🧠:[<think>Alright, let's see. So, the problem is about angles around a point, right? We have angles AOB, BOC, and COD that are equal. Then angle AOD is three times smaller than each of them. All rays OA, OB, OC, OD are distinct. We need to find the measure of angle AOD, listing all possible options.First, let's visualize this. Imagine point O, and around it, four rays: OA, OB, OC, OD. These rays form angles AOB, BOC, COD, and then angle AOD. Wait, angle AOD—is that the angle from A to O to D? So, depending on how the rays are arranged, angle AOD might be the remaining angle around the point?But the problem says all rays are distinct. So, OA, OB, OC, OD are four different rays. The angles between OA and OB, OB and OC, OC and OD are equal. Then angle AOD is three times smaller than each of those. Hmm. Let me clarify.First, angles AOB, BOC, COD are equal. Let's denote their measure as x degrees each. Then angle AOD is three times smaller, so that would be x/3 degrees.But how are these angles arranged? If OA, OB, OC, OD are in order around the point O, then the total angles around a point should sum to 360 degrees. So, angles AOB + BOC + COD + DOA = 360 degrees. Wait, but angle AOD is given. Wait, maybe I need to check the exact notation.Angle AOB is between OA and OB. Then BOC is between OB and OC. COD is between OC and OD. Then angle AOD—if that's the angle from OA to OD, then depending on how the rays are placed, that could be the sum of AOB + BOC + COD, which would be 3x. But if angle AOD is three times smaller than each of the angles AOB, BOC, COD, then angle AOD would be x/3. But if angle AOD is formed by OA and OD, then in the case where the rays are placed in the order OA, OB, OC, OD around the point, angle AOD would be the sum of the three angles AOB + BOC + COD = 3x. However, the problem states that angle AOD is three times smaller than each of them, so 3x = x/3? That can't be. That suggests a contradiction. So maybe the rays are arranged differently?Wait, perhaps the rays are not in the order OA, OB, OC, OD? Since all rays are distinct, but they could be arranged in different orders. For example, OA, OB, OC, OD might not be in a circular order. Let me think.Alternatively, maybe angle AOD is not the sum of the three angles but a different angle. For example, if OA and OD are adjacent rays, then angle AOD would be another angle. But the problem says that OA, OB, OC, OD are distinct rays. So, there must be four distinct rays, so the angles between them are each AOB, BOC, COD, and then perhaps other angles?Wait, maybe the notation is such that angle AOD is the angle from OA to OD, passing through another ray? Hmm. Wait, in angle notation, the middle letter is the vertex. So, angle AOD is the angle at point O between rays OA and OD. So, regardless of the order of the rays, angle AOD is the angle between OA and OD. So, depending on the positions of OA and OD, that angle could be different.Let me try to diagram this mentally. Suppose we have four rays from point O: OA, OB, OC, OD. The angles between OA and OB is x, between OB and OC is x, between OC and OD is x. Then, the angle between OA and OD would be 3x, if the rays are placed in order. But the problem states that angle AOD is three times smaller than each of the other angles. So angle AOD is x/3. But if OA and OD are separated by three segments each of x, then their angle is 3x. So 3x = x/3? That leads to x = 0, which is impossible.Alternatively, perhaps the arrangement is different. Maybe OA and OD are adjacent? So, the order is OA, OD, OB, OC or something. Let's try.If OA and OD are adjacent, then angle AOD is x/3. Then the other angles AOB, BOC, COD are each x. But then, how are the rays arranged? Let's suppose OA, OD, OB, OC. Then angle AOD is x/3, angle AOB would be the angle from OA to OB, which would go through OD? Wait, that might complicate things.Alternatively, maybe the rays are arranged such that OA, OB, OC, OD are not in order. So, for example, OA, OB, OC are in one direction, and OD is somewhere else. So angle AOD is separate.Wait, but all angles around point O must add up to 360 degrees. So, the sum of all angles around O is 360. So, if we have angles AOB, BOC, COD, and AOD, but maybe there are other angles as well? Wait, no. If we have four rays, OA, OB, OC, OD, then they divide the plane into four angles. Wait, no. With four rays from a point, you can have up to four angles if they are in pairs, but if they are all distinct, they can create more angles. Wait, actually, four distinct rays will divide the circle into four angles if they are arranged in order. Wait, no. Let me clarify.If you have four rays emanating from a point, arranged around the circle in order OA, OB, OC, OD, then the angles between consecutive rays are four angles: AOB, BOC, COD, and DOA. Wait, but that's four angles. So if OA, OB, OC, OD are arranged around the circle in order, then the angles between them are AOB, BOC, COD, and DOA. Their sum is 360 degrees. But the problem mentions angles AOB, BOC, COD, and AOD. So, maybe AOD is not the same as DOA? Wait, angle AOD is the angle from OA to OD, which in this case would be the sum of AOB + BOC + COD = 3x. But the problem says angle AOD is three times smaller than each of them. So 3x = x/3? Which is impossible. So, this suggests that the rays are not arranged in order OA, OB, OC, OD.Alternatively, maybe OD is between OA and OB? Let me try to think. If OD is between OA and OB, then angle AOD is part of angle AOB. Let's see. Suppose the rays are ordered OA, OD, OB, OC. Then angle AOB is split into angles AOD and DOB. If angle AOB is equal to BOC and COD, which are each x. Then angle AOD is x/3. So angle AOB is x, which is equal to angle BOC and COD. Then angle AOD is x/3, so angle DOB would be x - x/3 = (2x)/3. But then, what about the other angles?Wait, angle BOC is x. If the rays are ordered OA, OD, OB, OC, then angle BOC is between OB and OC. Then angle COD is between OC and OD? But OD is between OA and OB, so COD would be from OC to OD, which would wrap around. That complicates things.Alternatively, maybe the rays are arranged OA, OB, OC, OD in such a way that OD is in a different position. Hmm. This is getting a bit confusing. Let me try to approach this algebraically.Let’s denote the measure of angles AOB, BOC, COD as x each. Then angle AOD is x/3. Now, we need to consider the total angles around point O. The sum of all angles around a point is 360 degrees. But we have three angles of x each, and angle AOD is x/3. However, depending on how these angles are arranged, there might be other angles as well.Wait, perhaps angle AOD is overlapping with some of the other angles? For example, if OD is between OA and OB, then angle AOD is part of angle AOB. So, angle AOB is x, angle AOD is x/3, which would imply that angle DOB is x - x/3 = (2x)/3. Then, angle BOC is x, angle COD is x. But then, where is ray OD located? If OD is between OA and OB, then OC must be somewhere else. Let me try to map this:- OA, OD, OB, OC: So, starting from OA, we have OD, then OB, then OC. Then the angles would be: - AOD: x/3 - DOB: (2x)/3 - BOC: x - COA: Wait, but where is OC? If OC is after OB, then the angle from OC back to OA would be a large angle. Wait, this might not cover the full circle.Wait, perhaps the total angles would be AOD (x/3), DOB (2x/3), BOC (x), COD (x), and then the remaining angle from D to A? Wait, no. If rays are OA, OD, OB, OC, then the angles between them would be:1. Between OA and OD: x/32. Between OD and OB: 2x/3 (since AOB is x, and AOD is x/3)3. Between OB and OC: x4. Between OC and OA: Hmm, but where is OC located? If OC is after OB, then the angle between OC and OA would be the remaining angle. Let's compute the sum.Sum of angles around point O must be 360 degrees. So:x/3 (AOD) + 2x/3 (DOB) + x (BOC) + angle COA = 360 degrees.But angle COD is supposed to be x. Wait, where is COD? If OC is after OB, and OD is between OA and OB, then angle COD would be between OC and OD. But since OD is between OA and OB, and OC is after OB, angle COD would be from OC to OD, which wraps around. So angle COD would be angle COD = angle COA + angle AOD. Wait, this is getting too convoluted. Maybe my initial assumption about the arrangement is wrong.Alternatively, maybe the rays are arranged such that OA, OB, OC, OD are in a different order, leading to angle AOD being x/3. Let's think differently. Let’s suppose that OA, OB, OC, OD are arranged around point O such that there are three angles of x and one angle of x/3. But the sum of all angles around a point is 360, so:3x + (x/3) = 360Multiply both sides by 3 to eliminate the denominator:9x + x = 108010x = 1080x = 108 degreesThen angle AOD would be x/3 = 36 degrees. But is this possible? Wait, this would imply that the four angles around point O are 108°, 108°, 108°, and 36°, which sum to 108*3 +36 = 360. So that works. But the problem states that all rays are distinct. So, in this case, if we have four rays creating four angles: three of 108°, and one of 36°, then the rays must be arranged such that three large angles and one small angle. But would the angles AOB, BOC, COD each be 108°, and angle AOD be 36°?Wait, if OA, OB, OC, OD are arranged such that the three angles AOB, BOC, COD are each 108°, then angle AOD would be the remaining angle? But in that case, angle AOD would be 360 - 3*108 = 360 - 324 = 36°, which matches. So, that works. So, in this arrangement, OA, OB, OC, OD are placed such that between OA and OB is 108°, between OB and OC is 108°, between OC and OD is 108°, and between OD and OA is 36°. So, angle AOD is 36°, which is three times smaller than 108°. That seems to fit.But the problem says "all rays OA, OB, OC, OD are distinct." In this arrangement, the rays are distinct, so that's okay. So, is 36° the only possible measure?Wait, but maybe there's another arrangement where angle AOD is x/3, but the other angles are not consecutive. For example, angle AOD could be between two of the other angles. Let's explore.Suppose the rays are arranged in a different order, such that angle AOD is not adjacent to the other angles. For example, OA, OB, OD, OC. Then angle AOD could be x/3, and angles AOB, BOC, COD could each be x. But how would that work?Wait, let's denote angle AOB = x, BOC = x, COD = x, and angle AOD = x/3. Then, we need to figure out the total sum. But depending on the arrangement, the angles could overlap or combine differently.Alternatively, perhaps angle AOD is one of the angles between OA and OD, and the other angles are arranged such that the sum of all angles is 360. Let me consider different cases.Case 1: The three angles AOB, BOC, COD are consecutive, each equal to x, and angle AOD is the remaining angle, equal to x/3. Then:3x + x/3 = 360Multiply by 3: 9x + x = 1080 => 10x = 1080 => x = 108. So angle AOD = 36°. This is the previous solution.Case 2: Maybe angle AOD is not the remaining angle but is interspersed between other angles. For example, suppose the order of rays is OA, OD, OB, OC. Then angle AOD = x/3, angle DOB = x, angle BOC = x, angle COA = x. But then the sum would be x/3 + x + x + x = 360 => (10x)/3 = 360 => x = 108. Then angle AOD would be 36°, same as before.Wait, but in this case, angle COA is x, which would be the angle from OC to OA, which is actually the large angle. But if rays are arranged OA, OD, OB, OC, then the angle from OC to OA would indeed be a large angle. However, the problem states that angles AOB, BOC, COD are equal. In this arrangement, angle COD is from OC to OD. If the rays are OA, OD, OB, OC, then COD is from OC to OD, which is a different angle. Wait, let's check.If the order is OA, OD, OB, OC, then:- Angle AOD: between OA and OD = x/3- Angle DOB: between OD and OB = x- Angle BOC: between OB and OC = x- Angle COA: between OC and OA = ?But angle COD would be between OC and OD. If the rays are OA, OD, OB, OC, then COD is the angle from OC to OD, which would be the sum of angles COA and AOD. Wait, angle COD is supposed to be equal to x, but in this case, angle COD would be angle COA + angle AOD. If angle COA is x, and angle AOD is x/3, then COD = x + x/3 = 4x/3. But the problem states that COD is equal to x. Therefore, this arrangement doesn't satisfy the problem's conditions.So, this suggests that the only valid arrangement is when the three equal angles AOB, BOC, COD are consecutive, each equal to x, and angle AOD is the remaining angle equal to x/3. Thus, leading to x = 108°, angle AOD = 36°.But let's check another possibility. Suppose that the rays are arranged in such a way that angle AOD is adjacent to one of the equal angles. For example, OA, OB, OC, OD arranged such that angle AOD is next to angle AOB.Wait, but if OA, OB, OC, OD are arranged around O, and angles AOB, BOC, COD are equal, then the fourth angle is DOA. If angle AOD is three times smaller, then DOA is x/3, but angle AOD is the same as DOA? Wait, angle AOD and DOA are the same angle, just measured in the opposite direction. So, angle AOD = angle DOA. So, if angle AOD is x/3, then angle DOA is x/3, and the total sum is 3x + x/3 = 360°, leading to x=108°, angle AOD=36°, same as before.Therefore, in all cases, the only possible measure for angle AOD is 36°. Wait, but the problem says "list all possible options". Maybe there's another configuration?Wait, maybe there's a different configuration where angle AOD is three times smaller, but not the remaining angle. For example, if angle AOD is part of one of the other angles. Suppose that OD is between OA and OB, making angle AOD = x/3 and angle DOB = x - x/3 = 2x/3. Then, angles BOC = x, COD = x. Then, how do the other angles add up?Let's compute the total angles around O:- Angle AOD: x/3- Angle DOB: 2x/3- Angle BOC: x- Angle COD: x- The remaining angle would be from D to A, which is angle COA? Wait, no. If the rays are OA, OD, OB, OC, then after OC, the angle back to OA would be the remaining angle.Wait, let's break it down step by step.If the rays are ordered OA, OD, OB, OC, then the angles between consecutive rays are:1. OA to OD: x/3 (angle AOD)2. OD to OB: 2x/3 (angle DOB)3. OB to OC: x (angle BOC)4. OC to OA: ? (angle COA)The sum of all these angles must be 360°, so:x/3 + 2x/3 + x + angle COA = 360°Simplify:(x/3 + 2x/3) = xSo, x + x + angle COA = 360°Therefore, 2x + angle COA = 360°But angle COD is given as x. Wait, angle COD is between OC and OD. If the rays are ordered OA, OD, OB, OC, then angle COD is from OC to OD, which would be angle COA + angle AOD = angle COA + x/3.But the problem states angle COD is equal to x. Therefore:angle COD = angle COA + x/3 = xSo, angle COA = x - x/3 = 2x/3.Substituting back into the total angles:2x + 2x/3 = 360°Multiply all terms by 3 to eliminate denominators:6x + 2x = 1080°8x = 1080°x = 135°Then angle AOD = x/3 = 45°, angle COA = 2x/3 = 90°, angle DOB = 2x/3 = 90°, angle BOC = x = 135°, angle COD = x = 135°.Wait, let's verify the total sum:angle AOD = 45°, angle DOB = 90°, angle BOC = 135°, angle COD = 135°, angle COA = 90°. Wait, but that's five angles? How?Wait, no. If the rays are OA, OD, OB, OC, then the angles between consecutive rays are:1. OA to OD: 45°2. OD to OB: 90°3. OB to OC: 135°4. OC to OA: 90°So, 45 + 90 + 135 + 90 = 360°, which works. But angle COD is given as x = 135°, which is the angle from OC to OD. But in this arrangement, angle COD is from OC to OD, which is the sum of angles COA (90°) and AOD (45°), so 135°, which matches. So, in this case, angle AOD is 45°, which is x/3 where x=135°. So, this gives another possible measure for angle AOD: 45°.But wait, the problem states that angles AOB, BOC, COD are equal. In this arrangement, angle AOB is the angle from OA to OB, which is angle AOD + angle DOB = 45° + 90° = 135°, which equals x. Similarly, angle BOC is 135°, angle COD is 135°, so they are all equal. Therefore, this configuration also satisfies the problem's conditions. Therefore, angle AOD can be 45° as well.Therefore, there are two possible configurations: one where angle AOD is 36°, and another where it's 45°. Wait, so the answer is 36° and 45°? Let me verify.Wait, let's check both cases:Case 1: The three equal angles are consecutive, each 108°, and angle AOD is 36°.Sum: 3*108 + 36 = 360°, which works.Case 2: The three equal angles are 135°, with angle AOD being 45°, and other angles adjusting accordingly. Let's verify:Angles:- AOB: 135° (from OA to OB, through OD: 45° + 90°)- BOC: 135°- COD: 135° (from OC to OD, through OA: 90° + 45°)- The remaining angles (DOB=90°, COA=90°) but these are not mentioned in the problem. The problem only specifies angles AOB, BOC, COD, and AOD. So, in this case, AOD=45°, and AOB, BOC, COD are 135° each. Therefore, this also satisfies the problem's conditions.Therefore, angle AOD can be either 36° or 45°, depending on the arrangement. Wait, but how?Wait, in the first case, the three equal angles (108°) are adjacent, and angle AOD is the remaining angle (36°). In the second case, the three equal angles (135°) are not all adjacent, but arranged such that angle AOD is part of angle AOB, and the remaining angles adjust accordingly. So, both configurations are possible, leading to two different measures for angle AOD: 36° and 45°.But wait, is there another possible configuration? For instance, if angle AOD is placed elsewhere, could we get another measure?Let me try another arrangement. Suppose the rays are ordered OA, OB, OD, OC. Then:- Angle AOB = x- Angle BOD = ?- Angle DOC = x- Angle COA = ?But angle COD is x. Hmm. Let's see.If angle AOB = x, angle BOD = y, angle DOC = x, angle COA = z. Then, the total sum is x + y + x + z = 360°. Also, angle AOD is x/3. Angle AOD is the angle from OA to OD, which would be angle AOB + angle BOD = x + y. So x + y = x/3 => y = -2x/3. Which is impossible because angles can't be negative. Therefore, this arrangement is invalid.Alternatively, if angle AOD is angle between OA and OD, not passing through OB, which would be the reflex angle? Wait, no. In standard terms, angles are measured as the smaller arc between two rays, so angle AOD would be the smaller angle between OA and OD, less than or equal to 180°. So, if the actual angle between OA and OD is larger than 180°, we still consider the smaller one. Therefore, angle AOD must be less than or equal to 180°.Therefore, in the previous case where angle AOD was 45°, that's valid. If angle AOD was 36°, also valid.Wait, but are there other possibilities? Let's suppose that angle AOD is part of two different angles.Wait, maybe three angles AOB, BOC, COD are not consecutive. For example, OA, OB, OC, OD arranged such that AOB = x, BOC = x, and COD = x, but with OD not adjacent to OC. However, with four rays, if three angles are equal and one is different, the different angle can be placed in different positions. But given the problem states that angle AOD is three times smaller, which relates OA and OD, their positions affect the measure.Alternatively, if we have three angles of x and one angle of x/3, arranged in different orders. Let's consider all permutations.But since there are four angles, three of x and one of x/3, the sum is 3x + x/3 = 10x/3 = 360°, so x = 108°, leading to angle AOD = 36°. However, in the previous case, we found another configuration where x was 135°, but that seems to contradict the sum. Wait, no. Wait, in the second case, the angles were not three times x plus x/3. Instead, angle AOD was part of angle AOB, leading to different total.Wait, perhaps the confusion is arising because the problem mentions angles AOB, BOC, COD, and AOD. Depending on the arrangement, some angles may be combinations of others.Wait, the problem says: "Angles AOB, BOC and COD are equal to each other, while angle AOD is three times smaller than each of them." So, angles AOB, BOC, COD are equal (each is x), and angle AOD is x/3. Now, regardless of how the rays are arranged, the sum of all angles around O is 360°. However, depending on how these specific angles are placed, the total sum may vary.But if we assume that angles AOB, BOC, COD, and AOD are four distinct angles around point O, then their sum is 360°. Therefore:x + x + x + x/3 = 360Which simplifies to:(3x + 3x + 3x + x)/3 = 360(10x)/3 = 360x = 108Therefore, angle AOD = 108/3 = 36°However, in the previous example, we had another configuration where angle AOD was 45°, but that was because angle AOD was considered as part of angle AOB, which was divided into two angles. However, in that case, angle AOB wasn't equal to x but was equal to x/3 + another angle. Wait, but the problem states that angles AOB, BOC, COD are equal. Therefore, in that case, angle AOB must be equal to x. Therefore, if angle AOD is part of angle AOB, then angle AOB would be angle AOD + another angle, which must equal x. So, if angle AOD is x/3, then the remaining part of angle AOB is x - x/3 = 2x/3. Then, proceeding as before, we get:Sum of angles around point O:angle AOD = x/3angle DOB = 2x/3 (remaining part of AOB)angle BOC = xangle COD = xangle COA = ?But angle COD is supposed to be x. If COD is from OC to OD, and if the rays are arranged OA, OD, OB, OC, then angle COD is angle COA + angle AOD = angle COA + x/3 = x. Therefore, angle COA = x - x/3 = 2x/3.Then total angles:x/3 (AOD) + 2x/3 (DOB) + x (BOC) + x (COD) + 2x/3 (COA) = ?Wait, but hold on. If the rays are OA, OD, OB, OC, the angles between consecutive rays are:1. OA to OD: x/32. OD to OB: 2x/33. OB to OC: x4. OC to OA: 2x/3But angle COD is from OC to OD, which is angle COA + angle AOD = 2x/3 + x/3 = x. So that checks out. Then the total sum is:x/3 + 2x/3 + x + 2x/3 = (x/3 + 2x/3) + x + 2x/3 = x + x + 2x/3 = 2x + 2x/3 = 8x/3But this must equal 360°, so:8x/3 = 360°x = (360° * 3)/8 = 135°Therefore, x = 135°, angle AOD = x/3 = 45°Thus, in this configuration, angle AOD is 45°, and angles AOB, BOC, COD are each 135°. This works because angle AOB is composed of angles AOD (45°) and DOB (90°), totaling 135°, which is equal to x. Similarly, angle COD is composed of angles COA (90°) and AOD (45°), totaling 135°, equal to x. So this is a valid configuration.Therefore, there are two possible configurations:1. All three equal angles (x = 108°) are consecutive, with angle AOD = 36° as the remaining angle.2. The three equal angles (x = 135°) are arranged such that angle AOD = 45° is part of angle AOB and COD, with other angles adjusting to satisfy the sum.Therefore, angle AOD can be either 36° or 45°.But wait, the problem states "all rays OA, OB, OC, OD are distinct." In both configurations, the rays are distinct, so both are valid. Therefore, the possible measures of angle AOD are 36° and 45°.But let me check if there are other configurations. Suppose we have angle AOD overlapping with another angle. For example, if OD is between OC and OA, but that might not yield a different measure.Alternatively, suppose the three equal angles are separated by the smaller angle. For example, x, x/3, x, x/3, x, x/3, but with four rays, this would not fit.Alternatively, is there a configuration where angle AOD is reflex (greater than 180°), but since angle measures are typically taken as the smaller arc, angle AOD would still be the smaller angle, so it must be less than 180°. Therefore, only the two possibilities we found are valid.Thus, the measure of angle AOD can be either 36° or 45°.Final AnswerThe measure of angle ( AOD ) is boxed{36} degrees or boxed{45} degrees.</think>Given the problem, we need to find the measure of angle ( AOD ) where angles ( AOB ), ( BOC ), and ( COD ) are equal, and angle ( AOD ) is three times smaller than each of them. All rays ( OA ), ( OB ), ( OC ), and ( OD ) are distinct.1. First Configuration: - Let the measure of angles ( AOB ), ( BOC ), and ( COD ) be ( x ) each. - Angle ( AOD ) is given as ( frac{x}{3} ). - The sum of all angles around point ( O ) is ( 360^circ ). - Therefore, we have the equation: [ 3x + frac{x}{3} = 360^circ ] - Solving for ( x ): [ frac{10x}{3} = 360^circ implies 10x = 1080^circ implies x = 108^circ ] - Thus, angle ( AOD ) is: [ frac{108^circ}{3} = 36^circ ]2. Second Configuration: - Consider another arrangement where angle ( AOD ) is part of angle ( AOB ) and ( COD ). - Let the measure of angles ( AOB ), ( BOC ), and ( COD ) be ( x ) each. - Angle ( AOD ) is ( frac{x}{3} ). - The remaining part of angle ( AOB ) is ( x - frac{x}{3} = frac{2x}{3} ). - The sum of all angles around point ( O ) is: [ frac{x}{3} + frac{2x}{3} + x + x + frac{2x}{3} = 360^circ ] - Simplifying the equation: [ frac{8x}{3} = 360^circ implies 8x = 1080^circ implies x = 135^circ ] - Thus, angle ( AOD ) is: [ frac{135^circ}{3} = 45^circ ]Both configurations are valid, leading to the possible measures of angle ( AOD ). Final AnswerThe measure of angle ( AOD ) is boxed{36} degrees or boxed{45} degrees.,<think>Okay, let me try to solve this problem step by step. So, we have angles AOB, BOC, and COD that are equal to each other. Then, angle AOD is three times smaller than each of them. All rays OA, OB, OC, OD are distinct. We need to find the measure of angle AOD, considering all possible options.First, let me visualize the problem. There's a point O, and four rays OA, OB, OC, OD coming out from it. The angles between OA and OB, OB and OC, and OC and OD are all equal. Then, angle AOD, which is the angle between OA and OD, is three times smaller than each of those angles. Hmm, but angles AOB, BOC, COD are equal, so let's denote each of those angles as x. Then angle AOD is x/3.But wait, angle AOD is the sum of angles AOB, BOC, and COD if all these angles are adjacent and follow each other around the point O. So, in that case, angle AOD would be x + x + x = 3x. But the problem states that angle AOD is three times smaller than each of them. That seems conflicting. If AOD is three times smaller, then AOD = x/3, but if AOD is composed of three angles each equal to x, then AOD = 3x. So, 3x = x/3? That would lead to 3x = x/3, which simplifies to 9x = x, then 8x = 0, so x = 0. But that's impossible because angles can't be zero if the rays are distinct.Hmm, maybe my initial assumption is wrong. Maybe the angles AOB, BOC, COD are not all adjacent? But the problem says "rays OA, OB, OC, OD are distinct." So, OA, OB, OC, OD are all different rays. If all the angles between OA and OB, OB and OC, OC and OD are equal, but angle AOD is three times smaller. So, perhaps the configuration is not straightforward.Wait, maybe the rays are arranged in such a way that angle AOD is not the sum of the other three angles. For instance, if OA, OB, OC, OD are arranged around the point O in some order where the angles between consecutive rays (like AOB, BOC, COD) are equal, but AOD skips some rays. But how?Alternatively, maybe the rays are arranged in a different order. For example, OA, OC, OB, OD? Hmm, but the problem mentions angles AOB, BOC, COD. So, OA to OB to OC to OD. So, the order is OA, OB, OC, OD around the circle. So, consecutive angles. Then angle AOD is the angle from OA to OD, which would pass through OB and OC. So, angle AOD would be AOB + BOC + COD = x + x + x = 3x. But the problem states that angle AOD is three times smaller than each of them. So, 3x = x/3. Which again gives x = 0, which isn't possible.This suggests that perhaps the rays are arranged differently. Maybe not in the order OA, OB, OC, OD around the circle. Maybe the order is OA, OB, OD, OC or something else? Let me think.Alternatively, maybe the angles AOB, BOC, COD are not adjacent. For example, OA and OB form angle x, OB and OC form angle x, and OC and OD form angle x, but OA and OD could be in a different position. Wait, but OA, OB, OC, OD are distinct rays. So, if OA, OB, OC, OD are in a different order around point O, then the angle AOD might be a different combination.Alternatively, perhaps angle AOD is a reflex angle? But the problem probably refers to the smaller angle unless specified. So, if AOD is three times smaller, perhaps the reflex angle is being considered? Let's check.Suppose OA, OB, OC, OD are arranged such that each consecutive angle is x, so AOB = x, BOC = x, COD = x. Then, the total angle around the point would be 3x plus the remaining angle. But around a point, the total is 360 degrees. So, 3x + remaining angle = 360. But angle AOD is supposed to be three times smaller than x, so AOD = x/3. But how does AOD relate to the remaining angle?Wait, perhaps angle AOD is the remaining angle? If so, then 3x + AOD = 360. But if AOD is x/3, then 3x + x/3 = 360. Let's solve this equation:3x + x/3 = 360Multiply both sides by 3 to eliminate the denominator:9x + x = 108010x = 1080x = 108 degreesThen, angle AOD is x/3 = 108 / 3 = 36 degrees.But wait, in this case, the remaining angle would be AOD = 36 degrees. But if the rays are arranged as OA, OB, OC, OD, then angle AOD would be the sum of AOB, BOC, COD, which is 3x = 324 degrees, which is a reflex angle, but the smaller angle between OA and OD would be 360 - 324 = 36 degrees. So, in this case, angle AOD is 36 degrees. So, that seems possible.But in this scenario, the problem states that angle AOD is three times smaller than each of the angles AOB, BOC, COD. Since each of those angles is 108 degrees, then three times smaller would mean 108 / 3 = 36 degrees, which matches. So, that works.But is there another possible configuration?Wait, perhaps the angles AOB, BOC, COD are not all in the same direction. For example, OA, OB, OC might be in one direction, and OD could be in another direction. But the problem says all rays are distinct, but doesn't specify the order. So, maybe OD is between OA and OB? Let's explore.Suppose OA, OD, OB, OC. Then angles AOD, DOB, BOC. But the problem states angles AOB, BOC, COD. If OD is between OA and OB, then angle AOB is split into AOD and DOB. But angle AOB is given to be equal to BOC and COD. So, if AOB is equal to x, but it's split into two parts, AOD and DOB, then unless AOD is equal to DOB, but the problem states angle AOD is three times smaller than AOB. So, if angle AOB is x, then angle AOD is x/3. Then angle DOB would be x - x/3 = 2x/3. But then angle BOC is x, and angle COD is x. But if COD is x, then the angle between OC and OD is x. If OC is after OB, then OD would have to be somewhere else. Wait, this is getting complicated. Maybe there are multiple configurations.Alternatively, let's think of the angles in terms of rotations. Starting from OA, rotating x degrees to OB, then another x degrees to OC, another x degrees to OD. So, OA to OB is x, OB to OC is x, OC to OD is x, so OA to OD would be 3x. But angle AOD is given as three times smaller than each of the angles, so angle AOD = x/3. Therefore, 3x = x/3. Which is impossible unless x = 0. So, that can't be. Therefore, this configuration doesn't work.Alternatively, maybe the direction of rotation is different. For example, if some angles are measured clockwise and others counterclockwise? But in standard angle measurement, angles around a point are typically considered as the smallest angle between two rays, so between 0 and 180 degrees. If the angle is more than 180, it's a reflex angle, but usually, the smaller angle is considered unless specified.Wait, but the problem says "angle AOD is three times smaller than each of them." So, if AOD is three times smaller than x, then AOD = x/3. So, if the sum of angles AOB + BOC + COD = 3x, but AOD could be the other angle around the circle. Since the total around a point is 360 degrees, then 3x + AOD = 360. If AOD = x/3, then 3x + x/3 = 360. Solving:Multiply both sides by 3: 9x + x = 1080 => 10x = 1080 => x = 108. So, AOD = 108 / 3 = 36 degrees. That seems valid.But could there be another configuration where angle AOD is not the remaining angle but part of the other angles? For example, if the rays are arranged such that OD is between OA and OB. Then angle AOD is x/3, angle DOB is x - x/3 = 2x/3. Then angle BOC is x, angle COD is x. But angle COD would be the angle between OC and OD. If OC is after OB, then COD would be between OC and OD, which might be a different configuration.Wait, this is getting complicated. Let me try to diagram this mentally.Case 1: OA, OB, OC, OD are in order around the circle, each consecutive angle is x. Then angle AOD would be 3x. But problem states angle AOD is three times smaller, so 3x = x/3, which is impossible.Case 2: OA, OB, OC, OD are arranged such that angle AOD is a different part. For instance, OA, OD, OB, OC. Then, angles AOD = x/3, angles DOB = y, angles BOC = x, angles COD = x. But then we have to ensure all angles add up to 360. However, the problem states that angles AOB, BOC, COD are equal. If AOB is the angle between OA and OB, which in this case would be AOD + DOB = x/3 + y. If AOB is equal to BOC = x, then x/3 + y = x. Therefore, y = 2x/3. Then angle COD is x, which is between OC and OD. If OC is after OB, then COD would be between OC and OD. Let's see: OA, OD, OB, OC.So, from OA to OD is x/3, then from OD to OB is 2x/3, then from OB to OC is x, and from OC to OD is x? Wait, OC to OD is x, but OC is after OB. So, from OC back to OD would be another angle. This is confusing.Alternatively, if OA, OB, OC, OD are arranged such that OA to OB is x, OB to OC is x, OC to OD is x, but OD is not adjacent to OA, so there's a gap between OD and OA. Then the total angles would be 3x + angle AOD = 360. Then angle AOD is x/3, so 3x + x/3 = 360. Solving gives x = 108, angle AOD = 36. So, in this case, angle AOD is 36 degrees. So, that's the previous case.Alternatively, if angle AOD is not the remaining angle, but part of another configuration. Suppose OD is in between OA and OB. Then angle AOD is x/3, angle DOB is something else. But angle AOB would then be AOD + DOB = x/3 + DOB. But the problem states that angle AOB is equal to x, so x/3 + DOB = x. Therefore, DOB = 2x/3. Then, angle BOC is x, angle COD is x. So, from OC to OD is x. But where is OC? If OA, OD, OB, OC are arranged around the circle, then from OA to OD is x/3, OD to OB is 2x/3, OB to OC is x, OC to OD is x. Wait, but OC to OD would be a reflex angle unless OC is between OB and OD? Wait, no.This seems very convoluted. Let me try to add up all the angles. In this configuration: OA to OD is x/3, OD to OB is 2x/3, OB to OC is x, OC to OD is x. Wait, but OC to OD would be x, but OC is after OB. So, from OC to OD, going the other way around, would be 360 - (OA to OC) - (OC to OD). Hmm, perhaps this is not the right approach.Alternatively, perhaps there are two possible configurations: one where angle AOD is the small angle (36 degrees) and one where it's the reflex angle (324 degrees). But the problem says angle AOD is three times smaller than each of the angles AOB, BOC, COD. If angle AOD is the reflex angle, then 324 = x/3, so x = 972 degrees, which is impossible because angles around a point can't exceed 360. So, that can't be. Therefore, angle AOD must be the smaller angle, 36 degrees.Is there any other possible configuration?Suppose the angles AOB, BOC, COD are not all in the same rotational direction. For example, OA to OB is x clockwise, OB to OC is x counterclockwise, creating a zig-zag. But in that case, the rays might overlap, but the problem states all rays are distinct. This seems too complicated and likely not intended.Alternatively, maybe the problem allows for the angles to be in different planes, but since all rays are from a common point O, it's assumed to be in a plane unless stated otherwise.So, considering all possibilities, the only feasible solution seems to be 36 degrees. Let's verify again.If angles AOB, BOC, COD are each 108 degrees, then summing them gives 324 degrees. The remaining angle around point O is 360 - 324 = 36 degrees, which is angle AOD. Since 36 is three times smaller than 108 (108 / 3 = 36), this fits the condition. Also, all rays OA, OB, OC, OD are distinct because each angle is non-zero and they are spaced around the point.Therefore, the measure of angle AOD is 36 degrees. Are there any other possible measures?Wait, let's consider if the angles are measured in different directions. For example, if angle AOD is the reflex angle, but as I thought before, that would require x to be 972 degrees, which is impossible. So, no.Alternatively, could there be another arrangement where angle AOD is x/3 but the angles AOB, BOC, COD are arranged such that they don't sum up to 3x? For example, overlapping angles? But the problem states all rays are distinct, so overlapping angles would mean rays coincide, which is not allowed.Hence, the only possible measure for angle AOD is 36 degrees.Final AnswerThe measure of angle (AOD) is boxed{36} degrees.</think>Given angles ( angle AOB ), ( angle BOC ), and ( angle COD ) are equal to each other, and angle ( angle AOD ) is three times smaller than each of them. All rays ( OA ), ( OB ), ( OC ), and ( OD ) are distinct. We need to find the measure of angle ( angle AOD ).1. Let each of the angles ( angle AOB ), ( angle BOC ), and ( angle COD ) be denoted as ( x ).2. The problem states that ( angle AOD ) is three times smaller than each of these angles, so ( angle AOD = frac{x}{3} ).3. The sum of the angles around point ( O ) must be ( 360^circ ).4. If ( angle AOB ), ( angle BOC ), and ( angle COD ) are adjacent and sum up to ( 3x ), then the remaining angle ( angle AOD ) must be ( 360^circ - 3x ).5. According to the problem, ( angle AOD = frac{x}{3} ). Therefore, we set up the equation: [ 3x + frac{x}{3} = 360^circ ]6. Solving this equation: [ 3x + frac{x}{3} = 360^circ ] Multiply through by 3 to clear the fraction: [ 9x + x = 1080^circ ] [ 10x = 1080^circ ] [ x = 108^circ ]7. Substituting ( x = 108^circ ) back into ( angle AOD ): [ angle AOD = frac{108^circ}{3} = 36^circ ]Thus, the measure of angle ( angle AOD ) is (boxed{36}) degrees.]

🔑:# Solution.Angles A O B, B O C and C O D follow each other in the same direction (since no rays coincide). Their sum can be less than 360^{circ} (see Fig. 1) and more than 360^{circ} (see Fig. 2).Let the measure of angle AOD be denoted by x. Then each of the angles AOB, BOC, and COD is equal to 3x. In the first case, we get 3x + 3x + 3x + x = 360^{circ}, from which x = 36^{circ}. In the second case, 3x + 3x + 3x - x = 360^{circ}, from which x = 45^{circ}.Fig. 1Fig. 2Answer: 36^{circ}, 45^{circ}## Criteria:- For one of the answers 36^{circ} or 45^{circ}, accompanied by explanations, even in the form of a diagram - 3 points.- Only one of the answers without explanations - 1 point. Both answers are written, but there are no explanations - 2 points.

🧠:[<think>Okay, so I need to find the greatest common divisor (GCD) of 663 and 182, and also find integers u and v such that the GCD can be expressed as a linear combination of these two numbers. Hmm, I remember that the Euclidean algorithm is the way to go for finding GCDs, and then the extended Euclidean algorithm can help find the coefficients u and v. Let me start by recalling how the Euclidean algorithm works.First, the Euclidean algorithm involves dividing the larger number by the smaller one, then replacing the larger number with the smaller number and the smaller number with the remainder from the division. You repeat this process until the remainder is zero, and the last non-zero remainder is the GCD. Alright, let's apply that to 663 and 182.So, step 1: Divide 663 by 182. Let me compute that. 182 times 3 is 546, and 663 minus 546 is 117. So, the remainder is 117. Now, replace the numbers: the new pair is 182 and 117.Step 2: Divide 182 by 117. 117 times 1 is 117, subtracting that from 182 gives 65. So the remainder is 65. Now the pair is 117 and 65.Step 3: Divide 117 by 65. 65 times 1 is 65, subtracting gives 52. Remainder is 52. Now the pair is 65 and 52.Step 4: Divide 65 by 52. 52 times 1 is 52, subtracting gives 13. Remainder is 13. Now the pair is 52 and 13.Step 5: Divide 52 by 13. 13 times 4 is 52, subtracting gives 0. Remainder is 0. So, since the remainder is 0, the last non-zero remainder is 13. Therefore, the GCD of 663 and 182 is 13.Okay, so that gives me d = 13. Now, I need to find integers u and v such that 13 = 663u + 182v. For this, I need to use the extended Euclidean algorithm. From what I remember, this involves working backwards through the steps of the Euclidean algorithm, expressing each remainder as a linear combination of the original two numbers.Let me jot down the steps we did earlier with the remainders:1. 663 = 182*3 + 1172. 182 = 117*1 + 653. 117 = 65*1 + 524. 65 = 52*1 + 135. 52 = 13*4 + 0So starting from the last non-zero remainder, which is 13, we can express that in terms of the previous equation. From step 4: 13 = 65 - 52*1. Now, we need to express 52 in terms of the previous remainders. From step 3: 52 = 117 - 65*1. Substitute this into the equation for 13:13 = 65 - (117 - 65*1)*1= 65 - 117 + 65*1= 65*2 - 117*1Now, 65 can be expressed from step 2: 65 = 182 - 117*1. Substitute this into the equation:13 = (182 - 117*1)*2 - 117*1= 182*2 - 117*2 - 117*1= 182*2 - 117*3Next, express 117 from step 1: 117 = 663 - 182*3. Substitute that in:13 = 182*2 - (663 - 182*3)*3= 182*2 - 663*3 + 182*9= 182*(2 + 9) - 663*3= 182*11 - 663*3So, putting it all together, we have 13 = (-3)*663 + 11*182. Therefore, the coefficients are u = -3 and v = 11.Wait a second, let me check my arithmetic here to make sure I didn't make any mistakes. Let's compute -3*663 and 11*182 and see if their sum is 13.First, -3*663: 3*600 = 1800, 3*63 = 189, so 1800 + 189 = 1989, hence -3*663 = -1989.Then, 11*182: 10*182 = 1820, 1*182 = 182, so total is 1820 + 182 = 2002.Adding -1989 + 2002 gives 13, which matches the GCD. So that checks out. Therefore, the coefficients are correct.But just to recap the steps again for clarity:1. Start with GCD(663, 182) and apply Euclidean algorithm to find GCD = 13.2. Then, use the extended algorithm by backtracking:- Express 13 as 65 - 52*1 (from step 4).- Replace 52 with 117 - 65*1 (from step 3): 13 = 65 - (117 - 65) = 2*65 - 117.- Replace 65 with 182 - 117*1 (from step 2): 13 = 2*(182 - 117) - 117 = 2*182 - 3*117.- Replace 117 with 663 - 182*3 (from step 1): 13 = 2*182 - 3*(663 - 182*3) = 2*182 - 3*663 + 9*182 = 11*182 - 3*663.Hence, u = -3 and v = 11. That seems right. Let me verify once more by substituting:663*(-3) + 182*11 = -1989 + 2002 = 13. Perfect.Is there another pair of integers u and v that would also satisfy the equation? Well, infinitely many, since once you have one solution, you can add multiples of 182/d to u and subtract multiples of 663/d from v (or vice versa), but the question just asks for integers u and v, so our solution is one such pair.Alternatively, if I wanted to express the GCD as a positive combination, I could adjust the coefficients, but since the problem allows u and v to be any integers, negative values are acceptable.Therefore, the answer is d = 13, with u = -3 and v = 11.Final AnswerThe GCD is boxed{13}, with integers ( u = -3 ) and ( v = 11 ) such that ( 13 = 663u + 182v ). Thus, the final answer is (boxed{13}).</think>To find the GCD of 663 and 182 and integers (u) and (v) such that (d = 663u + 182v), we use the Euclidean algorithm and the extended Euclidean algorithm.First, we apply the Euclidean algorithm:1. (663 = 182 cdot 3 + 117)2. (182 = 117 cdot 1 + 65)3. (117 = 65 cdot 1 + 52)4. (65 = 52 cdot 1 + 13)5. (52 = 13 cdot 4 + 0)The last non-zero remainder is 13, so the GCD is 13.Next, we use the extended Euclidean algorithm to express 13 as a linear combination of 663 and 182:1. From step 4: (13 = 65 - 52)2. Substitute (52) from step 3: (52 = 117 - 65) [ 13 = 65 - (117 - 65) = 2 cdot 65 - 117 ]3. Substitute (65) from step 2: (65 = 182 - 117) [ 13 = 2 cdot (182 - 117) - 117 = 2 cdot 182 - 3 cdot 117 ]4. Substitute (117) from step 1: (117 = 663 - 182 cdot 3) [ 13 = 2 cdot 182 - 3 cdot (663 - 182 cdot 3) = 2 cdot 182 - 3 cdot 663 + 9 cdot 182 = 11 cdot 182 - 3 cdot 663 ]Thus, the GCD (d = 13) with integers (u = -3) and (v = 11) such that (13 = 663u + 182v).[boxed{13}],<think>Okay, so I need to find the greatest common divisor (GCD) of 663 and 182, and then also find integers u and v such that the GCD can be expressed as a linear combination of these two numbers, right? Hmm, let's start with the GCD part. I remember that the Euclidean algorithm is the standard way to find the GCD of two numbers. Let me recall how that works.The Euclidean algorithm involves dividing the larger number by the smaller one, then replacing the larger number with the smaller number and the smaller number with the remainder from the division. You repeat this process until the remainder is zero, and the last non-zero remainder is the GCD. Let me try applying that here.First, the numbers are 663 and 182. Since 663 is larger than 182, I divide 663 by 182. Let me calculate 182 times 3. 182 times 3 is 546. Subtract that from 663: 663 - 546 = 117. So, the remainder is 117. Now, I replace the numbers with 182 and 117.Next step: divide 182 by 117. 117 times 1 is 117. Subtract that from 182: 182 - 117 = 65. So, the remainder is 65. Now, the numbers are 117 and 65.Divide 117 by 65. 65 times 1 is 65. Subtract: 117 - 65 = 52. Remainder is 52. Now, numbers are 65 and 52.Divide 65 by 52. 52 times 1 is 52. Subtract: 65 - 52 = 13. Remainder is 13. Now, numbers are 52 and 13.Divide 52 by 13. 13 times 4 is 52. Subtract: 52 - 52 = 0. Remainder is 0. So, the last non-zero remainder is 13. Therefore, the GCD of 663 and 182 is 13.Okay, so that's the GCD part. Now, I need to find integers u and v such that 13 = 663u + 182v. This part is the extended Euclidean algorithm, right? Where we work backwards through the steps we took to find the GCD. Let me see if I can reconstruct that.Starting from the bottom of the Euclidean algorithm steps:We had:1. 663 = 182 * 3 + 1172. 182 = 117 * 1 + 653. 117 = 65 * 1 + 524. 65 = 52 * 1 + 135. 52 = 13 * 4 + 0So, the GCD is 13, which comes from step 4: 65 = 52 * 1 + 13. Let's rearrange that equation to express 13 as a combination of 65 and 52:13 = 65 - 52 * 1But 52 is from step 3: 117 = 65 * 1 + 52, which can be rearranged as 52 = 117 - 65 * 1. Substitute that into the equation above:13 = 65 - (117 - 65 * 1) * 113 = 65 - 117 + 65 * 113 = (65 + 65) - 11713 = 65 * 2 - 117 * 1Now, 65 comes from step 2: 182 = 117 * 1 + 65, which can be rearranged as 65 = 182 - 117 * 1. Substitute that into the equation:13 = (182 - 117 * 1) * 2 - 117 * 113 = 182 * 2 - 117 * 2 - 117 * 113 = 182 * 2 - 117 * 3Now, 117 comes from step 1: 663 = 182 * 3 + 117, which can be written as 117 = 663 - 182 * 3. Substitute that into the equation:13 = 182 * 2 - (663 - 182 * 3) * 313 = 182 * 2 - 663 * 3 + 182 * 913 = (182 * 2 + 182 * 9) - 663 * 313 = 182 * 11 - 663 * 3So, putting that all together, we have 13 expressed as 182*11 + 663*(-3). Therefore, u is -3 and v is 11.Wait, but the problem says d = 663u + 182v. In my final equation, it's 182*11 + 663*(-3), which is the same as 663*(-3) + 182*11. So, yes, that matches. Therefore, u = -3 and v = 11.Let me double-check the arithmetic to make sure I didn't make any substitution errors.Starting from the first substitution:13 = 65 - 52*1Then 52 = 117 - 65*1, so:13 = 65 - (117 - 65) = 65 - 117 + 65 = 65*2 - 117*1. That seems correct.Then 65 = 182 - 117*1, substituting:13 = (182 - 117)*2 - 117*1 = 182*2 - 117*2 - 117*1 = 182*2 - 117*3. Correct.Then 117 = 663 - 182*3, substituting:13 = 182*2 - (663 - 182*3)*3 = 182*2 - 663*3 + 182*9 = 182*(2 + 9) - 663*3 = 182*11 - 663*3. Yes, that's right.So, substituting back, the coefficients are u = -3 and v = 11. Let me verify this by plugging into the original equation:663*(-3) + 182*11 = -1989 + 2002 = 13. Which matches the GCD, so that's correct.Therefore, the GCD is 13, and the integers u and v are -3 and 11 respectively.Final AnswerThe GCD is boxed{13}, with integers ( u = -3 ) and ( v = 11 ). Thus, ( d = 663u + 182v ) gives boxed{13} = ( 663 times boxed{-3} + 182 times boxed{11} ).</think>To find the GCD of 663 and 182 using the Euclidean algorithm:1. Divide 663 by 182: ( 663 = 182 times 3 + 117 )2. Divide 182 by 117: ( 182 = 117 times 1 + 65 )3. Divide 117 by 65: ( 117 = 65 times 1 + 52 )4. Divide 65 by 52: ( 65 = 52 times 1 + 13 )5. Divide 52 by 13: ( 52 = 13 times 4 + 0 )The last non-zero remainder is 13, so the GCD is 13.Next, we use the extended Euclidean algorithm to express 13 as a linear combination of 663 and 182:Starting from the step where the GCD is found:[ 13 = 65 - 52 times 1 ]Substitute ( 52 = 117 - 65 times 1 ):[ 13 = 65 - (117 - 65 times 1) ][ 13 = 65 times 2 - 117 times 1 ]Substitute ( 65 = 182 - 117 times 1 ):[ 13 = (182 - 117 times 1) times 2 - 117 times 1 ][ 13 = 182 times 2 - 117 times 3 ]Substitute ( 117 = 663 - 182 times 3 ):[ 13 = 182 times 2 - (663 - 182 times 3) times 3 ][ 13 = 182 times 11 - 663 times 3 ]Thus, we have ( 13 = 663 times (-3) + 182 times 11 ).The GCD is (boxed{13}), with integers ( u = -3 ) and ( v = 11 ). Therefore, the final answer is:[boxed{13} = 663 times boxed{-3} + 182 times boxed{11}]]

🔑:Let's apply the 2-step algorithm. First, we set a=182 (the smaller of the two numbers) and b=663 (the larger).Step 0: a_{0}=b=663, a_{1}=a=182, u_{0}=v_{1}=0 and v_{0}=u_{1}=1Step 1: a_{0}=663=3 times 182+117 so we set a_{2}=117. With q=3, we apply the formula of the algorithm to find u_{2}=u_{0}-q u_{1}=-3 and v_{2}=v_{0}-q v_{1}=1.Step 2: a_{1}=182=1 times 117+65. We set a_{3}=65, q=1, u_{3}=u_{1}-q u_{2}=4 and v_{3}=v_{1}-q v_{2}=-1.Step 3: a_{2}=117=1 times 65+52. We set a_{4}=52, q=1, u_{4}=u_{2}-q u_{3}=-7 and v_{4}=v_{2}-q v_{3}=2.Step 4: a_{3}=65=1 times 52+13. We set a_{5}=13, q=1, u_{5}=u_{3}-q u_{4}=11 and v_{5}=v_{3}-q v_{4}=-3.Step 5: a_{4}=52=4 times 13+0. Since a_{6}=0, we have found the GCD d=a_{5}=13 and we know that it can be written as13=d=a_{5}=a u_{5}+b v_{5}=11 times 182-3 times 663Thus, d=13 and the solution pair (u, v)=(11,-3).

🧠:[<think>Okay, so I need to find the largest term in the sequence ( a_n = frac{2016^n}{n!} ) where ( n ) is a natural number. Hmm, let's see. I remember that sequences involving factorials and exponentials can have maximum points somewhere before the factorial starts to dominate and the terms decrease. But how exactly do I find where the maximum occurs?First, maybe I should try to understand how the terms behave as ( n ) increases. When ( n ) is 1, the term is ( 2016^1 / 1! = 2016 ). For ( n = 2 ), it's ( 2016^2 / 2! = (2016 times 2016)/2 ). Wait, that's actually larger than the first term. So maybe the sequence increases at first and then decreases? That makes sense because the factorial in the denominator grows faster than the exponential in the numerator as ( n ) gets large. So there should be some value of ( n ) where the sequence reaches its maximum term.To find this maximum, I think I need to compare consecutive terms and see when the sequence changes from increasing to decreasing. That is, find the smallest ( n ) for which ( a_{n+1} < a_n ). That point would be where the maximum occurs.Let me formalize that. The ratio of ( a_{n+1} ) to ( a_n ) is:[frac{a_{n+1}}{a_n} = frac{2016^{n+1}}{(n+1)!} times frac{n!}{2016^n} = frac{2016}{n+1}]So, the ratio ( frac{2016}{n+1} ) tells me whether the next term is larger or smaller than the current term. If this ratio is greater than 1, the next term is larger; if it's less than 1, the next term is smaller.Therefore, the sequence increases when ( frac{2016}{n+1} > 1 ), which implies ( n+1 < 2016 ), so ( n < 2015 ). Conversely, the sequence decreases when ( n+1 > 2016 ), so ( n > 2015 ).Therefore, the sequence increases up to ( n = 2015 ), and then decreases after that. Wait, but hold on. Let me check that again.If ( frac{2016}{n+1} > 1 ), then solving for ( n ):[2016 > n + 1 implies n < 2015]So, when ( n < 2015 ), ( a_{n+1} > a_n ), meaning the terms are increasing. When ( n = 2015 ), the ratio is ( frac{2016}{2016} = 1 ), so ( a_{2016} = a_{2015} ). Then, for ( n > 2015 ), the ratio is less than 1, so the terms are decreasing.Wait, that seems important. If ( n = 2015 ), then ( a_{2016} = a_{2015} times frac{2016}{2016} = a_{2015} ). So actually, the terms at ( n = 2015 ) and ( n = 2016 ) are equal?Let me compute that. Let's take ( a_{2015} = frac{2016^{2015}}{2015!} ) and ( a_{2016} = frac{2016^{2016}}{2016!} ). Then,[a_{2016} = a_{2015} times frac{2016}{2016} = a_{2015} times 1 = a_{2015}]So indeed, ( a_{2015} = a_{2016} ). Therefore, both terms are equal and are the maximum terms of the sequence. So, the largest terms are at ( n = 2015 ) and ( n = 2016 ).But wait, the problem says "the largest term". If there are two consecutive terms that are equal and both are maximum, do we need to report both? The problem says "the largest term", maybe just one, but perhaps both. Let me check the problem statement again.It says "Determine the largest term of the sequence ( a_{n} = frac{2016^{n}}{n!}, n in mathbb{N} )." So, if there are two terms that are equal and both are the maximum, then both are the largest terms. But maybe the question expects the value of the largest term, not the positions. Wait, but maybe the problem is expecting the term (i.e., the value) or the value of ( n ) where it occurs.Wait, the problem says "the largest term of the sequence". So, in mathematical terms, the term is the value ( a_n ). So, if two terms are equal and maximum, then the largest term is that value. So, even if there are two different ( n )s where the maximum occurs, the largest term is that value. However, sometimes in problems like this, they might want both ( n ) values. But in the original problem statement, it's translated as "the largest term", which might refer to the value. But maybe in the context of exams, sometimes they expect the term's position. Hmm.Wait, let me check the original problem statement again: "Determine the largest term of the sequence ( a_{n} = frac{2016^{n}}{n!}, n in mathbb{N} )." So, "term" can refer to the term's value. So if two consecutive terms are equal and are both the maximum, then the largest term is that value. So the answer is ( frac{2016^{2015}}{2015!} ) which is equal to ( frac{2016^{2016}}{2016!} ).But maybe the problem expects the value in terms of ( n ). Wait, the problem might be expecting the answer as the value, but maybe expressed as ( frac{2016^{2016}}{2016!} ), but both expressions are equal. Let me verify:[frac{2016^{2016}}{2016!} = frac{2016 times 2016^{2015}}{2016 times 2015!} = frac{2016^{2015}}{2015!}]Yes, they are equal. Therefore, the largest term is equal at ( n = 2015 ) and ( n = 2016 ), and the value is ( frac{2016^{2015}}{2015!} ).But wait, the problem says "the largest term", so maybe they just need the value. However, in math problems like this, sometimes they want the value of ( n ) where the maximum occurs. Wait, the problem says "the largest term", which is the maximum value of the sequence. So the answer is that maximum value. But since the problem is in Chinese (given the original instruction mentioned Chinese), and translated as "largest term", maybe in Chinese math problems, they might expect the term's position or both. Wait, but the user hasn't specified. Hmm.Wait, the user instruction here is in English, so maybe the problem is translated, but according to the user's problem statement: "Determine the largest term of the sequence ( a_n = frac{2016^n}{n!}, n in mathbb{N} )." So "largest term" would be the maximum value in the sequence. However, given that the maximum occurs at two consecutive values of ( n ), but the term values are equal. Therefore, the largest term is ( frac{2016^{2015}}{2015!} ), which is equal to ( frac{2016^{2016}}{2016!} ).But maybe they want to express it in terms of ( n ), like stating that the largest term occurs at ( n = 2015 ) and ( n = 2016 ). Hmm. Let me check.In mathematical terminology, the "largest term" refers to the maximum value attained by the sequence. So in this case, since both ( a_{2015} ) and ( a_{2016} ) are equal and are the maximum, the largest term is that value. But in some contexts, "term" might refer to the term itself, i.e., ( a_n ), so both ( a_{2015} ) and ( a_{2016} ). However, since the problem asks for "the largest term", not "all largest terms", maybe they just want the value. But in the answer, we need to box the answer. The user instruction says "put your final answer within boxed{}". So, if the answer is the value, how to represent it? It might be messy to write the actual value, as it's a huge number. Alternatively, maybe express it as ( frac{2016^{2015}}{2015!} ). But perhaps in the problem's context, since both terms are equal, they might accept either expression or state both ( n ). Wait, but the problem says "the largest term", so maybe just the value.Alternatively, maybe the question expects ( n ) where the maximum occurs. Wait, but how to be sure? Let me see. If I look for similar problems, for example, when you have ( a_n = frac{x^n}{n!} ), the maximum term occurs around ( n = x ), and if ( x ) is an integer, then ( n = x ) and ( n = x-1 ) might give the same value? Wait, no. Wait, let me check with a smaller number. Let's say ( a_n = frac{3^n}{n!} ). Let's compute terms:- ( a_1 = 3/1 = 3 )- ( a_2 = 9/2 = 4.5 )- ( a_3 = 27/6 = 4.5 )- ( a_4 = 81/24 = 3.375 )So here, maximum terms are at ( n=2 ) and ( n=3 ), both equal to 4.5. So similar to this problem, when the parameter is an integer, the maximum occurs at ( n = x-1 ) and ( n = x ), where ( x ) is the base in the numerator. Wait, in this case, with ( x=3 ), the maximum is at ( n=2 ) and ( n=3 ).Similarly, if we have ( a_n = frac{4^n}{n!} ):- ( a_1 = 4 )- ( a_2 = 16/2 = 8 )- ( a_3 = 64/6 ≈ 10.6667 )- ( a_4 = 256/24 ≈ 10.6667 )- ( a_5 = 1024/120 ≈ 8.5333 )So here, maximum at ( n=3 ) and ( n=4 ). So when ( x=4 ), maximum is at ( n=3 ) and ( n=4 ). Therefore, the general pattern seems that when the numerator is ( k^n ), the maximum terms occur at ( n = k-1 ) and ( n = k ), if ( k ) is an integer.Therefore, in our problem, with ( k = 2016 ), the maximum occurs at ( n = 2015 ) and ( n = 2016 ). Therefore, the largest terms are ( a_{2015} ) and ( a_{2016} ), which are equal. Therefore, if the problem is expecting the answer as the term, which is equal at both these ( n ), but since the problem is in Chinese, perhaps in their terminology, they might accept either ( n=2016 ) or both. Wait, but in the problem statement, the answer is expected to be boxed, which suggests a single answer. Hmm. Wait, in Chinese math problems, when the maximum occurs at two consecutive terms, sometimes they might expect both values of ( n ). However, given that the answer is to be boxed, which usually expects a single expression. Hmm.Alternatively, maybe the problem is expecting ( n = 2016 ), but as we saw, ( a_{2015} = a_{2016} ). Therefore, the answer is both. But how to write that in a boxed notation? Maybe box both numbers. But maybe the problem just wants the value. However, the value is too big to compute, so the answer is expressed in terms of factorials and exponentials. But in the problem statement's original form, perhaps the answer is ( boxed{a_{2016}} ), but since ( a_{2016} = a_{2015} ), they are equal. Alternatively, the answer could be ( boxed{frac{2016^{2016}}{2016!}} ).Alternatively, maybe the problem expects the position ( n ). So, given that both ( n=2015 ) and ( n=2016 ) give the maximum term, but in some textbooks, the answer might just be ( n=2016 ), since it's the point where the terms start decreasing. Hmm, but in reality, both terms are equal. Therefore, strictly speaking, both terms are the maximum. So, perhaps the answer should be that the largest terms are ( a_{2015} ) and ( a_{2016} ), which are equal. However, given that the problem says "the largest term", maybe it's expecting both ( n=2015 ) and ( n=2016 ), but how to present that?Alternatively, since the problem is from an exam, maybe they consider the term with the higher index as the maximum. Wait, but they are equal. So, in that case, both are acceptable. Hmm. Maybe the problem is designed so that even if there are two terms, you just need to state the value, which is equal. So, writing either expression would be acceptable, but they are the same. So, in terms of the problem's answer, writing ( boxed{dfrac{2016^{2015}}{2015!}} ) or ( boxed{dfrac{2016^{2016}}{2016!}} ) is correct. However, since the problem is presented in the form ( frac{2016^n}{n!} ), the answer can be written in either form, but perhaps using the larger factorial in the denominator might be preferred for some reason. But in reality, they are equal. So, to write both, but in the box, you can't write both. Therefore, perhaps the problem expects one of them, and since ( n=2016 ) is an integer, maybe they prefer that. But actually, since the user hasn't specified, but the problem says "the largest term", and since both terms are equal, maybe just writing one of them is acceptable.Alternatively, maybe we can check using calculus. Wait, even though ( n ) is discrete, we can approximate it as a continuous function and find the maximum. Let me try that.Let ( f(x) = frac{2016^x}{x!} ). But the factorial function is defined for integers, but we can use the Gamma function for real numbers, so ( f(x) = frac{2016^x}{Gamma(x+1)} ). To find the maximum of this function, take the derivative and set it to zero. However, differentiating the Gamma function might be complicated, but perhaps we can use logarithmic differentiation.Let ( ln f(x) = x ln 2016 - ln Gamma(x+1) ).Differentiating with respect to ( x ):[frac{f'(x)}{f(x)} = ln 2016 - frac{Gamma'(x+1)}{Gamma(x+1)}]The logarithmic derivative of the Gamma function is the digamma function ( psi(x+1) ). So,[frac{f'(x)}{f(x)} = ln 2016 - psi(x+1)]Setting this equal to zero for critical points:[psi(x+1) = ln 2016]The digamma function ( psi(n) ) for integer ( n ) is approximately ( ln(n) - gamma ), where ( gamma ) is Euler-Mascheroni constant. But for large ( n ), ( psi(n) approx ln(n) - frac{1}{2n} ). So, approximating ( psi(x+1) approx ln(x+1) - frac{1}{2(x+1)} ).Setting equal to ( ln 2016 ):[ln(x+1) - frac{1}{2(x+1)} approx ln 2016]Ignoring the ( frac{1}{2(x+1)} ) term for large ( x ), we get ( x+1 approx 2016 ), so ( x approx 2015 ). Therefore, the maximum occurs around ( x = 2015 ). Therefore, for integer ( n ), the maximum is at ( n = 2015 ) and ( n = 2016 ).Therefore, confirming our previous result. So, the answer is that the largest term occurs at both ( n = 2015 ) and ( n = 2016 ), and the value is ( frac{2016^{2015}}{2015!} ).But given that the problem says "the largest term", and in the context of such problems, even if there are two terms, sometimes people just state the value. Since both terms are equal, either expression is acceptable. However, if the answer requires the term, then both ( a_{2015} ) and ( a_{2016} ) are correct. However, since the problem is presented in Chinese (assuming from the original instruction), and given that in Chinese math problems, they might expect the value expressed in the original terms, so either ( frac{2016^{2015}}{2015!} ) or ( frac{2016^{2016}}{2016!} ). But these are equal. So, either one is acceptable.Alternatively, if the problem expects the value of ( n ), then both ( n = 2015 ) and ( n = 2016 ). But the problem says "the largest term", which is the value. Hmm. So perhaps writing ( boxed{dfrac{2016^{2016}}{2016!}} ) would be appropriate, but it's the same as ( boxed{dfrac{2016^{2015}}{2015!}} ).But let me check with the ratio again. If ( a_{n+1}/a_n = 2016/(n+1) ), then when ( n+1 = 2016 ), the ratio is 1. Therefore, ( a_{2016} = a_{2015} ). So, both terms are equal. Therefore, the maximum term is both. However, since the problem says "the largest term", which is singular, but in reality, there are two terms. Maybe in the problem's original language, the wording might account for that, but in English, it's ambiguous. If forced to write one answer, I would write both ( n = 2015 ) and ( n = 2016 ), but given that the answer needs to be boxed, maybe box both numbers. But in standard practice, boxed answers are usually single expressions.Alternatively, maybe the problem considers the term with the greater ( n ), which is 2016, as the maximum. But in reality, they are equal. Alternatively, the problem might want both ( n )s as the answer. Since I can't be sure, but given that mathematical definitions would recognize both as correct, perhaps the answer is that the largest term occurs at both ( n = 2015 ) and ( n = 2016 ), and the value is equal. However, given the need to box the answer, perhaps box the two values: ( boxed{2015} ) and ( boxed{2016} ). But the user might expect one boxed answer. Alternatively, state both in a single box: ( boxed{2015 text{ and } 2016} ). But I need to check common conventions.In mathematical literature, when a sequence has two consecutive terms that are equal maxima, it's common to state both indices. For example, in the sequence ( a_n = frac{k^n}{n!} ), the maximum occurs at ( n = k-1 ) and ( n = k ). Therefore, in this case, the answer is both ( 2015 ) and ( 2016 ). However, if the problem is from an exam expecting a single answer, perhaps they just want ( 2016 ). But since they are equal, strictly speaking, both are correct. Given the ambiguity, I think the correct answer is both ( n = 2015 ) and ( n = 2016 ), but since the problem says "the largest term", not "all largest terms", maybe the term itself is the answer, which is equal at both positions. But writing the term in the box would be too cumbersome. Alternatively, in some textbooks, they might just report the value of ( n ) where the maximum occurs, which in this case is both 2015 and 2016.Alternatively, perhaps the problem expects the answer in terms of ( n ), so stating both values. But how to write that in a box? Maybe separate boxes, but the user instruction says "put your final answer within boxed{}", so probably a single box.Alternatively, maybe the problem expects just the value, which can be written as ( frac{2016^{2015}}{2015!} ), but that's a huge number. Alternatively, since the problem is multiple choice, but the user hasn't indicated options, so perhaps the answer is supposed to be the value, but since it's too large, the answer is expressed as ( boxed{frac{2016^{2015}}{2015!}} ).Alternatively, in the Chinese curriculum, when they ask for the largest term, they might expect the term number, i.e., ( n ), so 2015 and 2016. But given that the problem statement is in English here, maybe it's better to go with the mathematical convention. Since the question says "Determine the largest term", which is the maximum value of ( a_n ), so the answer is ( frac{2016^{2015}}{2015!} ), which is equal to ( frac{2016^{2016}}{2016!} ). Therefore, either expression is acceptable, but since ( frac{2016^{2016}}{2016!} ) might look more straightforward because the exponent and factorial have the same index, maybe that's preferred.But to check, let's compute ( a_{2015} ) and ( a_{2016} ):( a_{2015} = frac{2016^{2015}}{2015!} )( a_{2016} = frac{2016^{2016}}{2016!} = frac{2016 times 2016^{2015}}{2016 times 2015!} = frac{2016^{2015}}{2015!} )Therefore, they are indeed equal. So, the answer is ( frac{2016^{2015}}{2015!} ), which is equal to ( frac{2016^{2016}}{2016!} ). Therefore, either form is correct, but perhaps ( frac{2016^{2016}}{2016!} ) is more symmetric.Alternatively, the problem might accept the answer as ( boxed{2016} ), but that would be incorrect because the term at ( n = 2015 ) is equal. However, depending on the problem's expectation, maybe they just want the integer part of ( 2016 ), but that's not right. Alternatively, perhaps there's a rounding consideration. Wait, in some cases, if the maximum of a continuous function is at ( x = 2015.5 ), then the discrete maximum would be at ( n = 2015 ) and ( n = 2016 ). But that's exactly what we found here.Therefore, given the problem statement and standard mathematical conventions, the correct answer is that the largest term occurs at both ( n = 2015 ) and ( n = 2016 ), and the value is ( frac{2016^{2015}}{2015!} ). However, given the instruction to put the final answer in a box, and considering the possible expectations, I think the problem expects the value. But since it's equal at both positions, writing either expression is acceptable. Alternatively, if we are to follow the most reduced form, ( frac{2016^{2015}}{2015!} ) is simpler because ( 2016! = 2016 times 2015! ), so writing ( frac{2016^{2015}}{2015!} ) is slightly more simplified. Therefore, I will present that as the answer.Final AnswerThe largest term of the sequence is boxed{dfrac{2016^{2015}}{2015!}}.</think>To determine the largest term of the sequence ( a_n = frac{2016^n}{n!} ) where ( n in mathbb{N} ), we analyze the ratio of consecutive terms:[frac{a_{n+1}}{a_n} = frac{2016^{n+1}}{(n+1)!} times frac{n!}{2016^n} = frac{2016}{n+1}]This ratio tells us whether the sequence is increasing or decreasing. The sequence increases when ( frac{2016}{n+1} > 1 ) (i.e., ( n < 2015 )) and decreases when ( frac{2016}{n+1} < 1 ) (i.e., ( n > 2015 )). When ( n = 2015 ), the ratio is 1, meaning ( a_{2016} = a_{2015} ).Thus, the terms ( a_{2015} ) and ( a_{2016} ) are equal and are the largest terms in the sequence. Calculating these terms:[a_{2015} = frac{2016^{2015}}{2015!}][a_{2016} = frac{2016^{2016}}{2016!} = frac{2016 times 2016^{2015}}{2016 times 2015!} = frac{2016^{2015}}{2015!}]Both terms are equal, confirming that the largest term is (frac{2016^{2015}}{2015!}).[boxed{dfrac{2016^{2015}}{2015!}}],<think>Okay, so I need to find the largest term in the sequence ( a_n = frac{2016^n}{n!} ) where ( n ) is a natural number. Hmm, let's see. I remember that sequences involving factorials and exponentials can sometimes have a maximum term before they start decreasing. But I need to recall how to determine exactly where that maximum occurs.First, maybe I can look at the ratio of consecutive terms. That often helps in sequences like these. So, let's compute ( frac{a_{n+1}}{a_n} ) and see how it behaves. If the ratio is greater than 1, the sequence is increasing, and if it's less than 1, the sequence is decreasing. The maximum term should occur around when this ratio changes from greater than 1 to less than 1.Let me write that ratio out:[frac{a_{n+1}}{a_n} = frac{2016^{n+1}}{(n+1)!} times frac{n!}{2016^n} = frac{2016}{n+1}]Okay, that simplifies nicely. So the ratio is ( frac{2016}{n+1} ). Now, we need to find when this ratio transitions from being greater than 1 to less than 1. That is, solve for ( n ) when ( frac{2016}{n+1} = 1 ). Let's do that:[frac{2016}{n+1} = 1 implies n + 1 = 2016 implies n = 2015]Wait, so when ( n = 2015 ), the ratio is exactly 1. That means that ( a_{2016} = a_{2015} times frac{2016}{2015 + 1} = a_{2015} times 1 ). So the terms ( a_{2015} ) and ( a_{2016} ) are equal? Hmm, that's interesting. But if the ratio is 1 at ( n = 2015 ), then before that, when ( n < 2015 ), the ratio ( frac{2016}{n+1} ) is greater than 1, meaning the sequence is increasing. And after ( n = 2015 ), the ratio becomes less than 1, so the sequence starts decreasing. Therefore, the maximum term(s) occur at ( n = 2015 ) and ( n = 2016 ), since those two terms are equal.But wait, the problem asks for the largest term. If two consecutive terms are equal and they are the maximum, then both are the largest. But the problem says "the largest term", so maybe they are equal? Let me check with actual numbers. Let's compute ( a_{2015} ) and ( a_{2016} ).But computing factorials of numbers that large is impractical. Maybe there's another way to confirm they are equal. Let's see:( a_{2016} = frac{2016^{2016}}{2016!} )( a_{2015} = frac{2016^{2015}}{2015!} )Let's compute the ratio ( frac{a_{2016}}{a_{2015}}} = frac{2016}{2016} = 1 ). So yes, they are indeed equal. Therefore, both ( a_{2015} ) and ( a_{2016} ) are the largest terms. However, sometimes in such problems, they might expect the smallest n where the maximum occurs or just one term. Let me check if the problem allows for multiple answers or expects a single one. The problem says "the largest term", so perhaps both terms are acceptable. But I need to check if they are indeed equal.Wait, let's think again. The ratio ( a_{n+1}/a_n = 2016/(n+1) ). So when n = 2015, this ratio is 2016/2016 = 1, so ( a_{2016} = a_{2015} ). Therefore, both terms are equal. Therefore, the maximum term occurs at both n = 2015 and n = 2016. However, depending on the problem's wording, sometimes people refer to the term with the higher index as the peak, but in reality, both are maximums. But since the problem says "the largest term", and both are equally large, perhaps both should be mentioned. But maybe I made a mistake here. Let me verify with a simpler example.Suppose we have ( a_n = frac{2^n}{n!} ). Let's compute terms:n=1: 2/1=2n=2: 4/2=2n=3: 8/6≈1.333So here, both n=1 and n=2 give the maximum term. Similarly, for 2016, since when n=2015 and n=2016, the terms are equal, both are maximum. Therefore, the answer should be both n=2015 and n=2016. However, the problem says "the largest term", which is a bit ambiguous. Maybe in the context of natural numbers, they might expect the term where the sequence stops increasing, which is n=2015, but since the next term is equal, both are maximum.Wait, but in the problem statement, it's specified as n ∈ ℕ. Depending on the convention, sometimes ℕ starts at 1, sometimes at 0. But since n is in the factorial, n=0 would give 0! =1, but 2016^0=1, so term would be 1. But in this case, n is likely starting from 1. However, even if n starts at 1, the maximum terms are at 2015 and 2016.Wait, but let me confirm with the ratio. Let's suppose n=2015:a_{2015} = 2016^{2015}/2015!a_{2016} = 2016^{2016}/2016! = (2016^{2015} * 2016)/(2015! * 2016) = a_{2015}Yes, exactly, they are equal. So both terms are equal and the largest. So the answer is that both terms at n=2015 and n=2016 are the largest.But the problem says "the largest term". Hmm. Maybe in the problem's context, even though they are equal, the answer expects both n=2015 and n=2016. However, in some textbooks or resources, when this happens, they might say the maximum occurs at n=2016 or n=2015? Wait, no, because they are equal.Wait, let me check with another example. If the ratio is 1 at n=2015, then a_{n+1} = a_n. So when the ratio is exactly 1, the terms are equal. So in such a case, both terms are maximum. Therefore, the answer is that the largest terms are a_{2015} and a_{2016}. But the problem says "the largest term", so maybe they are equal, so either one is acceptable. But perhaps the answer is n=2016? Wait, but according to the calculation, they are equal.Wait, perhaps I need to check when the sequence stops increasing. Since the ratio is greater than 1 when n+1 < 2016, so when n < 2015, the ratio is greater than 1, so the sequence is increasing up to n=2015. Then at n=2015, the ratio is 1, so the term stays the same. Then for n >=2016, the ratio is less than 1, so the terms decrease. So the maximum terms are at n=2015 and n=2016. Therefore, both are the largest.But in the problem statement, it's possible that the answer is n=2016, but since they are equal, both are correct. However, maybe in the problem's original source or in the problem's expectation, the answer is n=2016? Hmm, but according to the calculation, both are equally maximum.Wait, perhaps there's a mistake in my ratio calculation. Let me check again:( a_{n} = frac{2016^n}{n!} )Then ( frac{a_{n+1}}{a_n} = frac{2016^{n+1}}{(n+1)!} times frac{n!}{2016^n} = frac{2016}{n+1} ). That's correct.So when n + 1 = 2016, the ratio is 1, which is when n = 2015. Therefore, a_{2015} and a_{2016} are equal. Therefore, the maximum term is at both n=2015 and n=2016. But perhaps the problem expects the answer as n=2016? Wait, perhaps due to the integer nature of n, but if both terms are equal, they should both be considered.Alternatively, maybe the problem is expecting the value of the largest term, not the index. Wait, but the problem says "Determine the largest term of the sequence", so perhaps they want the value. Wait, but the original problem (1A) says "Determine the largest term of the sequence a_n = 2016^n /n! , n ∈ ℕ." So they might want the term, which would be a_{2015} (or a_{2016}), but since they are equal, either one is correct. But perhaps in the answer, we need to specify the term, but how?Wait, the problem is in Chinese maybe? Wait, no, the user wrote the problem in English. So, given that, the answer is that the largest term is at n=2015 and n=2016, both being equal. But in the answer, they might need to put both in boxes? But the user instruction says "put your final answer within boxed{}". So perhaps the answer is n=2016? Or n=2015 and 2016?Wait, maybe I need to check for possible miscalculations here. Let me verify with n=2014, n=2015, n=2016.Compute the ratio at n=2014: 2016/(2014 +1) = 2016/2015 ≈1.00049..., which is greater than 1, so a_{2015} > a_{2014}At n=2015: ratio is 2016/2016 =1, so a_{2016} = a_{2015}At n=2016: ratio is 2016/2017 ≈0.9995..., which is less than 1, so a_{2017} < a_{2016}Therefore, the terms increase up to n=2015, then stay the same at n=2016, then decrease. Therefore, both a_{2015} and a_{2016} are the maximum terms.But the problem says "the largest term", which is in singular. Hmm. Maybe in this case, the term refers to the value, not the index. Since both terms have the same value, it's just one value. So the largest term is the value at both n=2015 and n=2016. However, since the problem asks for the term, perhaps they are expecting the answer as the value, but since the value is the same for both, it's just one value. So maybe the answer is the value, which is equal for both n=2015 and n=2016.But how do we compute that value? It's ( frac{2016^{2015}}{2015!} ) or ( frac{2016^{2016}}{2016!} ), which are equal. But calculating these is obviously impractical. So in the answer, do we just state that the largest term occurs at n=2015 and n=2016, and they are equal? But the problem says "Determine the largest term", so maybe we need to express it as boxed{a_{2015}} and boxed{a_{2016}}? But the problem might expect just the index. Wait, the problem says "the largest term", so term refers to the value. However, since the value is the same for both n=2015 and n=2016, we can write both, but the user instruction says "put your final answer within boxed{}". So perhaps the answer is that the largest term is achieved at both n=2015 and n=2016. But how to box that? Maybe box both numbers. But in the problem statement, if it's from an Olympiad or exam, sometimes they might accept either one, but in this case, since they are equal, both are correct.Alternatively, maybe the problem expects the answer to be n=2016, but according to the calculation, they are equal. Hmm. Wait, maybe in the problem statement, n is supposed to be a positive integer, so starting from 1. So n=2015 and 2016 are both valid. Therefore, the largest terms are at both positions. But the problem says "the largest term", so maybe the answer is both n=2015 and n=2016, but since the problem is in Chinese (the user's instruction might be translated), the original problem might have a different expectation. Alternatively, perhaps the problem is expecting only n=2016. Let me check with another approach.Alternatively, sometimes, the maximum term occurs around n ≈ x, where x is the value where the ratio changes. In this case, x=2016. But when dealing with factorial sequences, the maximum term is usually around n ≈ x, where x is the parameter in the exponential. So in this case, 2016. However, because of the discrete nature of n, sometimes it's either floor(x) or ceil(x). But in this case, when x is exactly 2016, then n=2015 and 2016 give the same term. So in this case, both are maximum.Alternatively, perhaps using calculus for continuous variables. If we treat n as a continuous variable, take the derivative and set it to zero. Let's try that for the sake of thoroughness.Let me consider the function ( f(n) = frac{2016^n}{n!} ). To find its maximum, take the derivative with respect to n. But since n is discrete, this is an approximation. Using the gamma function for factorial, Γ(n+1) = n! for integer n. However, differentiating the gamma function is complicated, but we can use the ratio approach, which we already did.Alternatively, using Stirling's approximation: ( n! approx n^n e^{-n} sqrt{2 pi n} ). So, substituting into ( a_n ):[a_n approx frac{2016^n}{n^n e^{-n} sqrt{2 pi n}} = frac{(2016 e)^n}{n^n sqrt{2 pi n}} = left( frac{2016 e}{n} right)^n frac{1}{sqrt{2 pi n}}]Taking logarithm to make it easier:[ln a_n approx n ln left( frac{2016 e}{n} right) - frac{1}{2} ln(2 pi n)]To find the maximum, take derivative with respect to n:Let ( f(n) = n ln left( frac{2016 e}{n} right) - frac{1}{2} ln(2 pi n) )Then,[f'(n) = ln left( frac{2016 e}{n} right) + n times left( -frac{1}{n} right) - frac{1}{2} times frac{1}{n}]Simplifying:[f'(n) = ln left( frac{2016 e}{n} right) - 1 - frac{1}{2n}]Set derivative to zero:[ln left( frac{2016 e}{n} right) - 1 - frac{1}{2n} = 0]Simplify:[ln left( frac{2016 e}{n} right) = 1 + frac{1}{2n}][ln(2016) + ln(e) - ln(n) = 1 + frac{1}{2n}][ln(2016) + 1 - ln(n) = 1 + frac{1}{2n}]Subtract 1 from both sides:[ln(2016) - ln(n) = frac{1}{2n}][lnleft( frac{2016}{n} right) = frac{1}{2n}]This equation is transcendental and can't be solved exactly, but we can approximate. Since n is around 2016, ( frac{1}{2n} ) is very small. Therefore, approximately:[lnleft( frac{2016}{n} right) approx 0]Which implies ( frac{2016}{n} approx 1 implies n approx 2016 ). Therefore, the maximum is around n=2016. However, considering the previous ratio approach, when n=2015, the ratio ( frac{2016}{n+1} =1 ), so the maximum occurs at both n=2015 and n=2016. Therefore, the Stirling approximation gives n≈2016, but due to the discrete nature, the maximum is at both 2015 and 2016.Therefore, combining both methods, the answer is that the largest term occurs at both n=2015 and n=2016.But the problem says "the largest term", which is singular. But in the sequence, there are two terms with the same maximum value. Therefore, maybe the answer is both. However, in some contexts, the term with the higher index is considered the peak, but in reality, they are equal. So to be precise, both terms are the largest.But since the problem is asking for the largest term, not the index, but the value. However, the value is the same for both n=2015 and n=2016, so we can just express it as ( frac{2016^{2015}}{2015!} ), which is equal to ( frac{2016^{2016}}{2016!} ). But writing that in a box is not practical. Therefore, perhaps the answer is n=2016, but according to the ratio method, n=2015 and n=2016 give equal terms.Alternatively, maybe the problem expects the answer to be n=2016. Let me check online for similar problems. For example, in the case of the sequence ( a_n = frac{k^n}{n!} ), the maximum term occurs around n=k. When k is an integer, the maximum occurs at n=k-1 and n=k. For example, with k=2, we saw that n=1 and n=2 both have the same term. So yes, when k is an integer, the maximum occurs at n=k-1 and n=k. Therefore, in this case, since k=2016 is an integer, the maximum occurs at n=2015 and n=2016.Hence, the answer is that both terms at n=2015 and n=2016 are the largest. However, given that the problem asks for "the largest term", and in mathematical terminology, even if multiple terms are equally maximum, they are all considered the largest terms. Therefore, the answer is both n=2015 and n=2016.But how to present this in a box? The user instruction says "put your final answer within boxed{}". If we need to box two numbers, perhaps boxed{2015} and boxed{2016}. But the standard practice in such cases, when two terms are equal and maximum, is to state both. However, depending on the problem's expectation, they might require only one. But according to the mathematical reasoning, both are correct. Therefore, to be precise, the answer is both 2015 and 2016. So maybe box them together? Like boxed{2015} and boxed{2016}. But in some contexts, they might write boxed{2015} text{ and } boxed{2016}. But the user might expect a single box. Alternatively, if the question allows multiple answers, but since it's 1A, likely expects a single answer. Wait, perhaps the problem is in Chinese, maybe from an Asian exam, where sometimes for such sequences, they consider the term before the ratio becomes less than 1 as the maximum. But according to our ratio test, a_{2016} is equal to a_{2015}, so both are maximum.Alternatively, let's consider n=2015:a_{2015} = 2016^{2015}/2015!a_{2016} = 2016^{2016}/2016! = (2016/2016) * 2016^{2015}/2015! = a_{2015}Therefore, indeed, they are equal. Therefore, both are maximum. So the answer is both n=2015 and n=2016.But the problem says "the largest term", which is singular. So maybe there's a mistake in my calculation? Wait, no. If two terms are equal, they are both the largest. For example, in a sequence like 1, 3, 5, 5, 4, 3, the largest term is 5, occurring at positions 3 and 4. Similarly here.Therefore, the answer is that the largest terms are at positions 2015 and 2016. However, given the problem's phrasing, maybe the expected answer is 2016? But according to the math, both are correct. Hmm. Maybe the original problem had a non-integer parameter, leading to a single maximum, but in this case, since 2016 is an integer, it's two terms.Alternatively, since the problem is part 1A, maybe it's expecting 2016 as the answer. But in reality, the correct answer is both. But since the user is asking for the largest term, maybe they accept both. But how to present that in a boxed answer.Wait, perhaps in the problem's original source, the answer is 2016, but according to the mathematical analysis, both are correct. Therefore, to be precise, the correct answer is that both n=2015 and n=2016 give the largest term. However, given the problem's instructions, if the user expects a single boxed number, perhaps 2016. But since I need to be accurate, I should state both.But in the initial problem statement, the user wrote "put your final answer within boxed{}". If they expect a single value, but there are two, this is conflicting. However, in mathematical problems, when two terms are equal and maximum, both are considered correct. Therefore, the answer is both 2015 and 2016. Therefore, in the box, I should write both numbers. But how? Maybe separated by a comma.Alternatively, the problem might consider that the maximum term is achieved when n=2016, since after that it decreases. But according to the calculation, they are equal. So perhaps the answer is 2016. But I need to check with the original problem's intention.Wait, another approach: sometimes, in such sequences, even if two consecutive terms are equal, the later one is considered the peak. But in reality, they are the same. For example, in the simple case with a_n = 2^n /n!, we saw that a_1=2 and a_2=2. So both are peaks. So the answer would be n=1 and n=2.Therefore, similarly here, the answer is n=2015 and n=2016. Hence, the largest terms are at those positions.But since the user instruction says "put your final answer within boxed{}", maybe they accept multiple boxed numbers. So I can write boxed{2015} and boxed{2016}. Alternatively, if the system allows, separate boxes. But according to the problem's instruction, it's likely expecting a single answer, but given the mathematical truth, it's two answers.Hmm. Given that, perhaps the answer is 2016, but according to strict calculation, it's 2015 and 2016. So to be accurate, I need to state both.Alternatively, if the problem is in Chinese, maybe from the Chinese Olympiad, and the answer is 2016. But since my calculation shows they are equal, I need to go with the mathematical conclusion.Therefore, the answer is that the largest term occurs at both n=2015 and n=2016, so both are correct. Therefore, I will present both in boxed notation.Final AnswerThe largest terms occur at ( n = 2015 ) and ( n = 2016 ), so the answer is boxed{2015} and boxed{2016}.However, if the problem expects a single answer, given that ( a_{2015} = a_{2016} ), but the transition happens at ( n = 2015 ), then perhaps boxed{2016} is also acceptable. But strictly speaking, both are correct. Given the ambiguity in the problem's wording, but following mathematical rigor, both should be boxed.But since the user might expect a single value, and seeing that similar problems sometimes take the integer part when the parameter is a whole number, but in this case, since the ratio equals 1 exactly at n=2015, leading to equal terms at 2015 and 2016. Therefore, the correct answer is both. So I will present both boxed.Final Answerboxed{2015} text{ and } boxed{2016}But since LaTeX doesn't allow text inside the box, maybe separate boxes:Final AnswerThe largest terms are boxed{2015} and boxed{2016}.</think>To determine the largest term of the sequence ( a_n = frac{2016^n}{n!} ) where ( n in mathbb{N} ), we analyze the ratio of consecutive terms:[frac{a_{n+1}}{a_n} = frac{2016^{n+1}}{(n+1)!} times frac{n!}{2016^n} = frac{2016}{n+1}]We need to find when this ratio transitions from being greater than 1 to less than 1. Setting the ratio equal to 1:[frac{2016}{n+1} = 1 implies n + 1 = 2016 implies n = 2015]For ( n < 2015 ), the ratio ( frac{2016}{n+1} ) is greater than 1, indicating the sequence is increasing. For ( n geq 2015 ), the ratio is less than or equal to 1, indicating the sequence is decreasing. At ( n = 2015 ), the ratio is exactly 1, meaning ( a_{2016} = a_{2015} ). Therefore, the terms ( a_{2015} ) and ( a_{2016} ) are equal and are the largest terms in the sequence.Thus, the largest terms of the sequence occur at ( n = 2015 ) and ( n = 2016 ).[boxed{2015} text{ and } boxed{2016}]]

🔑:Solution. Odfrac{a_{n+1}}{a_{n}}=frac{2016}{n+1}it follows that the sequence grows up to the 2015th term which is equal to the 2016th, and then decreases. Therefore, the largest terms of the sequence are a_{2015} and a_{2016}.

🧠:[<think>Okay, so I need to find the zero of the function f(x) = 2^x - 8. Hmm, let's see. A zero of a function is where the function crosses the x-axis, right? That means the y-value is 0 at that point. So, I need to solve for x when f(x) = 0. Let me write that equation down: 2^x - 8 = 0. To solve for x, I can add 8 to both sides. That would give me 2^x = 8. Now, I need to figure out what exponent x makes 2 raised to that power equal to 8. I remember that 2^3 is 8 because 2*2*2 is 8. So, x should be 3. Let me check that again. If x is 3, then 2^3 is 8, and 8 - 8 is 0. Yeah, that works. So, the zero is at x = 3. Looking at the answer choices: A is 3, B is (3,0), C is 4, D is (4,0). Wait, the question is asking for the zero of the function. The zero is the x-value where the function is zero, so it's just x=3. But sometimes people might refer to the point as (3,0). Let me check the options again.Option A is 3, which is the x-intercept's x-coordinate. Option B is the point (3,0). The question says "the zero of the function," which in math usually refers to the x-value that makes the function zero. So, I think A is the correct answer. But maybe the problem considers the zero as the point. Let me think. In some contexts, especially in graphing, the zero might be referred to as the point where the graph crosses the x-axis. So that would be (3,0). But in algebra, when they say "find the zero," they often just want the x-value. Let me check the problem statement again. It says "the zero of the function f(x)=2^x -8 is (　)." The options include both numbers and points. If the answer is A:3, that's the x-value. If it's B:(3,0), that's the point. So which one is correct? Let me recall. The term "zero" of a function generally refers to the value of x for which f(x)=0. So the zero is x=3. Therefore, the answer should be A. But sometimes in certain textbooks or contexts, they might refer to the entire point. Hmm. Let me verify with an example. If we have f(x) = x - 2, the zero is x=2, right? So the x-intercept is (2,0). So, in that case, if the question is about the zero, it's x=2. If it's about the x-intercept, it's the point. Given that the question is about the zero, which is the root of the equation f(x)=0, the answer is x=3. So, option A. However, let's check the options again. Option B is a coordinate point. Maybe the question is from a source that conflates zeros and x-intercepts. Wait, the problem is presented as multiple-choice, so maybe I need to see the options. The function is given, and the answer is one of A, B, C, D. The options are A:3, B:(3,0), C:4, D:(4,0). So, x=3 is A, and the point is B. But the question says "the zero of the function." If the function is real-valued, then the zero is the x value. So, if the answer is 3, which is A. But maybe depending on the teacher or textbook, sometimes they might expect the coordinate. But in standard terminology, the zero is the x-value. For example, in precalculus, when you find zeros of a function, you solve for x when f(x)=0. So the zeros are x=..., which are numbers. The x-intercepts are points (x,0). So, maybe the answer is A. But let me verify once more. Let's solve 2^x -8=0. 2^x=8. 2^3=8, so x=3. So, x=3 is the solution, which is the zero. Therefore, the answer is A. But why is there an option C:4? Maybe someone might miscalculate. For example, if they thought 2^4 is 16, so 16-8=8, which is not zero. So 4 is not correct. Similarly, D is (4,0), which is also wrong. Therefore, the correct answer should be A:3. However, if the question is from a source where they consider the zero as the coordinate, then B would be correct. But in standard mathematical terms, the zero is x=3, so A is correct. Wait, but in some contexts, the "zero" can refer to the x-intercept, which is the point. For example, in graphing, you might say "plot the zeros," which would be points on the graph. But in algebraic terms, when solving equations, the zero is the x value. So the answer depends on the context. But since the question is presented in a multiple-choice format with both options, and the function is f(x)=2^x -8, perhaps the answer is A. Let me check similar problems. For example, if the function is f(x)=x - 5, the zero is x=5, so answer is 5, not (5,0). So, in that case, the answer is the x-value. Therefore, following that reasoning, the answer should be A:3. So, I think A is correct. Therefore, the answer is A.Final Answerboxed{A}</think>To find the zero of the function ( f(x) = 2^x - 8 ), we need to solve for ( x ) when ( f(x) = 0 ).Starting with the equation:[2^x - 8 = 0]Add 8 to both sides:[2^x = 8]Recognizing that ( 8 ) is ( 2^3 ), we rewrite the equation as:[2^x = 2^3]Since the bases are the same, the exponents must be equal:[x = 3]The zero of the function is the value of ( x ) that makes ( f(x) = 0 ), which is ( x = 3 ). Therefore, the correct answer is (boxed{A}).,<think>Okay, let's see. I need to find the zero of the function f(x) = 2^x - 8. Hmm, zero of a function usually means the x-value where the function equals zero, right? So I need to solve for x when 2^x - 8 = 0. Let me write that equation down: 2^x = 8. Wait, 2 raised to what power gives 8? Well, 2 cubed is 8 because 2*2=4, 4*2=8. So 2^3 = 8. That means x should be 3. Let me check that. If x is 3, then f(3) = 2^3 - 8 = 8 - 8 = 0. Yep, that works. So the zero is at x=3. Looking at the answer choices, option A is 3, B is (3,0), C is 4, D is (4,0). The question is asking for the zero, which is the x-intercept. But sometimes people get confused whether it's just the x-value or the coordinate point. If the function crosses the x-axis at x=3, then the point would be (3,0). But the term "zero" usually refers to the x-value that makes the function zero. Let me confirm that. In math, a zero of a function is a point where the function's value is zero. So it's the x-value, but sometimes people might refer to the point (x,0) as the zero. Wait, actually, no. The zero is the x-value, and the x-intercept is the point (x,0). Let me check that. For example, if we have a function f(x), the zeros are the solutions to f(x) = 0. So the zeros are x-values. The x-intercepts are the points (x, 0) on the graph. So maybe the question is using "zero" to mean the x-value. So if that's the case, then the answer should be A:3. But if they are considering the zero as the intercept point, then it's (3,0), which is B. Hmm. But let's look at the answer choices again. The options are A:3, B:(3,0), C:4, D:(4,0). The function is f(x) = 2^x -8. Let me verify again. Setting 2^x -8 =0, so 2^x =8, which is 2^3=8. So x=3. So if the question is asking for the zero, which is x=3, the answer is A. However, if they are asking for the x-intercept, which is a point, then it's (3,0). So why is there an option B and D as points? Wait, the question says: "The zero of the function f(x)=2^{x}-8 is (　)." So the wording is "the zero". In mathematics, the term "zero" can sometimes refer to the x-value that makes the function zero, but sometimes people might refer to the entire point. So which is it here? Let me check a math source. For example, according to standard definitions, a "zero" of a function is a value x such that f(x)=0. So the zero is x=3. But when graphing, the point (3,0) is where the graph crosses the x-axis, called an x-intercept. So maybe the answer is A. However, depending on how the question is phrased, sometimes in some contexts, people might refer to the intercept as the zero. But strictly speaking, the zero is the x-value. But wait, in the answer choices, they have both numbers and coordinate pairs. So since 3 is an option (A) and (3,0) is an option (B), and similarly for 4 and (4,0). So the problem is testing whether the student knows that the zero is the x-value, not the point. Therefore, the answer should be A:3. But wait, let me check if I did the calculation correctly. Maybe x=4? Let me compute 2^4. 2^4 is 16. 16-8=8, which is not zero. So x=4 is not a zero. Therefore, C and D are out. So between A and B. So in the problem statement, they say "the zero of the function". If they mean the x-intercept, then it's (3,0). If they mean the x-value, it's 3. Depending on the convention. But I think in most cases, the zero is the x-value. Let me check another example. For instance, if f(x) = x - 2, then the zero is x=2, and the x-intercept is (2,0). So in that case, the zero is 2. So similarly here, the zero should be 3, which is option A. But sometimes, teachers might use "zero" to mean the intercept. But given that the options are both, the answer is likely A. However, let me check some references. For example, according to the textbook Stewart Calculus, a zero of a function is a number x for which f(x) =0. Therefore, the zero is the x-value. Therefore, the answer is 3, which is option A. Wait, but the problem is presented in a multiple-choice format. Let me check the options again. If the answer is A:3, which is the x-value, but maybe the question is from a course where they teach that zeros are points? Hmm. Maybe not. Typically, zeros are x-values. So I think the answer is A:3. Alternatively, maybe I made a mistake in solving 2^x =8. Let me verify once more. 2^1=2, 2^2=4, 2^3=8. Yes, 2^3 is 8, so x=3. So the correct answer is 3. Therefore, option A. So why is option B there? Maybe as a distractor if someone confuses zero with x-intercept. So the answer is A.Final Answerboxed{A}</think>To find the zero of the function ( f(x) = 2^x - 8 ), we need to solve for ( x ) when ( f(x) = 0 ). Starting with the equation:[2^x - 8 = 0]we add 8 to both sides:[2^x = 8]Recognizing that 8 is ( 2^3 ), we rewrite the equation as:[2^x = 2^3]Since the bases are the same, the exponents must be equal:[x = 3]The zero of the function is the x-value that makes the function zero, which is ( x = 3 ). Although the x-intercept is the point ( (3, 0) ), the term "zero" refers to the x-value. Therefore, the correct answer is (boxed{A}).]

🔑:To find the zero of the function f(x)=2^{x}-8, we solve 2^{x}-8=0,and we get x=3.Therefore, the correct choice is boxed{A}.By utilizing the relationship between the zero of a function and the root of an equation, we can solve for the root of the equation.This question tests the application of the zero determination theorem and the method of finding the roots of equations, examining computational skills.

🧠:[<think>Okay, let me try to work through this problem step by step. So, the first thing I need to do is express the scalar product (dot product) of vectors r1 and r2 in coordinates. I remember that the dot product is calculated by multiplying the corresponding components of the vectors and then adding them up. So, if r1 is x1i + y1j + z1k and r2 is x2i + y2j + z2k, then their dot product should be x1*x2 + y1*y2 + z1*z2. Let me double-check that. Yeah, because each component is multiplied by the corresponding component of the other vector, and then summed. So that part seems straightforward.Next, I need to calculate cos φ, where φ is the angle between the vectors r1 and r2. Hmm, I recall that the formula for the cosine of the angle between two vectors is the dot product of the vectors divided by the product of their magnitudes. So, cos φ = (r1 · r2) / (|r1| |r2|). Right? Let me confirm that. The dot product formula can also be written as |r1||r2|cos φ, so solving for cos φ gives exactly that. So substituting the dot product we found earlier, this would be (x1x2 + y1y2 + z1z2) divided by (sqrt(x1² + y1² + z1²) * sqrt(x2² + y2² + z2²)). That makes sense.Now, the conditions for collinearity and perpendicularity. Let's start with collinearity. Collinear vectors mean they are scalar multiples of each other, right? So one vector is a scalar multiple of the other. In coordinates, this would mean that each component of r1 is the same scalar multiple of the corresponding component of r2. So, x1 = k*x2, y1 = k*y2, z1 = k*z2 for some scalar k. Alternatively, the cross product of the two vectors should be zero if they are collinear. But in coordinate terms, it's the proportionality of the components. So, the ratios x1/x2 = y1/y2 = z1/z2, assuming none of the denominators are zero. But if some components are zero, we have to be careful. For example, if x2 is zero, then x1 must also be zero, and similarly for the other components. So the general condition is that all corresponding components are proportional.For perpendicularity, that's when the dot product is zero. Since the dot product is x1x2 + y1y2 + z1z2, so the condition is x1x2 + y1y2 + z1z2 = 0. That's straightforward.Lastly, the distance between two points with coordinates (x1, y1, z1) and (x2, y2, z2). The distance formula in 3D space is derived from the Pythagorean theorem. So, the differences in each coordinate are squared, summed, and then take the square root. So, distance d = sqrt[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]. Let me check that. Yes, moving from one point to another, the differences along each axis form the sides of a rectangular box, and the space diagonal is the distance between the points, so the formula should be correct.Let me recap everything to make sure I didn't miss anything. The scalar product is component-wise multiplication and sum. Cos φ is that product divided by the product of the magnitudes. Collinear vectors have proportional components, and perpendicular vectors have a dot product of zero. The distance formula is the square root of the sum of squared differences in each coordinate. That seems to cover all parts of the question.Wait, just to be thorough, when they ask for the conditions for collinearity in coordinates, do they want the ratio form or the cross product equals zero? The problem says "in coordinates," so probably the ratio form. So writing that each component of r1 is a scalar multiple of r2's components. Alternatively, as x1/x2 = y1/y2 = z1/z2, but I need to note that this is when none of the denominators are zero. If any component is zero in r2, the corresponding component in r1 must also be zero. Maybe it's safer to express it as x1 = k x2, y1 = k y2, z1 = k z2 for some scalar k, which covers all cases. That way, even if x2 is zero, x1 has to be zero as well. So, in coordinate terms, the conditions for collinearity are that there exists a scalar k such that x1 = k x2, y1 = k y2, z1 = k z2.Similarly, perpendicularity is when the dot product is zero. So, x1x2 + y1y2 + z1z2 = 0.Distance formula is sqrt[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]. Yes, that's right.I think that's all. Let me check once more.1. Scalar product: sum of products of corresponding components. Correct.2. Cos φ: scalar product over product of magnitudes. Correct.3. Collinearity: components proportional. Expressed as ratios equal (with caveats) or scalar multiple. The scalar multiple expression is more general. So in coordinates, the relations x1 = kx2, y1 = ky2, z1 = kz2. Correct.4. Perpendicularity: dot product zero. Correct.5. Distance formula: sqrt of sum of squared differences. Correct.Yep, I think that's all. No steps skipped, and all parts addressed.Final AnswerThe scalar product is boxed{x_1x_2 + y_1y_2 + z_1z_2}. cos varphi is given by boxed{dfrac{x_1x_2 + y_1y_2 + z_1z_2}{sqrt{x_1^2 + y_1^2 + z_1^2} sqrt{x_2^2 + y_2^2 + z_2^2}}}. Vectors are collinear if boxed{dfrac{x_1}{x_2} = dfrac{y_1}{y_2} = dfrac{z_1}{z_2}} (provided x_2, y_2, z_2 neq 0) and perpendicular if boxed{x_1x_2 + y_1y_2 + z_1z_2 = 0}. The distance between the points is boxed{sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}.</think>The scalar product (dot product) of vectors (vec{r}_1) and (vec{r}_2) is calculated by multiplying the corresponding components and summing them up:[vec{r}_1 cdot vec{r}_2 = boxed{x_1x_2 + y_1y_2 + z_1z_2}]To find (cos varphi), where (varphi) is the angle between the vectors, we use the formula involving the dot product and the magnitudes of the vectors:[cos varphi = frac{vec{r}_1 cdot vec{r}_2}{|vec{r}_1| |vec{r}_2|} = boxed{dfrac{x_1x_2 + y_1y_2 + z_1z_2}{sqrt{x_1^2 + y_1^2 + z_1^2} sqrt{x_2^2 + y_2^2 + z_2^2}}}]For collinearity, the vectors must be scalar multiples of each other. This condition in coordinates is:[boxed{dfrac{x_1}{x_2} = dfrac{y_1}{y_2} = dfrac{z_1}{z_2}}](provided (x_2, y_2, z_2 neq 0); if any component is zero, the corresponding component must also be zero).For perpendicularity, the dot product must be zero:[boxed{x_1x_2 + y_1y_2 + z_1z_2 = 0}]The distance between the points ((x_1, y_1, z_1)) and ((x_2, y_2, z_2)) is given by the Euclidean distance formula:[boxed{sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}],<think>Okay, let's see. The problem is asking me to express the scalar product of two vectors, r₁ and r₂, in coordinates. Then I need to calculate cos φ, where φ is the angle between them. After that, I have to write the conditions for when the vectors are collinear and perpendicular, again in coordinates. Finally, I need to provide the formula for the distance between two points given their coordinates. Alright, let's take this step by step.First, scalar product, also known as the dot product. I remember that the scalar product of two vectors is the sum of the products of their corresponding components. So if r₁ is x₁i + y₁j + z₁k and r₂ is x₂i + y₂j + z₂k, then the dot product should be x₁x₂ + y₁y₂ + z₁z₂. Let me verify that. Yes, because when you multiply the vectors component-wise and add them up, that's exactly the definition of the dot product. So that part seems straightforward.Next, they want me to calculate cos φ, the cosine of the angle between the vectors. From what I recall, the formula for cos φ in terms of the vectors is the dot product of the vectors divided by the product of their magnitudes. So cos φ = (r₁ · r₂) / (|r₁| |r₂|). Let me check if that makes sense. The dot product formula is indeed |a||b|cos θ, so solving for cos θ gives the formula I just mentioned. Therefore, substituting the scalar product from the first part, it would be (x₁x₂ + y₁y₂ + z₁z₂) divided by the square roots of (x₁² + y₁² + z₁²) times (x₂² + y₂² + z₂²). Yep, that seems right.Now, conditions for collinearity. Collinear vectors mean they are scalar multiples of each other. So, if vector r₁ is collinear with r₂, then there exists some scalar λ such that r₁ = λ r₂. In coordinates, this would mean that each component of r₁ is λ times the corresponding component of r₂. So, x₁ = λ x₂, y₁ = λ y₂, and z₁ = λ z₂ for some λ. Alternatively, the ratios of the corresponding components must be equal, provided none of the denominators are zero. So, x₁/x₂ = y₁/y₂ = z₁/z₂. However, if any component of r₂ is zero, then the corresponding component in r₁ must also be zero. Hmm, so maybe a better way to write the condition is that all the cross ratios are zero. Wait, actually, another way to express collinearity is that the cross product of the vectors is the zero vector. So, r₁ × r₂ = 0. But since the problem asks for conditions in coordinates, let's stick to the component-wise ratios. So, if r₁ and r₂ are collinear, then x₁/x₂ = y₁/y₂ = z₁/z₂, provided that x₂, y₂, z₂ are not zero. If, say, x₂ is zero, then x₁ must also be zero, and similarly for the other components. So, the general condition is that each component of r₁ is proportional to the corresponding component of r₂ with the same proportionality constant.For perpendicularity, the condition is that the dot product is zero. Because if the vectors are perpendicular, the angle between them is 90 degrees, and cos 90° is zero. Therefore, the dot product r₁ · r₂ = 0. Which in coordinates is x₁x₂ + y₁y₂ + z₁z₂ = 0. That's straightforward.Lastly, the distance between two points with coordinates (x₁, y₁, z₁) and (x₂, y₂, z₂). The distance formula in three dimensions is an extension of the Pythagorean theorem. So, the distance d is the square root of the sum of the squares of the differences in each coordinate. Therefore, d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]. Let me verify that. Yes, because you can consider the vector from point 1 to point 2, which is (x₂ - x₁)i + (y₂ - y₁)j + (z₂ - z₁)k, and the magnitude of this vector is the distance between the points. The magnitude is the square root of the sum of the squares of its components, which gives the formula above.Let me recap to make sure I didn't miss anything. The scalar product is straightforward component multiplication and addition. Cos φ is the dot product over the product of magnitudes. Collinearity requires proportional components or cross product zero. Perpendicularity is dot product zero. Distance formula is the square root of the sum of squared differences. All these seem like standard results from vector algebra. I think that covers everything the problem is asking for.Final AnswerThe scalar product is boxed{x_1 x_2 + y_1 y_2 + z_1 z_2}.cos varphi is given by boxed{dfrac{x_1 x_2 + y_1 y_2 + z_1 z_2}{sqrt{x_1^2 + y_1^2 + z_1^2} sqrt{x_2^2 + y_2^2 + z_2^2}}}.Vectors vec{r}_1 and vec{r}_2 are collinear if boxed{dfrac{x_1}{x_2} = dfrac{y_1}{y_2} = dfrac{z_1}{z_2}} (assuming x_2, y_2, z_2 neq 0), and perpendicular if boxed{x_1 x_2 + y_1 y_2 + z_1 z_2 = 0}.The distance between the points is boxed{sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}.</think>The scalar product of vectors (vec{r}_1) and (vec{r}_2) is calculated by summing the products of their corresponding components:[vec{r}_1 cdot vec{r}_2 = x_1 x_2 + y_1 y_2 + z_1 z_2]Thus, the scalar product is (boxed{x_1 x_2 + y_1 y_2 + z_1 z_2}).To find (cos varphi), where (varphi) is the angle between the vectors, we use the formula:[cos varphi = frac{vec{r}_1 cdot vec{r}_2}{|vec{r}_1| |vec{r}_2|}]Substituting the scalar product and magnitudes of the vectors, we get:[cos varphi = frac{x_1 x_2 + y_1 y_2 + z_1 z_2}{sqrt{x_1^2 + y_1^2 + z_1^2} sqrt{x_2^2 + y_2^2 + z_2^2}}]Therefore, (cos varphi) is (boxed{dfrac{x_1 x_2 + y_1 y_2 + z_1 z_2}{sqrt{x_1^2 + y_1^2 + z_1^2} sqrt{x_2^2 + y_2^2 + z_2^2}}}).For collinearity, the vectors must be proportional. This condition in coordinates is:[frac{x_1}{x_2} = frac{y_1}{y_2} = frac{z_1}{z_2}](assuming (x_2, y_2, z_2 neq 0)). Hence, the condition for collinearity is (boxed{dfrac{x_1}{x_2} = dfrac{y_1}{y_2} = dfrac{z_1}{z_2}}).For perpendicularity, the dot product must be zero. Therefore, the condition is:[x_1 x_2 + y_1 y_2 + z_1 z_2 = 0]Thus, the condition for perpendicularity is (boxed{x_1 x_2 + y_1 y_2 + z_1 z_2 = 0}).The distance between the points ((x_1, y_1, z_1)) and ((x_2, y_2, z_2)) is given by the Euclidean distance formula:[sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}]Hence, the distance is (boxed{sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}).]

🔑:31. Using the distributive property of scalar multiplication of vectors, as well as the fact that overrightarrow{i^{2}}=overrightarrow{j^{2}}=vec{k}^{2}=1, vec{i} vec{j}=vec{j} vec{k}= =vec{k} vec{i}=0, quad we obtain: quad vec{r}_{1} vec{r}_{2}=r_{1} r_{2} cos varphi=x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2} . quad From this equality, it follows that: cos varphi=frac{x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}}{r_{1} r_{2}}, where r_{1}= =sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2}} ; r_{2}=sqrt{x_{2}^{2}+y_{2}^{2}+z_{2}^{2}}.If the vectors are perpendicular, then varphi=90^{circ} and cos varphi=0, x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}=0.If the vectors are collinear, then vec{r}_{2}=lambda vec{r}_{1}, quad x_{2} vec{i}+y_{2} vec{j}+z_{mathrm{varepsilon}} vec{k}= =lambda x_{1} vec{i}+lambda y_{1} vec{j}+lambda z_{1} vec{k} ; frac{x_{2}}{x_{1}}=frac{y_{2}}{y_{1}}=frac{z_{2}}{z_{1}}=lambda.Squaring the vector vec{r}_{2}-vec{r}_{1} scalarly, we obtain the square of the distance between the points with the corresponding coordinates: left(overrightarrow{r_{2}}-overrightarrow{r_{1}}right)^{2}=left(x_{2}-x_{1}right)^{2}+left(y_{2}-y_{1}right)^{2}+left(z_{2}-z_{1}right)^{2}.

🧠:[<think>Okay, so I need to figure out which of these numbers is irrational. The options are A: π, B: 1.010010001..., C: 3/5, and D: √9. Let me start by recalling what an irrational number is. If I remember right, an irrational number can't be expressed as a fraction of two integers. That means it's a number that goes on forever without repeating, right? So, like, it can't be written as a simple fraction. Let's go through each option one by one. Starting with option A: π. Hmm, π is the ratio of a circle's circumference to its diameter. I think π is a classic example of an irrational number. But wait, let me make sure. Is there any way to write π as a fraction? Well, you can approximate it with fractions like 22/7, but that's not exact. The actual value of π is non-repeating and non-terminating. So yeah, π is irrational. That might be the answer, but let me check the other options just to be safe.Option B: 1.010010001... So the number is 1.010010001 with a pattern of adding more zeros between the ones. The description says "real numbers" and this is written out with a decimal that continues. The question is, is this a repeating decimal or not? If it's non-repeating and non-terminating, then it's irrational. But if the pattern continues in a predictable way, maybe it's rational. Wait, but the way it's written, each time there's an extra zero. So like, 1.01, then 001, then 0001, etc. So the number of zeros between the ones increases each time. That means the decimal isn't repeating in a regular cycle. So even though there's a pattern, the pattern isn't periodic. Therefore, it doesn't repeat indefinitely in the same way, so it can't be expressed as a fraction. Wait, but actually, can a number with a non-repeating pattern be rational? I think only if the decimal either terminates or repeats periodically is it rational. So if the pattern is changing, like adding an extra zero each time, then the decimal doesn't terminate or repeat, so it must be irrational. Hmm, so maybe both A and B are irrational? But wait, let me check again.Wait, the problem gives four options and only one correct answer. So perhaps I made a mistake here. Let me double-check. The number B is 1.010010001... where the number of zeros increases each time. But is that considered a non-repeating, non-terminating decimal? Yes, because the pattern isn't the same each time. Each segment has an additional zero. So, for example, after the decimal, it's 0, then 1, then 00, then 1, then 000, then 1, etc. So the positions of the ones are at the 2nd, 5th, 9th, 14th, etc., decimal places, which is a pattern but not a repeating one. Therefore, this decimal neither terminates nor repeats, so it's irrational. But wait, the problem states "real numbers" and the options are given. But in that case, both A and B would be irrational. But the problem is presented as a multiple-choice with options A to D, and likely only one correct answer. That suggests I need to check my reasoning again.Wait, hold on. Let me confirm: π is definitely irrational. That's a known fact. So option A is irrational. For option B, if the decimal is written as 1.010010001... with the increasing number of zeros, but is that a specific number? If the ellipsis indicates that the pattern continues indefinitely with more zeros each time, then yes, that's a non-repeating, non-terminating decimal, hence irrational. But sometimes, if a number has a non-repeating pattern but is still definable as a fraction, it could be rational. Wait, but how can you define such a number as a fraction? It's not periodic, so you can't express it as a ratio. So perhaps B is also irrational. But that can't be, because the problem is expecting one answer. Wait, maybe I misread the options. Let me check again.Option B is written as "1.010010001". Wait, maybe the number is terminating here. If the number is 1.010010001 exactly, with a finite number of decimal places, then it's a terminating decimal, which is rational. But the problem says "real numbers" and in the original problem statement, the number might have an ellipsis or not. Wait, looking back at the user's question: "B: 1.010010001". The user wrote it with three dots at the end, but in the original problem, how is it presented? Wait, in the problem statement here, it's written as "B: 1.010010001". The LaTeX code shows that there are no ellipsis; it's just 1.010010001. So maybe it's a finite decimal. Wait, but 1.010010001 is a finite decimal. So that's a rational number. Let me confirm: if a decimal terminates, it's rational because it can be written as a fraction with denominator a power of 10. For example, 1.010010001 is equal to 1010010001/1000000000. So that's definitely rational. But wait, the problem might have a typo. The original problem, if written as 1.010010001... with an ellipsis, then it's non-terminating, but in this case, the user wrote "1.010010001" with three periods at the end in the initial description? Wait, no, looking back: the user wrote "B: 1.010010001" in LaTeX, so that would be typeset as 1.010010001 without an ellipsis, right? Wait, actually, checking again: in the original problem presented by the user, the options are:A: πB: 1.010010001C: 3/5D: √9So in that case, option B is 1.010010001 with no ellipsis. So that's a terminating decimal. Therefore, it's rational. So then, why did I initially think it's non-terminating? Maybe I misread. If it's written as 1.010010001 without any continuation, then that's a finite decimal, which is rational. So then, only option A is irrational. But let me check the other options just in case.Option C is 3/5. That's clearly a fraction, so it's rational. 3 divided by 5 is 0.6, which is a terminating decimal. So that's definitely rational.Option D is √9. The square root of 9 is 3, which is an integer. Integers are rational numbers because they can be expressed as a fraction (3/1). So that's also rational.Therefore, among the options, only π (option A) is irrational. So the answer should be A.Wait, but let me check again because initially I thought maybe B was irrational if there was an ellipsis, but in the problem statement, as presented by the user, it's just 1.010010001 without any continuation. So it's 1.010010001, which terminates at that decimal. So it's a finite decimal, which is rational. So the only irrational number in the options is π. Therefore, the answer is A.Final Answerboxed{A}</think>To determine which of the given real numbers is irrational, we need to recall that an irrational number cannot be expressed as a fraction of two integers and has a non-repeating, non-terminating decimal expansion.- Option A: π π is a well-known irrational number. It cannot be expressed as a fraction of two integers and its decimal expansion is non-repeating and non-terminating. Therefore, π is irrational.- Option B: 1.010010001 This number is written as a terminating decimal (1.010010001), which can be expressed as the fraction ( frac{1010010001}{1000000000} ). Since it is a terminating decimal, it is rational.- Option C: ( frac{3}{5} ) This is a fraction of two integers, which simplifies to 0.6, a terminating decimal. Therefore, it is rational.- Option D: ( sqrt{9} ) ( sqrt{9} ) simplifies to 3, which is an integer. All integers are rational numbers. Therefore, ( sqrt{9} ) is rational.Among the given options, only π (option A) is irrational. boxed{A},<think>Okay, so I have this question here asking which of the given real numbers is irrational. The options are A: π, B: 1.010010001, C: 3/5, and D: √9. Hmm, let me remember what an irrational number is. From what I recall, an irrational number is a number that cannot be expressed as a fraction of two integers, right? So, it's a number that goes on forever without repeating. Like π or √2. But let me make sure I have this straight.First, let's go through each option one by one. Starting with option A: π. I know π is the ratio of a circle's circumference to its diameter. Wait, but isn't that a fraction? No, wait, π is actually an irrational number because it cannot be expressed exactly as a simple fraction. Its decimal representation never ends and never settles into a permanent repeating pattern. So, π is definitely irrational. But let me check the other options too to make sure there isn't a trick here.Option B: 1.010010001. This number is written out with a decimal that goes 1.010010001. Let me look at the pattern here. It seems like there's a 1, then some zeros, then another 1, more zeros, then another 1. The number of zeros between the ones might be increasing each time. The user wrote it as 1.010010001, so maybe the pattern is 1.01 001 0001... So between each 1, the number of zeros increases by one each time. If that's the case, then this decimal is non-repeating and non-terminating. Wait, but is that the case here? The way it's written, B is 1.010010001. The user might have written it as a finite decimal. Let me check. If it's written as 1.010010001 with the three dots at the end, that would indicate it's non-terminating, but here the problem statement just writes it as 1.010010001 without any ellipsis. So maybe that's a terminating decimal? Or is it a decimal with a specific pattern that repeats?Wait, the problem lists B as 1.010010001. If there's no ellipsis, then that's a terminating decimal. Because it ends at the last 1. So if it's terminating, then it's a rational number. Because any terminating decimal can be expressed as a fraction. For example, 1.010010001 could be written as 1010010001 divided by 10^9, right? Since there are 9 decimal places. Let me count: 1.010010001. After the decimal, the digits are 0,1,0,0,1,0,0,0,1. That's 9 digits. So yes, 1.010010001 is 1010010001 divided by 10^9. Which is a fraction of two integers. So that's a rational number. Therefore, B is rational.Option C: 3/5. Well, that's clearly a fraction. 3 divided by 5 is 0.6, which is a terminating decimal. Terminating decimals are rational numbers. So C is definitely rational.Option D: √9. The square root of 9. What's that? Well, 3 times 3 is 9, so √9 is 3. 3 is an integer, which is a rational number. So D is rational.So going back to the options, A is π, which is irrational. B is a terminating decimal, so rational. C is a fraction, rational. D is an integer, rational. Therefore, the answer must be A: π.Wait, but let me just verify once more. Did I misinterpret any of the options? For example, is there a chance that option B is supposed to be a non-terminating decimal? If the problem had written B as 1.010010001... with an ellipsis, indicating that the pattern continues indefinitely, then in that case, if the pattern is non-repeating, it could be irrational. But in the problem statement, it's written as 1.010010001 without an ellipsis, which implies that it's terminating. So unless there's a typo or something missing, I think B is rational. So A is the only irrational number among the options. Yeah, that makes sense. I don't see any other possible irrational numbers here. So yeah, A is the answer.Final Answerboxed{A}</think>To determine which of the given real numbers is irrational, we need to check each option:- Option A: π π is a well-known irrational number because it cannot be expressed as a fraction of two integers and its decimal representation is non-terminating and non-repeating.- Option B: 1.010010001 This number is written as a terminating decimal (1.010010001) without an ellipsis, indicating it ends. Terminating decimals can be expressed as fractions (e.g., ( frac{1010010001}{10^9} )), making it rational.- Option C: ( frac{3}{5} ) This is a fraction of two integers, which is a rational number. Its decimal form (0.6) is terminating.- Option D: ( sqrt{9} ) ( sqrt{9} = 3 ), which is an integer and therefore rational.After evaluating all options, π (Option A) is the only irrational number.[boxed{A}]]

🔑:To solve the problem, we evaluate each option to determine if it is rational or irrational.- Option A: pipi is a well-known mathematical constant that represents the ratio of a circle's circumference to its diameter. It is a non-repeating, non-terminating decimal, which cannot be expressed as a fraction of two integers. Therefore, pi is irrational.- Option B: 1.010010001This number is a finite decimal, meaning it has a finite number of digits after the decimal point. Finite decimals can always be expressed as a fraction of two integers, making them rational numbers.- Option C: frac{3}{5}This is a fraction of two integers, where the numerator is 3 and the denominator is 5. By definition, any number that can be expressed as a fraction of two integers (where the denominator is not zero) is a rational number.- Option D: sqrt{9}sqrt{9} simplifies to 3, since 3 times 3 = 9. The number 3 is an integer, and all integers are rational numbers because they can be expressed as a fraction (in this case, 3 can be written as frac{3}{1}).Given the above evaluations:- pi is the only option that is irrational.- 1.010010001, frac{3}{5}, and sqrt{9} are all rational.Therefore, the correct answer is: boxed{A}.