 + 7 + 8 = 20. Then, according to the formula, this sum equals the height. So the height h = 20. Then, since h = (√3/2)a, then a = (2h)/√3 = (40)/√3 = (40√3)/3. Which is answer E. So that's correct. So answer E is the correct one.But let me see why option C is 20. Maybe if someone thought that the sum of the distances is equal to the side length? But that's not correct. The sum is equal to the height. So unless they confused the height with the side length, but in that case, for an equilateral triangle, the height is (√3/2)a, so if h = 20, then a = (2*20)/√3 = 40/√3, which is not 20. Therefore, answer C is incorrect.Therefore, the answer is E.Final Answerboxed{E}</think>Given a point inside an equilateral triangle with distances 5, 7, and 8 units from the three sides, we need to find the side length of the triangle.1. Key Insight: For any point inside an equilateral triangle, the sum of the perpendicular distances to the three sides is equal to the height of the triangle.2. Sum of Distances: The given distances are 5, 7, and 8. Their sum is (5 + 7 + 8 = 20).3. Height of the Triangle: The height (h) of an equilateral triangle with side length (a) is given by (frac{sqrt{3}}{2}a). According to the key insight, this height is equal to the sum of the distances: [ frac{sqrt{3}}{2}a = 20 ]4. Solving for Side Length (a): [ a = frac{20 times 2}{sqrt{3}} = frac{40}{sqrt{3}} = frac{40sqrt{3}}{3} ] This simplifies to (frac{40}{3}sqrt{3}).Thus, the length of the side of the triangle is (boxed{E}).]

🔑:5. The answer is (E). Referring to the adjacent figure, the area of the equilateral triangle with unknown side length l can be determined in two ways:1) as the sum of the areas of three triangles with base l and heights 5, 7, 82) using the usual area formula, keeping in mind that the height of an equilateral triangle is l frac{sqrt{3}}{2}.Thus, frac{l cdot(5+7+8)}{2}=frac{1}{2} l^{2} frac{sqrt{3}}{2}, from which l=frac{40}{3} sqrt{3}.

🧠:[<think>Okay, let's see. I need to solve the equation 8^(2/x) - 2^((3x + 3)/x) + 12 = 0. Hmm, exponents can be tricky, but maybe I can rewrite the terms with the same base. I know that 8 is 2 cubed, so maybe that's a start. Let me try that.First, 8^(2/x) is the same as (2^3)^(2/x). When you raise a power to another power, you multiply the exponents. So that would be 2^(3*(2/x)) = 2^(6/x). Okay, so the first term becomes 2^(6/x).Now the second term is 2^((3x + 3)/x). Let me simplify the exponent here. The exponent is (3x + 3)/x. If I split that fraction, it's 3x/x + 3/x, which simplifies to 3 + 3/x. So the second term is 2^(3 + 3/x). That can also be written as 2^3 * 2^(3/x) because when you add exponents, you multiply the bases. So 2^(3 + 3/x) = 8 * 2^(3/x). Hmm, that seems useful.So substituting these back into the equation, we have:2^(6/x) - 8 * 2^(3/x) + 12 = 0Hmm, this looks like a quadratic equation in terms of 2^(3/x). Let me let y = 2^(3/x). Then 2^(6/x) is (2^(3/x))^2 = y^2. So substituting, the equation becomes:y^2 - 8y + 12 = 0Oh, that's a quadratic equation! Let's solve for y. The quadratic equation is y^2 - 8y + 12 = 0. Let's factor this. Looking for two numbers that multiply to 12 and add up to -8. Those numbers are -6 and -2. So factoring gives:(y - 6)(y - 2) = 0So the solutions are y = 6 or y = 2. But remember that y was defined as 2^(3/x). So:Case 1: 2^(3/x) = 6Case 2: 2^(3/x) = 2Let's solve each case separately.Case 1: 2^(3/x) = 6To solve for x here, we can take the logarithm of both sides. Let's take natural logarithm or log base 2. Maybe log base 2 is easier here.Taking log base 2 of both sides:log2(2^(3/x)) = log2(6)Simplify the left side:(3/x) * log2(2) = log2(6)Since log2(2) is 1, this simplifies to:3/x = log2(6)Therefore, x = 3 / log2(6)Hmm, log2(6) can be written as log2(2*3) = log2(2) + log2(3) = 1 + log2(3). So x = 3 / (1 + log2(3)). Alternatively, we can express this in terms of natural logs if needed, but maybe leave it as is for now.Case 2: 2^(3/x) = 2Again, take log base 2 of both sides:log2(2^(3/x)) = log2(2)Simplify left side:(3/x) * 1 = 1So 3/x = 1 => x = 3Alright, so potential solutions are x = 3 and x = 3 / (1 + log2(3)). Let me check if these solutions are valid by plugging them back into the original equation.First, check x = 3.Original equation: 8^(2/3) - 2^((3*3 + 3)/3) + 12 = 0Calculate each term:8^(2/3) = (8^(1/3))^2 = 2^2 = 4Second term: exponent is (9 + 3)/3 = 12/3 = 4, so 2^4 = 16So 4 - 16 + 12 = 0. 4 -16 is -12, -12 +12 = 0. So that works. x=3 is valid.Now check x = 3 / (1 + log2(3)). Let's denote log2(3) as ln(3)/ln(2) if needed, but perhaps first compute 3/x:3/x = 1 + log2(3)So 3/x = log2(2) + log2(3) = log2(2*3) = log2(6). Therefore, 2^(3/x) = 6, which was our case 1. Let's verify if 8^(2/x) - 2^((3x + 3)/x) +12 = 0.Compute each term:First term: 8^(2/x) = 2^(6/x) as before. Since 3/x = log2(6), then 6/x = 2*(3/x) = 2*log2(6) = log2(6^2) = log2(36). Therefore, 2^(6/x) = 2^(log2(36)) = 36.Second term: 2^((3x +3)/x) = 2^(3 + 3/x). We know 3/x = log2(6), so 3 + 3/x = 3 + log2(6). Therefore, 2^(3 + log2(6)) = 2^3 * 2^(log2(6)) = 8 * 6 = 48.So plugging back into the equation: 36 - 48 +12 = 0. 36 -48 = -12, -12 +12 =0. So that works too.Therefore, both solutions are valid.But wait, x is in the denominator in the original equation. So x cannot be zero. Also, since we have exponents, x must be a real number except zero. But since both solutions are x=3 and x=3/(1 + log2(3)), which are real numbers not zero, they are valid.Hence, both solutions are acceptable.But perhaps the problem expects the solutions in some specific form? Let's see. The answer x=3 is straightforward, and the other solution is x=3/(1 + log2(3)). Maybe they want it expressed using logarithms with base 2 or natural logarithms. Alternatively, maybe to write it in terms of ln.Alternatively, perhaps rationalizing or another form. Let me see. Alternatively, using the change of base formula: log2(6) = ln6/ln2, so x=3/(ln6/ln2) = 3*ln2 / ln6. But perhaps they want to leave it as is. Or maybe they want to write log2(6) as 1 + log2(3), so x=3/(1 + log2(3)). Either way is acceptable. The problem might want both solutions boxed.Alternatively, maybe to check if 3/(1 + log2(3)) can be simplified further. Let's see. Let me compute log2(3) ≈1.58496, so 1 + log2(3)≈2.58496, so 3 divided by that is approximately 1.158. But I don't think it simplifies to a nicer form. So likely, the solutions are x=3 and x=3/(1 + log2(3)).But let me confirm once again that both solutions are valid. Since we performed exponentiation and logarithmic steps, which are sometimes irreversible, but in this case, since we substituted variables and solved quadratics, but since both solutions lead to positive results in the exponents (since 2^(anything) is positive), and x is not zero, so both solutions are acceptable.Therefore, the solutions are x=3 and x=3/(1 + log2(3)). Alternatively, if written in natural logs, x=3/(ln6/ln2) = 3*ln2 / ln6. But perhaps in the problem's answer, they want it in base 2.Alternatively, to write log base 6 of 2 or something, but I think 3/(1 + log2(3)) is acceptable.Alternatively, the problem may prefer to write both solutions in terms of logarithms. But since one solution is x=3, which is a nice integer, and the other is in log form. So final answer would be x=3 and x=3/(1 + log2(3)), or maybe written as x=3 and x=3/(log2(6)), since log2(6) is 1 + log2(3). Either way is acceptable. Maybe the problem expects both answers in boxed form.Wait, the original problem says "put your final answer within boxed{}". Since there are two solutions, should I present both in separate boxes? Like boxed{3} and boxed{dfrac{3}{1 + log_2 3}} or something. Let me check the example. The example number 12, but in the problem statement, the example is given as Example 12. So maybe the user wants the answer as the example's solution. But in the original problem, the user just says "Solve the equation" and provides the equation. So standard practice would be to present all solutions. So since there are two solutions, we need to present both.Alternatively, maybe check if 3/(1 + log2(3)) can be expressed as log2(6)/2 or something, but I don't think so. Let me compute 3/(1 + log2(3)):Let me compute log2(3) ≈1.58496, so denominator ≈2.58496, and 3 divided by that ≈1.158. Alternatively, perhaps there's an exact expression. But I don't think so.Alternatively, let me see. Wait, 1 + log2(3) = log2(2) + log2(3) = log2(6). So 3/(log2(6)) is the same as 3 divided by log2(6). So, x=3/log2(6). Alternatively, using change of base formula, log2(6) = ln6/ln2, so x=3*ln2/ln6. But unless the problem specifies the form, either is acceptable.So depending on how the problem wants the answer. Since the original equation uses base 2 and 8, which is 2^3, probably expressing in terms of log base 2 is preferable.So the two solutions are x=3 and x=3/log2(6). But log2(6) is 1 + log2(3), so both forms are possible. Hmm. Wait, 3/(1 + log2(3)) vs 3/log2(6). They are the same. Since log2(6) =1 + log2(3). So either form is acceptable. But perhaps the problem prefers one over the other. Let me check which is simpler.If I write log2(6) instead of 1 + log2(3), it's shorter, but if they want it in terms of log2(3), then 3/(1 + log2(3)) is better. Hmm.Alternatively, maybe the problem expects both answers in boxed form. Let me check similar problems. Generally, when solving exponential equations, unless a substitution is obvious, but in this problem, the substitution led to two solutions. So, the two solutions are x=3 and x=3/(1 + log2(3)). Alternatively, since log2(6) =1 + log2(3), so maybe x=3/log2(6). Either way, both are correct.Since the problem didn't specify the form, I can present both solutions as x=3 and x=3 divided by (1 plus log base 2 of 3). So boxed, probably:The solutions are boxed{3} and boxed{dfrac{3}{1 + log_2 3}}.Alternatively, if combined into one box, but generally, separate boxes for separate solutions. So yes, two solutions.Final AnswerThe solutions are boxed{3} and boxed{dfrac{3}{1 + log_2 3}}.</think>To solve the equation (8^{2/x} - 2^{(3x+3)/x} + 12 = 0), we start by rewriting the terms with the same base. First, express (8) as (2^3):[8^{2/x} = (2^3)^{2/x} = 2^{6/x}]Next, simplify the exponent in the second term:[2^{(3x + 3)/x} = 2^{3 + 3/x} = 2^3 cdot 2^{3/x} = 8 cdot 2^{3/x}]Substituting these back into the equation, we get:[2^{6/x} - 8 cdot 2^{3/x} + 12 = 0]Let (y = 2^{3/x}). Then (2^{6/x} = y^2), transforming the equation into a quadratic:[y^2 - 8y + 12 = 0]Solving the quadratic equation, we factor it as:[(y - 6)(y - 2) = 0]This gives the solutions (y = 6) and (y = 2). Substituting back (y = 2^{3/x}):1. For (y = 6): [ 2^{3/x} = 6 implies frac{3}{x} = log_2 6 implies x = frac{3}{log_2 6} ] Since (log_2 6 = 1 + log_2 3), this can be written as: [ x = frac{3}{1 + log_2 3} ]2. For (y = 2): [ 2^{3/x} = 2 implies frac{3}{x} = 1 implies x = 3 ]Both solutions are verified to satisfy the original equation. Therefore, the solutions are:[boxed{3} quad text{and} quad boxed{dfrac{3}{1 + log_2 3}}],<think>Okay, so I need to solve this equation: 8^(2/x) minus 2^[(3x+3)/x] plus 12 equals 0. Hmm, let me write that down again to make sure I have it right. The equation is 8^{2/x} - 2^{(3x + 3)/x} + 12 = 0. Alright, first step, maybe simplify the exponents or express everything in terms of the same base. I notice that 8 is a power of 2, since 8 is 2^3. That could be helpful. Let me rewrite 8^{2/x} as (2^3)^{2/x} which simplifies to 2^{6/x}. That seems good. So substituting back into the equation, we get 2^{6/x} - 2^{(3x + 3)/x} + 12 = 0. Let me check the exponent on the second term. The exponent is (3x + 3)/x. Let's simplify that. Dividing numerator and denominator by x, that's 3 + 3/x. So the second term is 2^{3 + 3/x}. So now the equation is 2^{6/x} - 2^{3 + 3/x} + 12 = 0.Hmm, maybe let's set a substitution to make this equation easier. Let me let y = 2^{3/x}. Then, 2^{6/x} would be (2^{3/x})^2 = y^2. Similarly, 2^{3 + 3/x} is equal to 2^3 * 2^{3/x} which is 8 * y. So substituting these into the equation, we get y² - 8y + 12 = 0. Oh, that's a quadratic equation! Nice. So solving y² - 8y + 12 = 0. Let's factor this quadratic. Looking for two numbers that multiply to 12 and add up to -8. Hmm, factors of 12 are 1 & 12, 2 & 6, 3 & 4. 2 and 6 add up to 8, so if both are negative, they'd add up to -8. So the factors would be (y - 6)(y - 2) = 0. Therefore, y = 6 or y = 2. But remember, y was defined as 2^{3/x}. So, now we have two equations:1. 2^{3/x} = 62. 2^{3/x} = 2Let me solve each of these for x.Starting with the second equation: 2^{3/x} = 2. Since 2^1 = 2, this implies that 3/x = 1, so x = 3. That's straightforward.Now, the first equation: 2^{3/x} = 6. To solve for x here, I need to use logarithms. Let's take the natural logarithm of both sides. ln(2^{3/x}) = ln(6). Using the logarithm power rule, that's (3/x) * ln(2) = ln(6). Then, solving for x: 3/x = ln(6)/ln(2), so x = 3 * ln(2)/ln(6). Alternatively, using logarithm base change formula, that's x = 3 / log₂(6). Let me check if that's correct. Yes, because log₂(6) is ln(6)/ln(2), so 3 divided by that is 3 * ln(2)/ln(6). Alternatively, maybe we can express this as x = 3 / (log₂(2*3)) = 3 / (1 + log₂(3)). Not sure if that's necessary, but perhaps it's useful. However, for the answer, either form is acceptable unless specified otherwise. So now we have two solutions: x = 3 and x = 3 / log₂(6). Let me verify these solutions in the original equation to make sure they work. First, testing x = 3. Plugging into the original equation:8^{2/3} - 2^{(3*3 + 3)/3} + 12. Let's compute each term.8^{2/3} is the cube root of 8 squared. The cube root of 8 is 2, so squared is 4. Next, 2^{(9 + 3)/3} = 2^{12/3} = 2^4 = 16. So the equation becomes 4 - 16 + 12 = 0. 4 - 16 is -12, plus 12 is 0. Correct. So x = 3 is a valid solution.Now, testing x = 3 / log₂(6). Let me compute each term. First, 2/x would be 2 divided by (3 / log₂(6)) which is (2 * log₂(6))/3. So 8^{2/x} = 8^{(2 log₂(6))/3}. Let's simplify 8 as 2^3, so (2^3)^{(2 log₂(6))/3} = 2^{3 * (2 log₂(6))/3} = 2^{2 log₂(6)} = (2^{log₂(6)})^2 = 6^2 = 36.Next term: 2^{(3x + 3)/x}. Let's compute (3x + 3)/x when x = 3 / log₂(6). That's [3*(3 / log₂(6)) + 3] / (3 / log₂(6)).First, compute numerator: 9 / log₂(6) + 3 = (9 + 3 log₂(6)) / log₂(6). Then, divide by 3 / log₂(6): [(9 + 3 log₂(6))/log₂(6)] * [log₂(6)/3] = (9 + 3 log₂(6)) / 3 = 3 + log₂(6). Therefore, the second term is 2^{3 + log₂(6)} = 2^3 * 2^{log₂(6)} = 8 * 6 = 48.So substituting back into the equation: 36 - 48 + 12 = 0. 36 - 48 is -12, plus 12 is 0. Correct. So this solution works too.Therefore, both solutions are valid. Thus, the solutions are x = 3 and x = 3 / log₂(6). But maybe we can write the second solution in a different form. Since log₂(6) is log₂(2*3) = 1 + log₂(3), so x = 3 / (1 + log₂(3)). Alternatively, using natural logs, x = 3 ln 2 / (ln 6). Either form is acceptable. Depending on what the problem expects, but since the problem is presented with logs maybe in base 2, but it's not specified. Alternatively, maybe we can rationalize x = 3 / log₂(6). Let me compute log₂(6) ≈ log₂(6) ≈ 2.58496. So 3 divided by that is approximately 1.158. But exact form is better.Alternatively, if they want it expressed as log base 6 of 2, but no. Wait, x = 3 / log₂(6) is the same as 3 * log_6(2) because log_b(a) = 1 / log_a(b). So x = 3 * log_6(2). Hmm, that might be another way to write it, but not sure if it's simpler. Alternatively, writing both solutions in boxed form. The problem didn't specify to approximate, so we should present the exact answers. So x = 3 and x = 3 / log₂(6). Alternatively, if we want to write log base 2, we can use the notation, but maybe it's better to write in terms of ln. But unless specified, log is typically base 10, but in the context of exponents with base 2, maybe log₂ is acceptable.Alternatively, if we want to write x = 3 / (1 + log₂3), since log₂6 = log₂(2*3) = 1 + log₂3. So maybe x = 3 / (1 + log₂3). Which is also acceptable.I think the answer is two solutions: x = 3 and x = 3 / log₂6. So box both of them.Final AnswerThe solutions are boxed{3} and boxed{dfrac{3}{log_2 6}}.</think>To solve the equation ( 8^{2/x} - 2^{(3x+3)/x} + 12 = 0 ):1. Rewrite ( 8 ) as ( 2^3 ): [ 8^{2/x} = (2^3)^{2/x} = 2^{6/x} ]2. Simplify the exponent in the second term: [ 2^{(3x + 3)/x} = 2^{3 + 3/x} ]3. Substitute these into the original equation: [ 2^{6/x} - 2^{3 + 3/x} + 12 = 0 ]4. Let ( y = 2^{3/x} ). Then, ( 2^{6/x} = y^2 ) and ( 2^{3 + 3/x} = 8y ). The equation becomes: [ y^2 - 8y + 12 = 0 ]5. Factor the quadratic equation: [ (y - 6)(y - 2) = 0 ] Thus, ( y = 6 ) or ( y = 2 ).6. Substitute back ( y = 2^{3/x} ): - For ( y = 2 ): [ 2^{3/x} = 2 implies frac{3}{x} = 1 implies x = 3 ] - For ( y = 6 ): [ 2^{3/x} = 6 implies frac{3}{x} = log_2 6 implies x = frac{3}{log_2 6} ]7. Verify both solutions in the original equation and confirm they work.The solutions are (boxed{3}) and (boxed{dfrac{3}{log_2 6}}).]

🔑:The solution is as follows. Since2^{(3 x+3) / x}=2^{3+3 / x}=8 cdot 2^{3 / x}=8 cdotleft(2^{3}right)^{1 / x}=8 cdot 8^{1 / x},the equation (5) is equivalent to the equation8^{2 / x}-8 cdot 8^{1 / x}+12=0Let 8^{1 / x}=t; then the equation (5) is equivalent to the system of equations8^{1 / x}=6, quad 8^{1 / x}=2,where the numbers 6 and 2 are the roots of the quadratic equationt^{2}-8 t+12=0From this system, we find two roots of the equation (5): x_{1}=3 log _{6} 2, x_{2}=3.Solving exponential equations in which there are three powers with different bases that are consecutive terms of a geometric progression, and these bases are raised to the same power depending on x, reduces to solving quadratic equations. Such equations have the formalpha a^{f(x)}+beta b^{f(x)}+gamma c^{f(x)}=0where alpha neq 0, beta, gamma are real numbers, f(x) is some function, and the bases a, b and c satisfy the condition b^{2}=a c.Equations of this type are reduced to solving the system of exponential equations(a / b)^{f(x)}=t_{1}, quad(a / b)^{f(x)}=t_{2}where t_{1}, t_{2} are the roots of the quadratic equation alpha t^{2}+beta t+gamma=0.

🧠:[<think>Okay, let's see. The problem is about probability with three coins. Hmm. So, when three uniformly dense coins are thrown at once, the probability that the other two coins are also facing up when at least one coin is facing up. The options are A: 1/8, B:7/8, C:1/7, D:6/7.Wait, let me parse the question again. It says: "the probability that the other two coins are also facing up when at least one coin is facing up." So, rephrasing, given that at least one coin is facing up, what's the probability that the other two are also facing up? That sounds like a conditional probability problem.So, conditional probability formula: P(A|B) = P(A ∩ B) / P(B). Here, event A is "all three coins are facing up," and event B is "at least one coin is facing up." Therefore, we need to compute P(A|B) = P(all three up) / P(at least one up).First, let's compute P(A), which is the probability that all three coins are facing up. Since each coin has a 1/2 chance of landing heads or tails (assuming "facing up" means heads, which I think it does), the probability for all three is (1/2)^3 = 1/8.Then, P(B), the probability that at least one coin is facing up. The complement of "at least one up" is "all three are down." So, the probability of all three down is (1/2)^3 = 1/8. Therefore, P(B) = 1 - 1/8 = 7/8.Therefore, P(A|B) = (1/8) / (7/8) = 1/7. Hmm, so the answer is C: 1/7. But let me verify that.Wait, but maybe I misinterpreted the problem. The problem says "the other two coins are also facing up when at least one coin is facing up." Hmm. So maybe another way to think is, given that at least one is up, what's the probability that both of the other two are also up. Wait, but if there are three coins, and at least one is up, then "the other two" would refer to the remaining two coins, assuming that one is already up. But in the problem statement, it's not specifying which coin is up. So perhaps, the event is that if at least one is up, then the other two (i.e., the remaining two) are up. But in probability terms, since we don't know which one is up, we have to consider all possibilities.Alternatively, maybe the problem is phrased a bit awkwardly, but in probability terms, it's equivalent to: given that at least one coin is up, what's the probability that all three are up. Because if all three are up, then certainly the other two are up as well. So, in that case, the answer would be 1/7, as calculated above.But let me check again. Let's list all possible outcomes when tossing three coins. Each coin can be Heads (H) or Tails (T). So there are 2^3 = 8 possible outcomes:1. HHH2. HHT3. HTH4. HTT5. THH6. THT7. TTH8. TTTOut of these, the event "at least one H" excludes only the last outcome (TTT). So there are 7 outcomes in total for event B.The event A (all three up) is only the first outcome, HHH. So the probability is 1 out of 7. Therefore, the answer is C: 1/7. That seems correct.But let me check the other options. Option D is 6/7, which would be the probability of not all three being heads given at least one head. So 1 - 1/7 = 6/7. So that's for a different question. The options here are A:1/8, which is the probability of all three heads without any condition. B:7/8 is the probability of at least one head. C:1/7, which is our answer, and D:6/7. So yeah, answer is C.Wait, but hold on. Let me re-express the problem again. The problem says: "the probability that the other two coins are also facing up when at least one coin is facing up." So maybe the wording is confusing. Does it mean that given at least one is up, the probability that the other two are up? But "the other two" implies that one is already up. So if there's at least one up, then the remaining two must be up. But in reality, the "at least one" could be one, two, or three. So the problem is perhaps phrased as: given that at least one coin is up, what is the probability that the other two are also up. But "the other two" is a bit ambiguous here. If exactly one is up, then "the other two" would be the ones that are down. If two are up, then "the other two" would include the one that's down, but which two?Wait, maybe the problem is actually asking for the probability that all three are up given that at least one is up. Because if all three are up, then the other two (relative to any one of them) are up. So perhaps that's what the problem is getting at. But in that case, it's the same as we calculated. Because if all three are up, then of course the other two are up. If only one is up, then the other two are not. If two are up, then one of the other two is up and the other is down. But the problem says "the other two coins are also facing up". So, if exactly two coins are up, then when considering the third one, the "other two" would be the two that are up. Wait, this is getting confusing.Alternatively, perhaps the problem is phrased as: when you have three coins, and you know that at least one is facing up, what is the probability that both of the other two are also facing up. So maybe it's equivalent to: given that at least one is up, what's the probability that all three are up. Which would be 1/7. But another way, maybe it's: given that a specific coin is up, what is the probability that the other two are up. But the problem doesn't specify a particular coin. So perhaps that's the same as all three being up. Hmm.Alternatively, maybe the problem is considering that if you have three coins, and you observe that one of them is up, then what is the probability that the other two are up. But in probability terms, that's different because you have information that a specific coin is up. However, the problem says "when at least one coin is facing up". So in the general case, where we know at least one is up, but not which one. So in that case, the probability that all three are up is 1/7. So answer C.Alternatively, maybe someone could interpret the problem differently. Let's see: "the probability that the other two coins are also facing up when at least one coin is facing up". So when you have at least one up, the probability that the other two are up. But depending on how you interpret "the other two", if you fix one that's up, but since which one is up isn't specified, it's still equivalent to the probability that all three are up given at least one is up.Alternatively, if you have at least one head, the probability that the remaining two coins (other than the one that's known to be up) are also up. But since we don't know which one is up, we have to consider all possibilities. For example, in the case where exactly one coin is up, then the "other two" are both down. If two coins are up, then "the other two" (relative to either of the two) would include one up and one down. If all three are up, then the other two are up. So the only case where the "other two" are both up is when all three are up. Therefore, the problem reduces to the probability that all three are up given at least one is up, which is 1/7. So answer C.Therefore, I think the answer is C: 1/7. So the options given are A:1/8, B:7/8, C:1/7, D:6/7. Therefore, the correct answer should be C.But let me verify once again. The total number of possible outcomes where at least one coin is up is 7. The favorable outcome is only 1 (all three up). So 1/7. Yep, that's correct.Alternatively, someone might think that since one is up, the probability that the other two are up is 1/2 each, so 1/4. But 1/4 isn't among the options, so that would be wrong. But that line of thinking is incorrect because the problem is a conditional probability where we already know that at least one is up, so we have to restrict the sample space to those 7 outcomes, and only 1 is favorable. So the answer is indeed 1/7. Therefore, C is correct.Final Answerboxed{C}</think>Given three uniformly dense coins thrown at once, we need to find the probability that the other two coins are also facing up when at least one coin is facing up. This is a conditional probability problem.1. Identify the events: - Event ( A ): All three coins are facing up (HHH). - Event ( B ): At least one coin is facing up.2. Calculate the probabilities: - The probability of all three coins facing up (( P(A) )) is ( left(frac{1}{2}right)^3 = frac{1}{8} ). - The probability of at least one coin facing up (( P(B) )) is the complement of all coins facing down. The probability of all coins facing down is ( left(frac{1}{2}right)^3 = frac{1}{8} ), so ( P(B) = 1 - frac{1}{8} = frac{7}{8} ).3. Apply the conditional probability formula: - ( P(A|B) = frac{P(A cap B)}{P(B)} ). Since ( A ) is a subset of ( B ), ( P(A cap B) = P(A) ). - Therefore, ( P(A|B) = frac{frac{1}{8}}{frac{7}{8}} = frac{1}{7} ).4. Verify by enumerating outcomes: - All possible outcomes when tossing three coins: 8 outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT). - Outcomes with at least one head (event ( B )): 7 outcomes (excluding TTT). - Favorable outcome for all three heads (event ( A )): 1 outcome (HHH).Thus, the probability is ( frac{1}{7} ).[boxed{C}],<think>Okay, so the problem is about probability with three coins. Let me read it again: "When three uniformly dense coins are thrown at once, the probability that the other two coins are also facing up when at least one coin is facing up is ( )." The options are A to D. Hmm. Wait, maybe I need to parse this sentence more carefully.So, the question is asking: given that at least one coin is facing up (which I assume means at least one head), what is the probability that the other two coins are also facing up? Wait, "the other two coins are also facing up when at least one coin is facing up." That wording is a bit confusing. Let me try to rephrase it. Maybe it's asking for the probability that all three coins are facing up (i.e., all heads) given that at least one coin is facing up. Or perhaps, given that one specific coin is facing up, what's the probability the other two are as well? But the wording says "the other two coins are also facing up when at least one coin is facing up." Hmm. Maybe it's a conditional probability: given that at least one is heads, what's the probability that all three are heads?Wait, that would make sense. Let me check again. The problem says: "the probability that the other two coins are also facing up when at least one coin is facing up." So when we have at least one head, what's the probability the other two are also heads. But wait, if there are three coins, and we know at least one is heads, then the "other two" must refer to the remaining two coins besides the one that's already known to be heads. But the problem doesn't specify a particular coin. So maybe it's that given at least one head, what's the probability that all three are heads? Because if at least one is heads, the "other two" would imply the remaining two, but if the problem is not specifying a particular coin, then maybe it's the probability that all three are heads given at least one is heads. Let me verify.Alternatively, maybe the problem is phrased as: when you have three coins, and you know that at least one is heads, what is the probability that the other two (meaning the remaining two) are also heads. So, that would be the conditional probability P(all three heads | at least one head). Let me compute that.First, the total number of possible outcomes when tossing three coins is 2^3 = 8. The possible outcomes are: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. Now, the event "at least one head" is all outcomes except TTT. So that's 7 outcomes. The event "all three heads" is just HHH. So the conditional probability is P(HHH) / P(at least one head) = (1/8) / (7/8) = 1/7. So the answer would be C: 1/7. But let me check if I'm interpreting the problem correctly.Wait, the problem says: "the probability that the other two coins are also facing up when at least one coin is facing up". The wording is a bit odd. If we have at least one head, then "the other two" might refer to the two coins besides the one that is known to be heads. But since it's not specified which coin is the one that's facing up, maybe it's considering any one of the coins. So perhaps, given that at least one is heads, what's the probability that all three are heads? Which is indeed 1/7, as the only favorable case is HHH out of the 7 possible cases with at least one head.Alternatively, maybe the problem is mistranslated or awkwardly phrased. If it were given that a specific coin is heads, then the probability that the other two are heads would be 1/2 * 1/2 = 1/4. But that's not one of the options. Alternatively, if they mean "the probability that both of the other two coins are heads given that at least one is heads", but that's ambiguous. Wait, maybe I need to parse the wording again.Original problem: "the probability that the other two coins are also facing up when at least one coin is facing up is ( )." The wording "when at least one coin is facing up" could be a conditional: given that at least one is facing up, what's the probability the other two are also facing up. So in other words, given that there's at least one head, what's the probability the remaining two coins are heads. But "the other two" would only make sense if we're considering one coin as the one that's already known to be heads, but the problem doesn't specify which one. Therefore, perhaps the correct interpretation is the probability that all three are heads given that at least one is heads. Since in that case, if all three are heads, then certainly the other two (relative to any head) are heads. But since the problem doesn't fix a specific coin, the only case where all the other two coins are heads regardless of which coin is considered is when all three are heads.Therefore, the conditional probability is 1/7, which is answer C. However, let me check again the possible outcomes. The total possible outcomes with at least one head are 7: HHH, HHT, HTH, THH, HTT, THT, TTH. The only case where all three are heads is HHH. So the probability is 1/7. That seems correct.But let me think if there's another interpretation. Suppose someone misreads the problem as "given that exactly one coin is facing up, what's the probability the other two are also facing up", but that would make no sense because if exactly one is heads, the others are tails. But maybe not. Alternatively, if the problem is phrased as "the probability that the other two coins are also facing up when at least one is facing up", perhaps they mean the probability that, given that at least one is heads, all three are heads, which is 1/7. So the answer is C.But I want to verify once more. Let's see. The problem is a bit ambiguous, but given the answer choices, 1/7 is an option. Alternatively, if it's the probability that all three are heads given at least one head, that's 1/7. If someone mistakenly counts the number of favorable outcomes as 3 (thinking each coin could be the one that's heads), but no, because if all three are heads, then all cases where at least one is heads includes HHH only once. Therefore, the correct answer is C: 1/7.Final Answerboxed{C}</think>Given three uniformly dense coins are thrown at once, we need to find the probability that all three coins are facing up (heads) given that at least one coin is facing up (heads).First, we determine the total number of possible outcomes when tossing three coins, which is (2^3 = 8). The possible outcomes are: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.The event "at least one head" excludes the outcome TTT, leaving 7 outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH.The event "all three heads" is only the outcome HHH. The conditional probability is calculated as:[P(text{all three heads} mid text{at least one head}) = frac{P(text{all three heads})}{P(text{at least one head})} = frac{frac{1}{8}}{frac{7}{8}} = frac{1}{7}]Thus, the final answer is (boxed{C}).]

🔑:When three uniformly dense coins are thrown at once, and we are considering the scenario where at least one coin is facing up, we need to calculate the probability that the other two coins are also facing up under this condition.First, let's enumerate all possible outcomes when at least one coin is facing up. These outcomes are:1. Head Head Head2. Head Head Tail3. Head Tail Head4. Tail Head Head5. Head Tail Tail6. Tail Head Tail7. Tail Tail HeadIn total, there are 7 outcomes when at least one coin is facing up.Among these 7 outcomes, there is only 1 outcome where all three coins are facing up, which is the "Head Head Head" outcome.To find the probability that the other two coins are also facing up given that at least one coin is facing up, we calculate the ratio of the number of favorable outcomes to the total number of possible outcomes under the given condition. This gives us:[P = frac{text{Number of favorable outcomes}}{text{Total number of possible outcomes}} = frac{1}{7}]Therefore, the probability that the other two coins are also facing up when at least one coin is facing up is boxed{frac{1}{7}}.Hence, the correct answer is: boxed{C}.