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11.17. How many roots does the equation $\sin x=\frac{x}{100}$ have?
## 11.4. Sums of Sines and Cosines Related to Regular Polygons
In solving the problems of this section, the following geometric problem is useful. | 11.17. Answer: 63. First, note that the number of positive roots is equal to the number of negative roots, and there is also a root 0. Therefore, it is sufficient to verify that the number of positive roots is 31. If $\sin x = x / 100$, then $|x| = 100|\sin x| \leqslant 100$. Consider the graphs of the functions $y = x... | 63 | Calculus | math-word-problem | olympiads | 2e133df9-a038-55ff-844d-6a56f5c1b3b2 | [
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"<think>\nOkay, so I need to figure out how many roots the equation sin x = x/100 has. Hmm, let's see. I remember that when solving equations involving sine and linear functions, graphical methods are often useful. The idea is to find the points where the two functions intersect, right? So, in this case, I need to ... | [
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Okay, so I have this equation: sin x equals x over 100. And I need to find how many roots it has. Hmm, let's see. I remember that equations involving sine and linear functions usually have solutions where their graphs intersect. So maybe I can approach this by graphing both y = sin x and y = x/100 and see how m... | 35,712 | <|im_start|>system
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11.17. How many roots does the equation $\sin x=\frac{x}{100}$ have?
## 11.4. Sums of Sines and Cosines Related to Regular Polygons
In solving the problems of this section, the following geometric proble... | 521 |
"3 Given a positive integer $n$, find the smallest positive number $\\lambda$, such that for any $\\(...TRUNCATED) | "When $n=1$, $\\cos \\theta_{1}=\\left(1+\\tan ^{2} \\theta_{1}\\right)^{-\\frac{1}{2}}=\\frac{\\sqr(...TRUNCATED) | n-1 | Inequalities | math-word-problem | olympiads | 63b0d30d-0985-5266-bc29-e7bf1f530daf | [
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"Example 20 (22nd All-Soviet Union Mathematical Olympiad, 1988) Let $a$ and $d$ be non-negative numb(...TRUNCATED) | "If we simultaneously swap $a$ with $d$ and $b$ with $c$, the conditions and conclusions of the prob(...TRUNCATED) | \sqrt{2}-\frac{1}{2} | Inequalities | math-word-problem | olympiads | f2ed0f4e-6070-5113-ba64-31e21d50992e | [
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"1. A social club has $2 k+1$ members, each of whom is proficient in the same $k$ languages, and any(...TRUNCATED) | "1. The maximum value of $A$ is $\\frac{2 k(k-2)(2 k+1)}{3}$.\n\nConsider the social club as a compl(...TRUNCATED) | \frac{2 k(k-2)(2 k+1)}{3} | Combinatorics | math-word-problem | cn_contest | 3ef9e807-c9b4-52c2-acf1-ea64cb63c47d | [
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"8. (4 points) A triangle is divided into 1000 triangles. What is the minimum number of distinct poi(...TRUNCATED) | "Answer: 503\n\nSolution: The sum of the angles of a triangle is $180^{\\circ}$, so for a thousand t(...TRUNCATED) | 503 | Geometry | math-word-problem | olympiads | c1665e90-e27b-5802-be7c-4ca1e00d67a2 | [
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"5. Ms. Olga Ivanovna, the class teacher of 5B, is staging a \"Mathematical Ballet.\" She wants to a(...TRUNCATED) | "Answer: 20 girls.\n\nSolution: Let's select and fix two arbitrary boys - $\\mathrm{M}_{1}$ and $\\m(...TRUNCATED) | 20 | Combinatorics | math-word-problem | olympiads | 53f890e0-da4e-53e0-a40b-6282481e246a | [
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"During a secret meeting, 20 interns elect their favorite animator. Each votes for two animators. It(...TRUNCATED) | "Suppose there is no animator who has 20 votes (which is not a case that minimizes the winner's scor(...TRUNCATED) | 14 | Combinatorics | math-word-problem | olympiads | baeec787-c2a0-5c93-85c9-1d92f1becc02 | [
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"Suppose that $X_1, X_2, \\ldots$ are real numbers between 0 and 1 that are chosen independently and(...TRUNCATED) | "1. **Define the problem and notation:**\n We are given a sequence of real numbers \\( X_1, X_2, \(...TRUNCATED) | 2\sqrt{e} - 3 | Calculus | math-word-problem | aops_forum | e67aea14-93f7-5f5a-a3b4-2d76e9689aca | [
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"Authors: Dumanov I., Krotovsky R.\n\nDiscussing winter holidays in class, Sasha said: \"Now, after (...TRUNCATED) | "Evaluation. Suppose three points were enough. They define a plane. If it passes through the center (...TRUNCATED) | 4 | Logic and Puzzles | math-word-problem | olympiads | ce373a17-5997-5652-a0be-e2188efe2623 | [
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"(17) Let the quadratic function $f(x)$ have a maximum value of 12 on the interval $[-1,4]$, and the(...TRUNCATED) | "(17 (1) According to the problem, let\n$$\nf(x)=a x(x-5)(a>0),\n$$\n\nwhich is\n$$\nf(x)=a\\left(x-(...TRUNCATED) | (-\infty,-5)\cup(1,+\infty) | Algebra | math-word-problem | olympiads | 4bc8a97d-fac9-546a-b27f-48f4c12c437b | [
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