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Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$ | 1. Define the integral \( I_n(x) = \int_x^{2x} e^{-t^n} \, dt \). We need to find the value of \( x \) that maximizes \( I_n(x) \).
2. To find the maximum, we first compute the derivative of \( I_n(x) \) with respect to \( x \):
\[
I_n'(x) = \frac{d}{dx} \left( \int_x^{2x} e^{-t^n} \, dt \right)
\]
Using t... | -\ln 2 | Calculus | math-word-problem | aops_forum | 167ecbbb-4270-534f-afa5-75d01d968ab5 | [
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"<think>\nOkay, so I need to find the limit as n approaches infinity of the natural logarithm of a_n, where a_n is the value that maximizes the integral from x to 2x of e^{-t^n} dt for x >= 0. Hmm, let's break this down step by step.\n\nFirst, I remember that to find the value of x that maximizes a function, we tak... | [
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] | null | null | 4 | [
{
"content": "Let $a_n\\ (n\\geq 1)$ be the value for which $\\int_x^{2x} e^{-t^n}dt\\ (x\\geq 0)$ is maximal. Find $\\lim_{n\\to\\infty} \\ln a_n.$",
"role": "user"
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{
"content": "<think>\nOkay, so I need to find the limit as n approaches infinity of the natural logarithm of a_n, where a_n is the... | <think>
Okay, let me try to work through this problem step by step. The problem is about finding the limit of the natural logarithm of \( a_n \) as \( n \) approaches infinity, where \( a_n \) is the value that maximizes the integral \( \int_x^{2x} e^{-t^n} dt \) for \( x \geq 0 \). Hmm, that sounds a bit involved, but... | 11,853 | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$<|im_end|>
<|im_start|>assistant
<|im_start|>think
Okay, let me try to work thr... |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{3 / 2} \frac{x^{2} \cdot d x}{\sqrt{9-x^{2}}}
$$ | ## Solution
$$
\int_{0}^{3 / 2} \frac{x^{2} \cdot d x}{\sqrt{9-x^{2}}}=
$$
Substitution:
$$
\begin{aligned}
& x=3 \sin t \Rightarrow d x=3 \cos t d t \\
& x=0 \Rightarrow t=\arcsin \frac{0}{3}=0 \\
& x=\frac{3}{2} \Rightarrow t=\arcsin \frac{\left(\frac{3}{2}\right)}{3}=\arcsin \frac{1}{2}=\frac{\pi}{6}
\end{aligned... | \frac{3\pi}{4}-\frac{9\sqrt{3}}{8} | Calculus | math-word-problem | olympiads | 7721db8b-0313-545f-a292-7116ab4a5ec3 | [
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"<think>\nOkay, so I need to calculate the definite integral of x squared divided by the square root of (9 minus x squared) from 0 to 3/2. Hmm, let me think about how to approach this. Integrals with square roots in the denominator often suggest a trigonometric substitution, especially when there's a term like a^2 ... | [
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"content": "## Problem Statement\n\nCalculate the definite integral:\n\n$$\n\\int_{0}^{3 / 2} \\frac{x^{2} \\cdot d x}{\\sqrt{9-x^{2}}}\n$$",
"role": "user"
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{
"content": "<think>\nOkay, so I need to calculate the definite integral of x squared over the square root of (9 minus x squared) from 0 ... | <think>
Okay, so I need to calculate the definite integral of x squared over the square root of (9 minus x squared) from 0 to 3/2. Hmm, let me think. Integrals with square roots in the denominator often involve trigonometric substitution, right? The denominator here is sqrt(9 - x²), which resembles the form sqrt(a² - x... | 8,595 | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{3 / 2} \frac{x^{2} \cdot d x}{\sqrt{9-x^{2}}}
$$<|im_end|>
<|im_start|>assistant
<|im_start|>think
Okay, so I need to calculate the def... |
"Condition of the problem\n\nFind the derivative.\n\n$$\ny=\\frac{1}{24}\\left(x^{2}+8\\right) \\sqr(...TRUNCATED) | "## Solution\n\n$$\n\\begin{aligned}\n& y^{\\prime}=\\left(\\frac{1}{24}\\left(x^{2}+8\\right) \\sqr(...TRUNCATED) | \frac{x^{3}-x}{8\sqrt{x^{2}-4}}+\frac{x}{8}\cdot\arcsin\frac{2}{x} | Calculus | math-word-problem | olympiads | df4f8552-e64a-5168-a68f-2561adbeabef | [
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"The function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x(...TRUNCATED) | "1. **Given Conditions and Initial Analysis:**\n - The function \\( f: \\mathbb{R} \\rightarrow \\(...TRUNCATED) | \frac{7}{3} | Calculus | math-word-problem | aops_forum | 1617e146-e118-5933-a973-11c199667efe | [
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] | null | null | 2 | [{"content":"The function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $(...TRUNCATED) | "<think>\nOkay, so I need to find the greatest real number m such that f(x) is greater than or equal(...TRUNCATED) | 19,532 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"Example 1. Find the general solution of the linear differential equation\n\n$$\ny^{\\prime}+y-k x=0(...TRUNCATED) | "Solution. The given equation is of the form (9.20), where $p(x)=1, q(x)=k x$. Let $y=u v$, then $y^(...TRUNCATED) | k(x-1)+Ce^{-x} | Calculus | math-word-problem | olympiads | dd0ba5bf-0c4a-51d8-b3b0-df59d6c5cff7 | [
true,
true
] | ["<think>\nOkay, so I need to find the general solution to the differential equation y' + y - kx = 0(...TRUNCATED) | [
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false
] | null | null | 1 | [{"content":"Example 1. Find the general solution of the linear differential equation\n\n$$\ny^{\\pr(...TRUNCATED) | "<think>\nOkay, so I need to find the general solution to the differential equation y' + y - kx = 0.(...TRUNCATED) | 4,923 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"Let there be a figure with $9$ disks and $11$ edges, as shown below.\n\nWe will write a real number(...TRUNCATED) | "1. **Model the problem as an electrical circuit:**\n - We are given a graph with 9 disks (nodes) (...TRUNCATED) | \frac{20}{33} | Calculus | math-word-problem | aops_forum | 69191341-fb5c-54c6-bc21-3d80836f8875 | [
true,
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] | ["<think>\nAlright, so I have this problem here where there's a figure with 9 disks labeled A to I a(...TRUNCATED) | [
false,
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] | null | null | 1 | [{"content":"Let there be a figure with $9$ disks and $11$ edges, as shown below.\n\nWe will write a(...TRUNCATED) | "<think>\nOkay, so I need to find the minimum sum of the squares of the differences between numbers (...TRUNCATED) | 23,176 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"## Problem Statement\n\nCalculate the definite integral:\n\n$$\n\\int_{0}^{\\frac{\\pi}{2}} \\frac{(...TRUNCATED) | "## Solution\n\nWe will use the universal substitution:\n\n$$\nt=\\operatorname{tg} \\frac{x}{2}\n$$(...TRUNCATED) | \frac{(9-4\sqrt{3})\pi}{18} | Calculus | math-word-problem | olympiads | 7ea4c352-2181-570e-9b05-b1286a85e44d | [
true,
true
] | ["<think>\nOkay, so I have this integral to solve: the definite integral from 0 to π/2 of cos x div(...TRUNCATED) | [
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] | null | null | 2 | [{"content":"## Problem Statement\n\nCalculate the definite integral:\n\n$$\n\\int_{0}^{\\frac{\\pi}(...TRUNCATED) | "<think>\nOkay, so I need to calculate the definite integral of cos x divided by (2 + cos x) from 0 (...TRUNCATED) | 14,314 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"3. The monotonic increasing interval of the function $f(x)$ $=\\frac{(x+1)^{2}}{x^{2}+1}$ is $\\qqu(...TRUNCATED) | "$$\nf(x)=1+\\frac{2 x}{x^{2}+1}, f^{\\prime}(x)=\\frac{2\\left(x^{2}+1\\right)-2 x \\cdot 2 x}{\\le(...TRUNCATED) | (-1,1) | Calculus | math-word-problem | olympiads | f8725ea0-4110-52a7-b5c7-4d3ea686281a | [
true,
true
] | ["<think>\nOkay, so I need to find the monotonic increasing interval of the function f(x) = (x + 1)^(...TRUNCATED) | [
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] | null | null | 1 | [{"content":"3. The monotonic increasing interval of the function $f(x)$ $=\\frac{(x+1)^{2}}{x^{2}+1(...TRUNCATED) | "<think>\nOkay, so I need to find the monotonic increasing interval of the function f(x) = (x + 1)^2(...TRUNCATED) | 8,425 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"## Problem Statement\n\nCalculate the limit of the function:\n\n$\\lim _{x \\rightarrow 2} \\frac{\(...TRUNCATED) | "## Solution\n\nSubstitution:\n\n$x=y+2 \\Rightarrow y=x-2$\n\n$x \\rightarrow 2 \\Rightarrow y \\ri(...TRUNCATED) | \frac{2}{3\pi} | Calculus | math-word-problem | olympiads | 4b8968e3-32d7-5798-a146-11af4a9470d2 | [
true,
true
] | ["<think>\nOkay, let's see. I need to find the limit as x approaches 2 of arctg(x² - 2x) divided by(...TRUNCATED) | [
true,
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] | null | null | 2 | [{"content":"## Problem Statement\n\nCalculate the limit of the function:\n\n$\\lim _{x \\rightarrow(...TRUNCATED) | "<think>\nOkay, so I need to find the limit as x approaches 2 of arctg(x² - 2x) divided by sin(3πx(...TRUNCATED) | 6,875 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rota(...TRUNCATED) | "To find the ratio \\(\\frac{V_2}{V_1}\\), we need to compute the volumes \\(V_1\\) and \\(V_2\\) ac(...TRUNCATED) | k | Calculus | math-word-problem | aops_forum | 4fa5519d-227a-5a82-867b-a3e2e38e9c1b | [
true,
true
] | ["<think>\nOkay, so I need to find the ratio V2/V1, where V1 is the volume obtained by rotating the (...TRUNCATED) | [
true,
true
] | null | [
"stop",
"stop"
] | 2 | [{"content":"Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generat(...TRUNCATED) | "<think>\nOkay, so I need to find the ratio V2/V1, where V1 is the volume obtained by rotating the a(...TRUNCATED) | 15,060 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
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