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DachengLi

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"Find the maximal possible finite number of roots of the equation $|x-a_1|+\\dots+|x-a_{50}|=|x-b_1(...TRUNCATED)

"1. **Define the function \\( f(x) \\):**\n \\[\n f(x) = \\sum_{i=1}^{50} |x - a_i| - \\sum_{i=1(...TRUNCATED)

"<think>\nOkay, so I need to find the maximal possible finite number of roots for the equation |x - (...TRUNCATED)

"<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED)

"8. Expand the binomial $\\left(\\frac{\\sqrt{x+1}}{2 \\sqrt[4]{x}}\\right)^{n}$ in descending power(...TRUNCATED)

"3\n8.【Analysis and Solution】It is not difficult to find that the coefficients of the first thre(...TRUNCATED)

"<think>\nOkay, so I need to expand the binomial \\(\\left(\\frac{\\sqrt{x+1}}{2 \\sqrt[4]{x}}\\righ(...TRUNCATED)

"<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED)

"$$\n\\begin{array}{l}\n\\text { Three. (50 points) Given } a, b, c \\in \\mathbf{R}_{+}, a b c=1 \\(...TRUNCATED)

"Three, let $a=\\frac{y}{x}, b=\\frac{z}{y}, c=\\frac{x}{z}$. Then\n$\\frac{1}{b^{2}+b c}=\\frac{1}{(...TRUNCATED)

"<think>\nOkay, let's tackle problem three first. So we have three positive real numbers a, b, c suc(...TRUNCATED)

"<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED)

"A6. The equation\n$$\n(x-1)(x-2) \\cdots(x-2016)=(x-1)(x-2) \\cdots(x-2016)\n$$\nis written on the (...TRUNCATED)

"Answer. 2016.\nSolution. Since there are 2016 common linear factors on both sides, we need to erase(...TRUNCATED)

"<think>\nOkay, so I need to figure out the minimum number of linear factors to erase from both side(...TRUNCATED)

"<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED)

"1. We call polynomials $A(x)=a_{n} x^{n}+\\ldots+a_{1} x+a_{0}$ and $B(x)=b_{m} x^{m}+\\ldots+b_{1}(...TRUNCATED)

"1. Since $3^{2012} \\equiv 1(\\bmod 5)$, it holds that $Q\\left(3^{2012}\\right) \\equiv Q(1)=P(1) (...TRUNCATED)

"<think>\nAlright, so I need to solve this problem where I have two similar polynomials P(x) and Q(x(...TRUNCATED)

"<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED)

"Exercise 18. We say that a set $A$ of polynomials with real coefficients is beautiful if, whenever (...TRUNCATED)

"Solution to Exercise 18 Let $A$ be a magnificent set. Suppose that $A$ has 3 distinct elements, whi(...TRUNCATED)

"<think>\nOkay, so I need to figure out the maximal cardinality of a \"beautiful set\" A of polynomi(...TRUNCATED)

"<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED)

"A woman from a village started at 10:31 and arrived at village $B$ at 13:43, traveling at a constan(...TRUNCATED)

"I. solution: The woman completed the entire journey in 192 minutes, and the man in 160 minutes. Sin(...TRUNCATED)

"<think>\nOkay, let's try to tackle this problem step by step. It's about two people traveling betwe(...TRUNCATED)

"<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED)

"3.1. A transgalactic ship has encountered an amazing meteorite stream. Some of the meteorites are f(...TRUNCATED)

"3.1. A transgalactic ship has encountered an amazing meteor shower. Some of the meteors are flying (...TRUNCATED)

"<think>\nOkay, so there's this problem about a transgalactic ship encountering meteorites. Let me t(...TRUNCATED)

"<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED)

"Write on the board the equation\n\n$$\n(x-1)(x-2) \\cdots(x-2016)=(x-1)(x-2) \\cdots(x-2016)\n$$\n\(...TRUNCATED)

"Nagy Kartal's solution. As the first step, we show that $k>2015$. If we delete fewer than 2016 line(...TRUNCATED)

"<think>\nOkay, so I need to figure out the smallest positive value of k such that if I omit exactly(...TRUNCATED)

"<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED)

"$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may t(...TRUNCATED)

"1. **Initial Setup**: We are given \\( N \\) different numbers on the blackboard, one of which is \(...TRUNCATED)

"<think>\nOkay, let me try to figure out this problem step by step. So, the problem is: we have N di(...TRUNCATED)

"<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED)