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2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation. | 2. This problem is equivalent to finding the smallest positive integer $b$, such that the equation $7 b^{2}+7 b+7=x^{4}$, (1) has an integer solution for $x$. Since 7 is a prime number, it follows from equation (1) that 7 is a divisor of $x$. Therefore, let $x=7 k$, then equation (1) becomes $b^{2}+b+1=7^{3} k^{4}$. Th... | 18 | Number Theory | math-word-problem | olympiads | 39a6c760-dfa3-5201-aa87-105127951e16 | [
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"<think>\nOkay, let's tackle this problem. So, N is an integer represented as 777 in base b, and we need to find the smallest positive integer b such that N is a fourth power of an integer in decimal. Hmm, interesting. Let me break this down step by step.\n\nFirst, I need to recall how numbers are represented in di... | [
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"content": "<think>\nOkay, let's tackle this problem. So,... | <think>
Okay, so I need to find the smallest positive integer base \( b \) such that the number represented as 777 in base \( b \) is a perfect fourth power in decimal. Hmm, let's break this down.
First, the number \( N \) in base \( b \) is 777. To convert that to decimal, I need to calculate \( 7 \times b^2 + 7 \tim... | 12,841 | 13 | 80 | 0.594159 | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
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2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation... |
"2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution th(...TRUNCATED) | "2. Solution. Since the desired integer solution $x$ is divisible by 10 and 12, it is divisible by 6(...TRUNCATED) | 9599 | Algebra | math-word-problem | olympiads | 622e4608-b0d7-5071-90bd-fa39367cb1ac | [
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"18.3.19 $\\star \\star$ Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+b^{2}+c^{(...TRUNCATED) | "Given $44<\\sqrt{2005}<45,25<\\sqrt{\\frac{2005}{3}}<26$, so\n$$\n26 \\leqslant c \\leqslant 44 .\n(...TRUNCATED) | (23,24,30),(12,30,31),(9,30,32),(4,30,33),(15,22,36),(9,18,40),(4,15,42) | Number Theory | math-word-problem | olympiads | 1e57f229-b82e-5c03-ab5e-e465aef6a830 | [
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"7. Given positive integers $a, b$ satisfy\n$$\n\\sqrt{\\frac{a b}{2 b^{2}-a}}=\\frac{a+2 b}{4 b} \\(...TRUNCATED) | "7.2012.\n\nNotice,\n$$\n\\begin{array}{l}\n\\sqrt{\\frac{a b}{2 b^{2}-a}}=\\frac{a+2 b}{4 b} \\\\\n(...TRUNCATED) | 2012 | Algebra | math-word-problem | cn_contest | 2e8a9edd-dfb2-5c52-a7c3-ce50ae674f3a | [
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"6. Given the function $f(x)=|| x-1|-1|$, if the equation $f(x)=m(m \\in \\mathbf{R})$ has exactly 4(...TRUNCATED) | "6. $(-3,0)$.\n\nThe graph of the function $f(x)=|| x-1|-1|$ is shown in Figure 2. From the graph, w(...TRUNCATED) | (-3,0) | Algebra | math-word-problem | olympiads | 8d3475c2-0f5b-5832-b805-8f83f719a910 | [
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] | ["<think>\nOkay, so I need to find the range of the product x₁x₂x₃x₄ where x₁, x₂, x₃,(...TRUNCATED) | [
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] | null | null | 2 | [{"content":"6. Given the function $f(x)=|| x-1|-1|$, if the equation $f(x)=m(m \\in \\mathbf{R})$ h(...TRUNCATED) | "<think>\nOkay, so I have this problem here: Given the function f(x) = ||x - 1| - 1|, and if the equ(...TRUNCATED) | 17,818 | 260 | 63 | 0.513651 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many(...TRUNCATED) | "1. **Identify the problem and the goal:**\n We need to determine all integers \\( n \\) for which(...TRUNCATED) | n = 4 | Geometry | math-word-problem | aops_forum | 84b967ad-0515-5261-b6ef-e7ad07e90929 | [
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] | 1 | [{"content":"To [i]dissect [/i] a polygon means to divide it into several regions by cutting along f(...TRUNCATED) | "<think>\nOkay, so I need to figure out for which integers n it's possible to dissect a regular n-si(...TRUNCATED) | 42,509 | 309 | 81 | 0.516286 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"Given a regular polygon with $n$ sides. It is known that there are $1200$ ways to choose three of t(...TRUNCATED) | "1. **Understanding the problem**: We need to find the number of sides \\( n \\) of a regular polygo(...TRUNCATED) | 50 | Geometry | math-word-problem | aops_forum | 65a25bad-4af1-58e3-b4c2-dacbffbeaa91 | [
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] | ["<think>\nOkay, so I need to find the number of sides, n, of a regular polygon where there are 1200(...TRUNCATED) | [
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"An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ra(...TRUNCATED) | "1. Given the centroid of the triangle is at \\((1,1)\\), we use the centroid formula:\n \\[\n \(...TRUNCATED) | 107 | Geometry | math-word-problem | aops_forum | 16f898da-1752-53b1-a929-32726b10ddc1 | [
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] | ["<think>\nOkay, so I have this problem here about a triangle with some given conditions. Let me try(...TRUNCATED) | [
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"Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white choco(...TRUNCATED) | "To solve this problem, we need to determine the probability that exactly 3 of the 5 trapezoidal pie(...TRUNCATED) | 39 | Combinatorics | math-word-problem | aops_forum | 4c505060-aa74-5805-a011-56534fe6adf6 | [
true,
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] | ["<think>\nOkay, let's see. So we have this chocolate bar that's a rectangle, half dark and half whi(...TRUNCATED) | [
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] | null | null | 2 | [{"content":"Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half(...TRUNCATED) | "<think>\nOkay, let's try to figure out this problem step by step. So, we have a rectangular chocola(...TRUNCATED) | 44,096 | 682 | 62 | 0.568543 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"Find the smallest positive integer $k$ such that \\[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) (...TRUNCATED) | "1. **Apply the Cauchy-Schwarz Inequality:**\n The Cauchy-Schwarz Inequality states that for any s(...TRUNCATED) | k = 1297 | Inequalities | math-word-problem | aops_forum | 3516c6c2-b131-53d6-964d-9b07cc1e286d | [
true,
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] | ["<think>\nOkay, so I need to find the smallest positive integer k such that the product (16a² + 36(...TRUNCATED) | [
true,
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] | null | null | 2 | [{"content":"Find the smallest positive integer $k$ such that \\[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b(...TRUNCATED) | "<think>\nOkay, so I need to find the smallest positive integer k such that the product (16a² + 36b(...TRUNCATED) | 39,154 | 748 | 77 | 0.596873 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
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