elichen-skymizer/qwen3-4b-thinking-aime · Datasets at Hugging Face

question stringlengths 80 1.73k	id int64 0 89	source stringclasses 2 values	input_ids listlengths 6.2k 30.8k	input_tokens_len int64 6.2k 30.8k	generated_texts stringlengths 17.6k 104k	generated_tokens_len int64 6.05k 30.7k	n_prefill_tokens int64 38 840	seed int64 1 1	labels listlengths 6.2k 30.8k
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, including $t$ minutes spent in the coffee shop. Suppose Aya walks at $s+\frac{1}{2}$ kilometers per hour. Find the number of minutes the walk takes her, including the $t$ minutes spent in the coffee shop.	60	aime24	[ 151644, 872, 198, 11510, 6556, 362, 7755, 5780, 369, 264, 400, 24, 3, 12, 85526, 20408, 23791, 4227, 323, 17933, 518, 264, 10799, 8061, 26807, 13, 3197, 1340, 22479, 518, 264, 6783, 4628, 315, 400, 82, 3, 40568, 817, 6460, 11, 279, 4227, 4990, 1059, 220, 19, 4115, 11, 2670, 400, 83, 3, 4420, 7391, 304, 279, 10799, 8061, 13, 3197, 1340, 22479, 400, 82, 10, 17, 3, 40568, 817, 6460, 11, 279, 4227, 4990, 1059, 220, 17, 4115, 323, 220, 17, 19, 4420, 11, 2670, 400, 83, 3, 4420, 7391, 304, 279, 10799, 8061, 13, 82610, 362, 7755, 22479, 518, 400, 82, 41715, 37018, 90, 16, 15170, 17, 31716, 40568, 817, 6460, 13, 7379, 279, 1372, 315, 4420, 279, 4227, 4990, 1059, 11, 2670, 279, 400, 83, 3, 4420, 7391, 304, 279, 10799, 8061, 13, 151645, 198, 151644, 77091, 198, 151667, 198, 1986, 374, 264, 6351, 476, 17003, 3405, 11, 323, 432, 374, 5000, 311, 3410, 264, 2118, 323, 4396, 4226, 13, 358, 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15170, 17, 92, 84500, 14085, 198, 82, 488, 1124, 37018, 90, 16, 15170, 17, 92, 284, 220, 17, 13, 20, 488, 220, 15, 13, 20, 284, 220, 18, 1124, 1318, 90, 13136, 7530, 532, 14085, 271, 78516, 882, 518, 220, 18, 13136, 7530, 1447, 14085, 198, 59, 37018, 90, 24, 15170, 18, 92, 284, 220, 18, 1124, 1318, 90, 4115, 92, 284, 220, 16, 23, 15, 1124, 1318, 90, 4420, 532, 14085, 271, 2212, 10799, 882, 1447, 14085, 198, 16, 23, 15, 488, 220, 17, 19, 284, 1124, 79075, 90, 17, 15, 19, 92, 1124, 1318, 90, 4420, 532, 14085, 271, 44364, 14374, 25521, 227, 13023, 21806, 1447, 14085, 198, 59, 79075, 90, 17, 15, 19, 532, 14085, 151645 ]	6,195	This is a complex or challenging question, and it is difficult to provide a direct and correct answer. I need to think about it. Well, let's tackle this problem step by step. So, Aya walks 9 kilometers every morning, stops at a coffee shop for t minutes, and the time she takes is the walking time plus the coffee shop time. Wait, actually, the problem says "the walk takes her 4 hours, including t minutes spent in the coffee shop." Hmm, maybe I need to clarify: is the total time (walking + coffee) equal to the time given? Let's check the problem statement again. "When she walks at a constant speed of s kilometers per hour, the walk takes her 4 hours, including t minutes spent in the coffee shop." Wait, that wording is a bit confusing. Maybe "the walk" refers to the entire trip? Like, she walks 9 km, then stops at coffee shop for t minutes, so total time is walking time + t minutes. But the problem says "the walk takes her 4 hours, including t minutes spent in the coffee shop." Wait, maybe "walk" here is being used loosely? Wait, no, let's parse it carefully. Wait, maybe better to define variables clearly. Let's let: - Distance walked: 9 km (this is fixed, right? She goes for a 9-km-long walk, so the walking distance is always 9 km) - Time spent walking: let's call this T1 when speed is s, T2 when speed is s+2, etc. - Time spent at coffee shop: t minutes (this is the same in both cases, as per problem: "including t minutes spent in the coffee shop" for both scenarios) - Total time for the trip (walking + coffee): for first case, 4 hours; second case, 2 hours 24 minutes. Oh! That must be it. The problem says "the walk takes her 4 hours, including t minutes spent in the coffee shop." Wait, maybe the problem is using "walk" to mean "the entire activity" (walking and coffee), but maybe not. Wait, no, let's see: if it's "the walk takes her 4 hours, including t minutes in coffee," that would imply that the coffee time is part of the walk time? But that seems odd. Wait, no, probably a translation issue or wording issue. Let's think mathematically. Let's denote: Let W be the time (in hours) she spends walking (not including coffee shop). Then, when she walks at speed s, time walking is W = 9 / s hours. Then total time for the trip (walking + coffee) is W + (t / 60) hours? Wait, but the problem says "the walk takes her 4 hours, including t minutes spent in the coffee shop." Wait, maybe "including" here means that the total time is 4 hours, which consists of the walking time and the coffee time. So total time = walking time + coffee time. Yes, that makes sense. So for the first scenario: Total time = 4 hours = (time walking at speed s) + (time at coffee shop, which is t minutes) For the second scenario: Total time = 2 hours 24 minutes = (time walking at speed s+2) + (time at coffee shop, which is still t minutes) Ah, that's probably the key. The coffee shop time is the same t minutes in both cases, so the difference in total times is equal to the difference in walking times. Let's formalize this with units to avoid confusion. Let's work in minutes for time, maybe? Wait, or hours, but convert t to hours at the end. Let's see. First, convert all times to hours to be consistent with speed in km per hour. First scenario: Total time: 4 hours Time spent walking: 9 / s hours (since distance = speed * time => time = distance / speed) Time spent at coffee shop: t minutes = t / 60 hours So total time = walking time + coffee time => 4 = (9 / s) + (t / 60) ...(1) Second scenario: Total time: 2 hours 24 minutes. Let's convert that to hours: 2 + 24/60 = 2 + 0.4 = 2.4 hours, or 12/5 hours (since 24/60 = 2/5, so 2 + 2/5 = 12/5). Maybe fractions are better to be precise. Time spent walking: 9 / (s + 2) hours (speed is s + 2 km/h) Time spent at coffee shop: still t minutes = t / 60 hours So total time = walking time + coffee time => 12/5 = (9 / (s + 2)) + (t / 60) ...(2) Okay, now we have two equations (1) and (2) with two unknowns: s and t. Our goal is to find, when she walks at speed s + 1/2 km/h, what's the total time (walking time + coffee time) in minutes. Let's write equations (1) and (2) again clearly: From (1): 4 = 9/s + t/60 --> let's solve for t/60 to make it easier, since t is the same in both, so the coffee time in hours is constant. Let me denote C = t / 60, so C is the coffee time in hours (so t = 60C minutes). Then equations become: (1): 4 = 9/s + C --> so C = 4 - 9/s ...(1a) (2): 12/5 = 9/(s + 2) + C --> substitute C from (1a) into (2): 12/5 = 9/(s + 2) + 4 - 9/s Okay, now we can solve for s here. Let's do that step by step. First, let's rearrange the equation to get all terms on one side, maybe subtract 4 from both sides: 12/5 - 4 = 9/(s + 2) - 9/s Compute 12/5 - 4: 4 is 20/5, so 12/5 - 20/5 = -8/5 So left side is -8/5, right side is 9/(s + 2) - 9/s Let's compute the right side: common denominator s(s + 2), so [9s - 9(s + 2)] / [s(s + 2)] = [9s - 9s - 18] / [s(s + 2)] = (-18) / [s(s + 2)] So now we have: -8/5 = (-18) / [s(s + 2)] Hmm, nice, the negatives cancel out if we multiply both sides by -1: 8/5 = 18 / [s(s + 2)] Now cross-multiply to solve for s(s + 2): 8 * s(s + 2) = 5 * 18 Calculate right side: 518=90 Left side: 8s(s + 2) = 8s² + 16s So 8s² + 16s = 90 Let's simplify this equation. Divide both sides by 2 to make numbers smaller: 4s² + 8s = 45 Bring all terms to left side: 4s² + 8s - 45 = 0 Now, quadratic in s. Let's try to solve using quadratic formula. For equation ax² + bx + c = 0, solutions x = [-b ± sqrt(b² - 4ac)]/(2a) Here, a = 4, b = 8, c = -45 Discriminant D = b² - 4ac = 64 - 44(-45) = 64 + 720 = 784 Oh, 784 is a perfect square! sqrt(784) = 28, good. So s = [-8 ± 28]/(24) = [-8 ± 28]/8 We have two solutions: first, take the plus: (-8 + 28)/8 = 20/8 = 5/2 = 2.5 Second, take the minus: (-8 - 28)/8 = (-36)/8 = -9/2 = -4.5. Speed can't be negative, so discard that. So s = 5/2 km/h? Wait, hold on, 5/2 is 2.5 km/h? Wait, but let's check if that makes sense with the walking times. Wait, if s = 2.5 km/h, then walking time at speed s is 9 / 2.5 = 3.6 hours, which is 3 hours 36 minutes. Then total time is 4 hours, so coffee time is 4 - 3.6 = 0.4 hours = 24 minutes. Let's check the second scenario: speed s + 2 = 4.5 km/h, walking time = 9 / 4.5 = 2 hours, total time 2 hours 24 minutes, so coffee time 24 minutes. Oh! Wait, that's exactly what the problem says: in both cases, including t minutes in coffee, so total time is walking time + t minutes. So in first case, walking time 3.6h = 216 min, total time 4h=240 min, so t=24 min. Second case, walking time 2h=120 min, total time 2h24m=144 min, so t=24 min. Perfect, that checks out. Wait, so maybe I didn't need to go through all the algebra with C, but anyway, we found s=2.5 km/h, which is 5/2 km/h, as we had. Wait, but let's just make sure we didn't misinterpret the problem. The problem says: "When she walks at a constant speed of s kilometers per hour, the walk takes her 4 hours, including t minutes spent in the coffee shop." So "the walk takes her 4 hours" – maybe "walk" here is the entire trip? Like, she walks (the 9 km) and then stops at coffee, so the entire activity is called "the walk" in this context? Maybe that's the confusion. But in our calculation, with s=2.5, it works: walking 9 km at 2.5 km/h takes 3.6 hours (216 minutes), plus 24 minutes at coffee, total 240 minutes = 4 hours. Then walking 9 km at 4.5 km/h takes 2 hours (120 minutes), plus 24 minutes at coffee, total 144 minutes = 2 hours 24 minutes. Perfect, that's exactly the problem statement. So our interpretation was correct: total time (walking + coffee) is given as 4 hours and 2h24m, with coffee time being t minutes in both cases, so the difference in total times is difference in walking times. Great, so now we know s=5/2 km/h, so s + 1/2 = 5/2 + 1/2 = 6/2 = 3 km/h. Wait, that's nice! s=2.5, so s + 0.5=3 km/h. Now, we need to find the total time when she walks at 3 km/h, including t minutes in coffee shop. We already found t=24 minutes from the check above, but let's confirm with equations to be thorough. From equation (1a): C = t/60 = 4 - 9/s. We have s=5/2, so 9/s=9/(5/2)=18/5=3.6 hours, so C=4 - 3.6=0.4 hours=24 minutes, so t=24 minutes. Correct. So when walking speed is 3 km/h, walking time is 9 / 3 = 3 hours. Then total time is walking time + coffee time = 3 hours + 24 minutes. Convert 3 hours to minutes: 360=180 minutes, plus 24 minutes is 204 minutes. Wait, hold on, is that it? Let's make sure we didn't skip any steps. Let's recap to check for errors. First, we set up equations based on total time = walking time + coffee time (in hours). Converted total times correctly: 2h24m=12/5 h=2.4h, yes. Then substituted coffee time as constant, solved for s, got s=2.5, which checks out with the time calculations (walking time 3.6h at 2.5km/h is 9km, correct; 9/4.5=2h, correct). Coffee time is 24 minutes in both cases, which is t=24. Then new speed: s + 1/2 = 2.5 + 0.5 = 3 km/h. Walking time: 9/3=3 hours=180 minutes. Total time: 180 + 24=204 minutes. Wait, but let's do it without converting to decimals to be precise, in case we made a decimal error. s=5/2 km/h (since we had s=20/8=5/2, yes, 5/2=2.5). s + 1/2 = 5/2 + 1/2 = 6/2 = 3 km/h, correct. Walking time at 3 km/h: 9 km / 3 km/h = 3 hours = 180 minutes. Coffee time t: from first scenario, total time 4 hours = walking time + t/60 hours. Walking time=9/s=9/(5/2)=18/5=3.6 hours=216 minutes, so t=460 - 216=240 - 216=24 minutes. Perfect, same as before. So total time: 180 + 24=204 minutes. Wait, but let me think again: is there any chance that the problem meant that "the walk takes her 4 hours" refers to just the walking time, and "including t minutes" is a misphrase? Let's check that interpretation, just to be safe, because that could be a critical mistake. Alternative interpretation: "When she walks at speed s, the walk (walking time) takes 4 hours, and she spends t minutes at coffee shop afterwards." But the problem says "the walk takes her 4 hours, including t minutes spent in the coffee shop." The word "including" suggests that t minutes are part of the 4 hours. So that interpretation would be wrong because "including" means the 4 hours consists of walking and coffee. Wait, another way: maybe "the walk" is the entire trip, so "the walk" is walking + coffee, so "takes her 4 hours" is the total time, which includes the coffee time. That's what we did, and it worked out with the numbers, so that's probably right. Let's test the other interpretation just in case. Suppose "the walk takes her 4 hours" is walking time only, so walking time=4 hours, then coffee time is t minutes, so total time=4h + t minutes. Then second scenario: walking time=2h24m, coffee time=t minutes, total time=2h24m + t minutes. But then the problem says "including t minutes spent in the coffee shop" – if "the walk takes her 4 hours" is walking time, then "including t minutes" would be confusing. The problem says "the walk takes her 4 hours, including t minutes...", so the "including" is modifying "4 hours", so the 4 hours includes the t minutes. So our first interpretation is correct. Also, when we did the calculation, the numbers worked out perfectly: walking time at s=2.5 is 3.6h, so total time 4h implies coffee time 0.4h=24m; walking time at s+2=4.5 is 2h, total time 2.4h implies coffee time 0.4h=24m. So t=24 is consistent, which is a good check. So then, for speed s + 1/2=3 km/h, walking time=9/3=3h=180m, coffee time=24m, total=204m. Wait, but let's make sure we didn't miscalculate the quadratic. Let's go back to the equation step to verify. We had from (1): 4 = 9/s + t/60 From (2): 12/5 = 9/(s+2) + t/60 Subtract equation (1) minus equation (2): 4 - 12/5 = 9/s - 9/(s+2) Wait, hold on! Wait a second, earlier I did (2) minus (1) with signs, but let's do this subtraction to see if we get the same. Left side: 4 - 12/5 = 20/5 - 12/5 = 8/5 Right side: 9/s - 9/(s+2) = 9[ (s+2 - s)/(s(s+2)) ] = 9[2/(s(s+2))] = 18/[s(s+2)] Ah! Here we go, this is a better way to subtract equations to avoid sign errors. So: 8/5 = 18/[s(s + 2)] Which is exactly what we had before (since earlier we had -8/5 = -18/[s(s+2)], so same equation). So that's correct, so 8s(s+2)=90, 4s²+8s-45=0, solutions s=(-8±28)/8, so 20/8=5/2, correct. So s=5/2 is definitely right. Therefore, s + 1/2=3, walking time=9/3=3 hours=180 minutes, t=24 minutes, total=204 minutes. Is there any chance the problem wants the answer in hours? No, it says "the number of minutes the walk takes her, including the t minutes", so minutes, so 204. Wait, let's just compute t another way to be 100% sure. From first scenario: total time 4 hours = 240 minutes. Walking time at s km/h: 9/s hours = (9/s)60 minutes. So coffee time t=240 - (960)/s. Second scenario: total time 2h24m=144 minutes. Walking time at s+2 km/h: 9/(s+2) hours=(960)/(s+2) minutes. Coffee time t=144 - (960)/(s+2). Set equal: 240 - 540/s = 144 - 540/(s+2) Let's solve this equation in minutes to see if we get the same s. 240 - 144 = 540/s - 540/(s+2) 96 = 540[ (s+2 - s)/(s(s+2)) ] = 540[2/(s(s+2))] = 1080/[s(s+2)] So 96 = 1080/[s(s+2)] => s(s+2)=1080/96 Simplify 1080/96: divide numerator and denominator by 24: 1080÷24=45, 96÷24=4, so 45/4. Wait, 2445=1080? 2440=960, 245=120, 960+120=1080, yes. 244=96, yes. So s(s+2)=45/4. Wait, hold on! Wait, in this equation, we have s(s+2)=45/4? Wait, but earlier we had 8s(s+2)=90, so s(s+2)=90/8=45/4, yes! Same thing. So s² + 2s - 45/4=0, multiply by 4: 4s² +8s -45=0, same quadratic. So s=(-8±sqrt(64 + 720))/8=(-8±28)/8, so s=20/8=5/2, correct. So t=240 - (960)/(5/2)=240 - (540)/(5/2)=240 - 540(2/5)=240 - 216=24 minutes. Perfect, matches. Then for speed s + 1/2=3 km/h, walking time in minutes: (9 km)/(3 km/h)=3 hours=180 minutes, so total time=180 +24=204 minutes. I think that's solid. The key was correctly interpreting that the total time given (4 hours, 2h24m) includes the coffee shop time, which is the same t minutes in both cases, so we can set up equations to solve for s and t, then compute the new total time. I think the potential pitfall here would be misinterpreting whether the coffee time is included in the total time or not, but since the problem says "including t minutes spent in the coffee shop", it's definitely part of the total time. Also, when we tested the numbers with s=2.5, it worked out perfectly with the coffee time being 24 minutes in both scenarios, so that gives confidence. Another way someone might mess up is by forgetting to convert minutes to hours or vice versa, but by working in minutes for the coffee time (as we did in the second approach), we could verify without dealing with hours for t, which is helpful. Let me just check once more: if she walks at 3 km/h, 9 km takes 3 hours, which is 180 minutes. She spends 24 minutes at coffee, so total time 204 minutes. 204 minutes is 3 hours and 24 minutes, which is 3.4 hours. Let's see if that's consistent with the problem's structure, but since we already have all the pieces, I think that's the answer. I don't see any mistakes in the reasoning now. Initially, I was a bit confused by the wording "the walk takes her 4 hours, including t minutes", but by testing with the numbers we got, it all checks out, so that interpretation must be correct. So the final answer should be 204 minutes. </think> To solve the problem, we interpret the key phrase: "the walk takes her 4 hours, including $ t $ minutes spent in the coffee shop." This means that the total time (walking + coffee) is given as 4 hours for a speed of $ s $ km/h, and 2 hours and 24 minutes for a speed of $ s + 2 $ km/h. The coffee shop time $ t $ (in minutes) is the same in both scenarios. --- ### Step 1: Convert Total Times to Hours - First scenario: $ 4 $ hours - Second scenario: $ 2 + \frac{24}{60} = \frac{12}{5} $ hours Let: - $ C = \frac{t}{60} $ be the coffee time in hours - Walking time at speed $ s $: $ \frac{9}{s} $ hours - Walking time at speed $ s + 2 $: $ \frac{9}{s+2} $ hours From the problem: $$ \begin{align} (1) & \quad 4 = \frac{9}{s} + C \\ (2) & \quad \frac{12}{5} = \frac{9}{s+2} + C \end{align} $$ Subtracting equation (1) from equation (2): $$ 4 - \frac{12}{5} = \frac{9}{s} - \frac{9}{s+2} $$ Simplify left-hand side: $$ \frac{20}{5} - \frac{12}{5} = \frac{8}{5} $$ Right-hand side: $$ \frac{9}{s} - \frac{9}{s+2} = 9 \left( \frac{1}{s} - \frac{1}{s+2} \right) = 9 \cdot \frac{(s+2) - s}{s(s+2)} = \frac{18}{s(s+2)} $$ So, $$ \frac{8}{5} = \frac{18}{s(s+2)} \Rightarrow s(s+2) = \frac{18 \cdot 5}{8} = \frac{90}{8} = \frac{45}{4} $$ Now solve the quadratic: $$ s^2 + 2s - \frac{45}{4} = 0 $$ Multiply through by 4: $$ 4s^2 + 8s - 45 = 0 $$ Use the quadratic formula: $$ s = \frac{-8 \pm \sqrt{8^2 + 4 \cdot 4 \cdot 45}}{2 \cdot 4} = \frac{-8 \pm \sqrt{64 + 720}}{8} = \frac{-8 \pm \sqrt{784}}{8} = \frac{-8 \pm 28}{8} $$ Take the positive root: $$ s = \frac{20}{8} = \frac{5}{2} = 2.5 \text{ km/h} $$ --- ### Step 2: Find Coffee Time $ t $ From equation (1): $$ C = 4 - \frac{9}{s} = 4 - \frac{9}{2.5} = 4 - 3.6 = 0.4 \text{ hours} $$ Convert to minutes: $$ t = 0.4 \times 60 = 24 \text{ minutes} $$ --- ### Step 3: Compute New Total Time at Speed $ s + \frac{1}{2} $ $$ s + \frac{1}{2} = 2.5 + 0.5 = 3 \text{ km/h} $$ Walking time at 3 km/h: $$ \frac{9}{3} = 3 \text{ hours} = 180 \text{ minutes} $$ Add coffee time: $$ 180 + 24 = \boxed{204} \text{ minutes} $$ --- ### ✅ Final Answer: $$ \boxed{204} $$<\|im_end\|>	6,052	143	1	[ -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, -100, 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"Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$(...TRUNCATED)	61	aime24	[151644,872,198,10061,400,25411,3,387,264,21495,1640,17433,304,12671,57960,32696,12947,6771,279,2187(...TRUNCATED)	26,425	"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)	26,309	116	1	[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)
"Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$(...TRUNCATED)	64	aime24	[151644,872,198,10061,400,79,3,387,279,3245,10250,1372,369,892,1052,6724,264,6785,7546,400,77,3,1741(...TRUNCATED)	22,935	"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)	22,860	75	1	[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)
"Let $ABCD$ be a tetrahedron such that $AB=CD= \\sqrt{41}$, $AC=BD= \\sqrt{80}$, and $BC=AD= \\sqrt{(...TRUNCATED)	65	aime24	[151644,872,198,10061,400,1867,6484,3,387,264,27853,32507,291,2248,1741,429,400,1867,28,6484,28,1124(...TRUNCATED)	13,610	"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)	13,449	161	1	[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)
"Let $\\mathcal{B}$ be the set of rectangular boxes with surface area $54$ and volume $23$. Let $r$ (...TRUNCATED)	66	aime24	[151644,872,198,10061,57960,10374,5416,90,33,31716,387,279,738,315,51424,14697,448,7329,3082,400,20,(...TRUNCATED)	16,066	"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)	15,960	106	1	[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)
"There exist real numbers $x$ and $y$, both greater than 1, such that $\\log_x\\left(y^x\\right)=\\l(...TRUNCATED)	67	aime24	[151644,872,198,3862,3000,1931,5109,400,87,3,323,400,88,54876,2176,7046,1091,220,16,11,1741,429,5796(...TRUNCATED)	8,098	"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)	8,039	59	1	[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)
"Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take tur(...TRUNCATED)	68	aime24	[151644,872,198,61686,323,14261,1486,279,2701,1809,13,362,5611,315,400,77,3,11211,15448,1573,1105,13(...TRUNCATED)	9,978	"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)	9,870	108	1	[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)
"Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers(...TRUNCATED)	69	aime24	[151644,872,198,41,268,28833,264,38239,553,21132,400,19,3,12460,5109,504,400,50,34433,90,16,11,17,11(...TRUNCATED)	6,587	"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)	6,453	134	1	[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)
"Consider the paths of length $16$ that follow the lines from the lower left corner to the upper rig(...TRUNCATED)	71	aime24	[151644,872,198,37175,279,12716,315,3084,400,16,21,3,429,1795,279,5128,504,279,4722,2115,9131,311,27(...TRUNCATED)	10,180	"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)	10,116	64	1	[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)
"Find the largest possible real part of \\[(75+117i)z+\\frac{96+144i}{z}\\]where $z$ is a complex nu(...TRUNCATED)	72	aime24	[151644,872,198,9885,279,7772,3204,1931,949,315,1124,9697,22,20,10,16,16,22,72,8,89,41715,37018,90,2(...TRUNCATED)	9,952	"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)	9,895	57	1	[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)

Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, including $t$ minutes spent in the coffee shop. Suppose Aya walks at $s+\frac{1}{2}$ kilometers per hour. Find the number of minutes the walk takes her, including the $t$ minutes spent in the coffee shop.

60

aime24

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6,195

This is a complex or challenging question, and it is difficult to provide a direct and correct answer. I need to think about it. Well, let's tackle this problem step by step. So, Aya walks 9 kilometers every morning, stops at a coffee shop for t minutes, and the time she takes is the walking time plus the coffee shop time. Wait, actually, the problem says "the walk takes her 4 hours, including t minutes spent in the coffee shop." Hmm, maybe I need to clarify: is the total time (walking + coffee) equal to the time given? Let's check the problem statement again. "When she walks at a constant speed of s kilometers per hour, the walk takes her 4 hours, including t minutes spent in the coffee shop." Wait, that wording is a bit confusing. Maybe "the walk" refers to the entire trip? Like, she walks 9 km, then stops at coffee shop for t minutes, so total time is walking time + t minutes. But the problem says "the walk takes her 4 hours, including t minutes spent in the coffee shop." Wait, maybe "walk" here is being used loosely? Wait, no, let's parse it carefully. Wait, maybe better to define variables clearly. Let's let: - Distance walked: 9 km (this is fixed, right? She goes for a 9-km-long walk, so the walking distance is always 9 km) - Time spent walking: let's call this T1 when speed is s, T2 when speed is s+2, etc. - Time spent at coffee shop: t minutes (this is the same in both cases, as per problem: "including t minutes spent in the coffee shop" for both scenarios) - Total time for the trip (walking + coffee): for first case, 4 hours; second case, 2 hours 24 minutes. Oh! That must be it. The problem says "the walk takes her 4 hours, including t minutes spent in the coffee shop." Wait, maybe the problem is using "walk" to mean "the entire activity" (walking and coffee), but maybe not. Wait, no, let's see: if it's "the walk takes her 4 hours, including t minutes in coffee," that would imply that the coffee time is part of the walk time? But that seems odd. Wait, no, probably a translation issue or wording issue. Let's think mathematically. Let's denote: Let W be the time (in hours) she spends walking (not including coffee shop). Then, when she walks at speed s, time walking is W = 9 / s hours. Then total time for the trip (walking + coffee) is W + (t / 60) hours? Wait, but the problem says "the walk takes her 4 hours, including t minutes spent in the coffee shop." Wait, maybe "including" here means that the total time is 4 hours, which consists of the walking time and the coffee time. So total time = walking time + coffee time. Yes, that makes sense. So for the first scenario: Total time = 4 hours = (time walking at speed s) + (time at coffee shop, which is t minutes) For the second scenario: Total time = 2 hours 24 minutes = (time walking at speed s+2) + (time at coffee shop, which is still t minutes) Ah, that's probably the key. The coffee shop time is the same t minutes in both cases, so the difference in total times is equal to the difference in walking times. Let's formalize this with units to avoid confusion. Let's work in minutes for time, maybe? Wait, or hours, but convert t to hours at the end. Let's see. First, convert all times to hours to be consistent with speed in km per hour. First scenario: Total time: 4 hours Time spent walking: 9 / s hours (since distance = speed * time => time = distance / speed) Time spent at coffee shop: t minutes = t / 60 hours So total time = walking time + coffee time => 4 = (9 / s) + (t / 60) ...(1) Second scenario: Total time: 2 hours 24 minutes. Let's convert that to hours: 2 + 24/60 = 2 + 0.4 = 2.4 hours, or 12/5 hours (since 24/60 = 2/5, so 2 + 2/5 = 12/5). Maybe fractions are better to be precise. Time spent walking: 9 / (s + 2) hours (speed is s + 2 km/h) Time spent at coffee shop: still t minutes = t / 60 hours So total time = walking time + coffee time => 12/5 = (9 / (s + 2)) + (t / 60) ...(2) Okay, now we have two equations (1) and (2) with two unknowns: s and t. Our goal is to find, when she walks at speed s + 1/2 km/h, what's the total time (walking time + coffee time) in minutes. Let's write equations (1) and (2) again clearly: From (1): 4 = 9/s + t/60 --> let's solve for t/60 to make it easier, since t is the same in both, so the coffee time in hours is constant. Let me denote C = t / 60, so C is the coffee time in hours (so t = 60C minutes). Then equations become: (1): 4 = 9/s + C --> so C = 4 - 9/s ...(1a) (2): 12/5 = 9/(s + 2) + C --> substitute C from (1a) into (2): 12/5 = 9/(s + 2) + 4 - 9/s Okay, now we can solve for s here. Let's do that step by step. First, let's rearrange the equation to get all terms on one side, maybe subtract 4 from both sides: 12/5 - 4 = 9/(s + 2) - 9/s Compute 12/5 - 4: 4 is 20/5, so 12/5 - 20/5 = -8/5 So left side is -8/5, right side is 9/(s + 2) - 9/s Let's compute the right side: common denominator s(s + 2), so [9s - 9(s + 2)] / [s(s + 2)] = [9s - 9s - 18] / [s(s + 2)] = (-18) / [s(s + 2)] So now we have: -8/5 = (-18) / [s(s + 2)] Hmm, nice, the negatives cancel out if we multiply both sides by -1: 8/5 = 18 / [s(s + 2)] Now cross-multiply to solve for s(s + 2): 8 * s(s + 2) = 5 * 18 Calculate right side: 5*18=90 Left side: 8s(s + 2) = 8s² + 16s So 8s² + 16s = 90 Let's simplify this equation. Divide both sides by 2 to make numbers smaller: 4s² + 8s = 45 Bring all terms to left side: 4s² + 8s - 45 = 0 Now, quadratic in s. Let's try to solve using quadratic formula. For equation ax² + bx + c = 0, solutions x = [-b ± sqrt(b² - 4ac)]/(2a) Here, a = 4, b = 8, c = -45 Discriminant D = b² - 4ac = 64 - 4*4*(-45) = 64 + 720 = 784 Oh, 784 is a perfect square! sqrt(784) = 28, good. So s = [-8 ± 28]/(2*4) = [-8 ± 28]/8 We have two solutions: first, take the plus: (-8 + 28)/8 = 20/8 = 5/2 = 2.5 Second, take the minus: (-8 - 28)/8 = (-36)/8 = -9/2 = -4.5. Speed can't be negative, so discard that. So s = 5/2 km/h? Wait, hold on, 5/2 is 2.5 km/h? Wait, but let's check if that makes sense with the walking times. Wait, if s = 2.5 km/h, then walking time at speed s is 9 / 2.5 = 3.6 hours, which is 3 hours 36 minutes. Then total time is 4 hours, so coffee time is 4 - 3.6 = 0.4 hours = 24 minutes. Let's check the second scenario: speed s + 2 = 4.5 km/h, walking time = 9 / 4.5 = 2 hours, total time 2 hours 24 minutes, so coffee time 24 minutes. Oh! Wait, that's exactly what the problem says: in both cases, including t minutes in coffee, so total time is walking time + t minutes. So in first case, walking time 3.6h = 216 min, total time 4h=240 min, so t=24 min. Second case, walking time 2h=120 min, total time 2h24m=144 min, so t=24 min. Perfect, that checks out. Wait, so maybe I didn't need to go through all the algebra with C, but anyway, we found s=2.5 km/h, which is 5/2 km/h, as we had. Wait, but let's just make sure we didn't misinterpret the problem. The problem says: "When she walks at a constant speed of s kilometers per hour, the walk takes her 4 hours, including t minutes spent in the coffee shop." So "the walk takes her 4 hours" – maybe "walk" here is the entire trip? Like, she walks (the 9 km) and then stops at coffee, so the entire activity is called "the walk" in this context? Maybe that's the confusion. But in our calculation, with s=2.5, it works: walking 9 km at 2.5 km/h takes 3.6 hours (216 minutes), plus 24 minutes at coffee, total 240 minutes = 4 hours. Then walking 9 km at 4.5 km/h takes 2 hours (120 minutes), plus 24 minutes at coffee, total 144 minutes = 2 hours 24 minutes. Perfect, that's exactly the problem statement. So our interpretation was correct: total time (walking + coffee) is given as 4 hours and 2h24m, with coffee time being t minutes in both cases, so the difference in total times is difference in walking times. Great, so now we know s=5/2 km/h, so s + 1/2 = 5/2 + 1/2 = 6/2 = 3 km/h. Wait, that's nice! s=2.5, so s + 0.5=3 km/h. Now, we need to find the total time when she walks at 3 km/h, including t minutes in coffee shop. We already found t=24 minutes from the check above, but let's confirm with equations to be thorough. From equation (1a): C = t/60 = 4 - 9/s. We have s=5/2, so 9/s=9/(5/2)=18/5=3.6 hours, so C=4 - 3.6=0.4 hours=24 minutes, so t=24 minutes. Correct. So when walking speed is 3 km/h, walking time is 9 / 3 = 3 hours. Then total time is walking time + coffee time = 3 hours + 24 minutes. Convert 3 hours to minutes: 3*60=180 minutes, plus 24 minutes is 204 minutes. Wait, hold on, is that it? Let's make sure we didn't skip any steps. Let's recap to check for errors. First, we set up equations based on total time = walking time + coffee time (in hours). Converted total times correctly: 2h24m=12/5 h=2.4h, yes. Then substituted coffee time as constant, solved for s, got s=2.5, which checks out with the time calculations (walking time 3.6h at 2.5km/h is 9km, correct; 9/4.5=2h, correct). Coffee time is 24 minutes in both cases, which is t=24. Then new speed: s + 1/2 = 2.5 + 0.5 = 3 km/h. Walking time: 9/3=3 hours=180 minutes. Total time: 180 + 24=204 minutes. Wait, but let's do it without converting to decimals to be precise, in case we made a decimal error. s=5/2 km/h (since we had s=20/8=5/2, yes, 5/2=2.5). s + 1/2 = 5/2 + 1/2 = 6/2 = 3 km/h, correct. Walking time at 3 km/h: 9 km / 3 km/h = 3 hours = 180 minutes. Coffee time t: from first scenario, total time 4 hours = walking time + t/60 hours. Walking time=9/s=9/(5/2)=18/5=3.6 hours=216 minutes, so t=4*60 - 216=240 - 216=24 minutes. Perfect, same as before. So total time: 180 + 24=204 minutes. Wait, but let me think again: is there any chance that the problem meant that "the walk takes her 4 hours" refers to just the walking time, and "including t minutes" is a misphrase? Let's check that interpretation, just to be safe, because that could be a critical mistake. Alternative interpretation: "When she walks at speed s, the walk (walking time) takes 4 hours, and she spends t minutes at coffee shop afterwards." But the problem says "the walk takes her 4 hours, including t minutes spent in the coffee shop." The word "including" suggests that t minutes are part of the 4 hours. So that interpretation would be wrong because "including" means the 4 hours consists of walking and coffee. Wait, another way: maybe "the walk" is the entire trip, so "the walk" is walking + coffee, so "takes her 4 hours" is the total time, which includes the coffee time. That's what we did, and it worked out with the numbers, so that's probably right. Let's test the other interpretation just in case. Suppose "the walk takes her 4 hours" is walking time only, so walking time=4 hours, then coffee time is t minutes, so total time=4h + t minutes. Then second scenario: walking time=2h24m, coffee time=t minutes, total time=2h24m + t minutes. But then the problem says "including t minutes spent in the coffee shop" – if "the walk takes her 4 hours" is walking time, then "including t minutes" would be confusing. The problem says "the walk takes her 4 hours, including t minutes...", so the "including" is modifying "4 hours", so the 4 hours includes the t minutes. So our first interpretation is correct. Also, when we did the calculation, the numbers worked out perfectly: walking time at s=2.5 is 3.6h, so total time 4h implies coffee time 0.4h=24m; walking time at s+2=4.5 is 2h, total time 2.4h implies coffee time 0.4h=24m. So t=24 is consistent, which is a good check. So then, for speed s + 1/2=3 km/h, walking time=9/3=3h=180m, coffee time=24m, total=204m. Wait, but let's make sure we didn't miscalculate the quadratic. Let's go back to the equation step to verify. We had from (1): 4 = 9/s + t/60 From (2): 12/5 = 9/(s+2) + t/60 Subtract equation (1) minus equation (2): 4 - 12/5 = 9/s - 9/(s+2) Wait, hold on! Wait a second, earlier I did (2) minus (1) with signs, but let's do this subtraction to see if we get the same. Left side: 4 - 12/5 = 20/5 - 12/5 = 8/5 Right side: 9/s - 9/(s+2) = 9[ (s+2 - s)/(s(s+2)) ] = 9[2/(s(s+2))] = 18/[s(s+2)] Ah! Here we go, this is a better way to subtract equations to avoid sign errors. So: 8/5 = 18/[s(s + 2)] Which is exactly what we had before (since earlier we had -8/5 = -18/[s(s+2)], so same equation). So that's correct, so 8s(s+2)=90, 4s²+8s-45=0, solutions s=(-8±28)/8, so 20/8=5/2, correct. So s=5/2 is definitely right. Therefore, s + 1/2=3, walking time=9/3=3 hours=180 minutes, t=24 minutes, total=204 minutes. Is there any chance the problem wants the answer in hours? No, it says "the number of minutes the walk takes her, including the t minutes", so minutes, so 204. Wait, let's just compute t another way to be 100% sure. From first scenario: total time 4 hours = 240 minutes. Walking time at s km/h: 9/s hours = (9/s)*60 minutes. So coffee time t=240 - (9*60)/s. Second scenario: total time 2h24m=144 minutes. Walking time at s+2 km/h: 9/(s+2) hours=(9*60)/(s+2) minutes. Coffee time t=144 - (9*60)/(s+2). Set equal: 240 - 540/s = 144 - 540/(s+2) Let's solve this equation in minutes to see if we get the same s. 240 - 144 = 540/s - 540/(s+2) 96 = 540[ (s+2 - s)/(s(s+2)) ] = 540[2/(s(s+2))] = 1080/[s(s+2)] So 96 = 1080/[s(s+2)] => s(s+2)=1080/96 Simplify 1080/96: divide numerator and denominator by 24: 1080÷24=45, 96÷24=4, so 45/4. Wait, 24*45=1080? 24*40=960, 24*5=120, 960+120=1080, yes. 24*4=96, yes. So s(s+2)=45/4. Wait, hold on! Wait, in this equation, we have s(s+2)=45/4? Wait, but earlier we had 8s(s+2)=90, so s(s+2)=90/8=45/4, yes! Same thing. So s² + 2s - 45/4=0, multiply by 4: 4s² +8s -45=0, same quadratic. So s=(-8±sqrt(64 + 720))/8=(-8±28)/8, so s=20/8=5/2, correct. So t=240 - (9*60)/(5/2)=240 - (540)/(5/2)=240 - 540*(2/5)=240 - 216=24 minutes. Perfect, matches. Then for speed s + 1/2=3 km/h, walking time in minutes: (9 km)/(3 km/h)=3 hours=180 minutes, so total time=180 +24=204 minutes. I think that's solid. The key was correctly interpreting that the total time given (4 hours, 2h24m) includes the coffee shop time, which is the same t minutes in both cases, so we can set up equations to solve for s and t, then compute the new total time. I think the potential pitfall here would be misinterpreting whether the coffee time is included in the total time or not, but since the problem says "including t minutes spent in the coffee shop", it's definitely part of the total time. Also, when we tested the numbers with s=2.5, it worked out perfectly with the coffee time being 24 minutes in both scenarios, so that gives confidence. Another way someone might mess up is by forgetting to convert minutes to hours or vice versa, but by working in minutes for the coffee time (as we did in the second approach), we could verify without dealing with hours for t, which is helpful. Let me just check once more: if she walks at 3 km/h, 9 km takes 3 hours, which is 180 minutes. She spends 24 minutes at coffee, so total time 204 minutes. 204 minutes is 3 hours and 24 minutes, which is 3.4 hours. Let's see if that's consistent with the problem's structure, but since we already have all the pieces, I think that's the answer. I don't see any mistakes in the reasoning now. Initially, I was a bit confused by the wording "the walk takes her 4 hours, including t minutes", but by testing with the numbers we got, it all checks out, so that interpretation must be correct. So the final answer should be 204 minutes. </think> To solve the problem, we interpret the key phrase: **"the walk takes her 4 hours, including $ t $ minutes spent in the coffee shop."** This means that the **total time** (walking + coffee) is given as 4 hours for a speed of $ s $ km/h, and 2 hours and 24 minutes for a speed of $ s + 2 $ km/h. The coffee shop time $ t $ (in minutes) is the same in both scenarios. --- ### Step 1: Convert Total Times to Hours - First scenario: $ 4 $ hours - Second scenario: $ 2 + \frac{24}{60} = \frac{12}{5} $ hours Let: - $ C = \frac{t}{60} $ be the coffee time in **hours** - Walking time at speed $ s $: $ \frac{9}{s} $ hours - Walking time at speed $ s + 2 $: $ \frac{9}{s+2} $ hours From the problem: $$ \begin{align*} (1) & \quad 4 = \frac{9}{s} + C \\ (2) & \quad \frac{12}{5} = \frac{9}{s+2} + C \end{align*} $$ Subtracting equation (1) from equation (2): $$ 4 - \frac{12}{5} = \frac{9}{s} - \frac{9}{s+2} $$ Simplify left-hand side: $$ \frac{20}{5} - \frac{12}{5} = \frac{8}{5} $$ Right-hand side: $$ \frac{9}{s} - \frac{9}{s+2} = 9 \left( \frac{1}{s} - \frac{1}{s+2} \right) = 9 \cdot \frac{(s+2) - s}{s(s+2)} = \frac{18}{s(s+2)} $$ So, $$ \frac{8}{5} = \frac{18}{s(s+2)} \Rightarrow s(s+2) = \frac{18 \cdot 5}{8} = \frac{90}{8} = \frac{45}{4} $$ Now solve the quadratic: $$ s^2 + 2s - \frac{45}{4} = 0 $$ Multiply through by 4: $$ 4s^2 + 8s - 45 = 0 $$ Use the quadratic formula: $$ s = \frac{-8 \pm \sqrt{8^2 + 4 \cdot 4 \cdot 45}}{2 \cdot 4} = \frac{-8 \pm \sqrt{64 + 720}}{8} = \frac{-8 \pm \sqrt{784}}{8} = \frac{-8 \pm 28}{8} $$ Take the positive root: $$ s = \frac{20}{8} = \frac{5}{2} = 2.5 \text{ km/h} $$ --- ### Step 2: Find Coffee Time $ t $ From equation (1): $$ C = 4 - \frac{9}{s} = 4 - \frac{9}{2.5} = 4 - 3.6 = 0.4 \text{ hours} $$ Convert to minutes: $$ t = 0.4 \times 60 = 24 \text{ minutes} $$ --- ### Step 3: Compute New Total Time at Speed $ s + \frac{1}{2} $ $$ s + \frac{1}{2} = 2.5 + 0.5 = 3 \text{ km/h} $$ Walking time at 3 km/h: $$ \frac{9}{3} = 3 \text{ hours} = 180 \text{ minutes} $$ Add coffee time: $$ 180 + 24 = \boxed{204} \text{ minutes} $$ --- ### ✅ Final Answer: $$ \boxed{204} $$<|im_end|>

6,052

143

1

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23, 15, 44510, 82, 1141, 10, 17, 7252, 589, 274, 1141, 10, 17, 11730, 16, 15, 23, 15, 14, 24, 21, 271, 50, 70206, 220, 16, 15, 23, 15, 14, 24, 21, 25, 21749, 63533, 323, 46912, 553, 220, 17, 19, 25, 220, 16, 15, 23, 15, 144422, 17, 19, 28, 19, 20, 11, 220, 24, 21, 144422, 17, 19, 28, 19, 11, 773, 220, 19, 20, 14, 19, 13, 13824, 11, 220, 17, 19, 9, 19, 20, 28, 16, 15, 23, 15, 30, 220, 17, 19, 9, 19, 15, 28, 24, 21, 15, 11, 220, 17, 19, 9, 20, 28, 16, 17, 15, 11, 220, 24, 21, 15, 10, 16, 17, 15, 28, 16, 15, 23, 15, 11, 9834, 13, 220, 17, 19, 9, 19, 28, 24, 21, 11, 9834, 13, 2055, 274, 1141, 10, 17, 11730, 19, 20, 14, 19, 382, 14190, 11, 3331, 389, 0, 13824, 11, 304, 419, 23606, 11, 582, 614, 274, 1141, 10, 17, 11730, 19, 20, 14, 19, 30, 13824, 11, 714, 6788, 582, 1030, 220, 23, 82, 1141, 10, 17, 11730, 24, 15, 11, 773, 274, 1141, 10, 17, 11730, 24, 15, 14, 23, 28, 19, 20, 14, 19, 11, 9834, 0, 25739, 3166, 13, 2055, 274, 29456, 488, 220, 17, 82, 481, 220, 19, 20, 14, 19, 28, 15, 11, 30270, 553, 220, 19, 25, 220, 19, 82, 29456, 488, 23, 82, 481, 19, 20, 28, 15, 11, 1852, 79151, 13, 2055, 274, 63242, 23, 37021, 26888, 7, 21, 19, 488, 220, 22, 17, 15, 30085, 23, 63242, 23, 37021, 17, 23, 5620, 23, 11, 773, 274, 28, 17, 15, 14, 23, 28, 20, 14, 17, 11, 4396, 382, 4416, 259, 28, 17, 19, 15, 481, 320, 24, 9, 21, 15, 24254, 20, 14, 17, 11730, 17, 19, 15, 481, 320, 20, 19, 15, 24254, 20, 14, 17, 11730, 17, 19, 15, 481, 220, 20, 19, 15, 6599, 17, 14, 20, 11730, 17, 19, 15, 481, 220, 17, 16, 21, 28, 17, 19, 4420, 13, 23239, 11, 9071, 382, 12209, 369, 4628, 274, 488, 220, 16, 14, 17, 28, 18, 13136, 7530, 11, 11435, 882, 304, 4420, 25, 320, 24, 13136, 24254, 18, 13136, 7530, 11730, 18, 4115, 28, 16, 23, 15, 4420, 11, 773, 2790, 882, 28, 16, 23, 15, 488, 17, 19, 28, 17, 15, 19, 4420, 382, 40, 1744, 429, 594, 6437, 13, 576, 1376, 572, 12440, 65644, 429, 279, 2790, 882, 2661, 320, 19, 4115, 11, 220, 17, 71, 17, 19, 76, 8, 5646, 279, 10799, 8061, 882, 11, 892, 374, 279, 1852, 259, 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488, 10799, 8, 374, 2661, 438, 220, 19, 4115, 369, 264, 4628, 315, 400, 274, 400, 13136, 7530, 11, 323, 220, 17, 4115, 323, 220, 17, 19, 4420, 369, 264, 4628, 315, 400, 274, 488, 220, 17, 400, 13136, 7530, 13, 576, 10799, 8061, 882, 400, 259, 400, 320, 258, 4420, 8, 374, 279, 1852, 304, 2176, 25283, 382, 44364, 14374, 14822, 220, 16, 25, 7169, 10657, 8523, 311, 29092, 271, 12, 5512, 15048, 25, 400, 220, 19, 400, 4115, 198, 12, 10440, 15048, 25, 400, 220, 17, 488, 1124, 37018, 90, 17, 19, 15170, 21, 15, 92, 284, 1124, 37018, 90, 16, 17, 15170, 20, 92, 400, 4115, 271, 10061, 510, 12, 400, 356, 284, 1124, 37018, 90, 83, 15170, 21, 15, 92, 400, 387, 279, 10799, 882, 304, 3070, 30382, 1019, 12, 39055, 882, 518, 4628, 400, 274, 400, 25, 400, 1124, 37018, 90, 24, 15170, 82, 92, 400, 4115, 198, 12, 39055, 882, 518, 4628, 400, 274, 488, 220, 17, 400, 25, 400, 1124, 37018, 90, 24, 15170, 82, 10, 17, 92, 400, 4115, 271, 3830, 279, 3491, 1447, 14085, 198, 59, 7265, 90, 6612, 9, 532, 7, 16, 8, 609, 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"Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$(...TRUNCATED)

61

aime24

[151644,872,198,10061,400,25411,3,387,264,21495,1640,17433,304,12671,57960,32696,12947,6771,279,2187(...TRUNCATED)

26,425

"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)

26,309

116

1

[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)

"Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$(...TRUNCATED)

64

aime24

[151644,872,198,10061,400,79,3,387,279,3245,10250,1372,369,892,1052,6724,264,6785,7546,400,77,3,1741(...TRUNCATED)

22,935

"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)

22,860

75

1

[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)

"Let $ABCD$ be a tetrahedron such that $AB=CD= \\sqrt{41}$, $AC=BD= \\sqrt{80}$, and $BC=AD= \\sqrt{(...TRUNCATED)

65

aime24

[151644,872,198,10061,400,1867,6484,3,387,264,27853,32507,291,2248,1741,429,400,1867,28,6484,28,1124(...TRUNCATED)

13,610

"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)

13,449

161

1

[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)

"Let $\\mathcal{B}$ be the set of rectangular boxes with surface area $54$ and volume $23$. Let $r$ (...TRUNCATED)

66

aime24

[151644,872,198,10061,57960,10374,5416,90,33,31716,387,279,738,315,51424,14697,448,7329,3082,400,20,(...TRUNCATED)

16,066

"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)

15,960

106

1

[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)

"There exist real numbers $x$ and $y$, both greater than 1, such that $\\log_x\\left(y^x\\right)=\\l(...TRUNCATED)

67

aime24

[151644,872,198,3862,3000,1931,5109,400,87,3,323,400,88,54876,2176,7046,1091,220,16,11,1741,429,5796(...TRUNCATED)

8,098

"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)

8,039

59

1

[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)

"Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take tur(...TRUNCATED)

68

aime24

[151644,872,198,61686,323,14261,1486,279,2701,1809,13,362,5611,315,400,77,3,11211,15448,1573,1105,13(...TRUNCATED)

9,978

"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)

9,870

108

1

[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)

"Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers(...TRUNCATED)

69

aime24

[151644,872,198,41,268,28833,264,38239,553,21132,400,19,3,12460,5109,504,400,50,34433,90,16,11,17,11(...TRUNCATED)

6,587

"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)

6,453

134

1

[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)

"Consider the paths of length $16$ that follow the lines from the lower left corner to the upper rig(...TRUNCATED)

71

aime24

[151644,872,198,37175,279,12716,315,3084,400,16,21,3,429,1795,279,5128,504,279,4722,2115,9131,311,27(...TRUNCATED)

10,180

"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)

10,116

64

1

[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)

"Find the largest possible real part of \\[(75+117i)z+\\frac{96+144i}{z}\\]where $z$ is a complex nu(...TRUNCATED)

72

aime24

[151644,872,198,9885,279,7772,3204,1931,949,315,1124,9697,22,20,10,16,16,22,72,8,89,41715,37018,90,2(...TRUNCATED)

9,952

"This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED)

9,895

57

1

[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED)