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[
{
"content": "Anna and Zs贸fi take turns rolling a die, and the number rolled is always added to the sum of the numbers rolled so far. The person who first makes the sum divisible by 4 wins. If Anna starts, what is the probability that she will win? Let's think step by step and output the final answer within \\b... |
[
{
"content": "Let $\\Gamma$ be a circle with center $O$ and $P$ a point outside the circle. Let $A \\in \\Gamma$ such that the line segment $PA$ is tangent to $\\Gamma$, and let $B$ be the point diametrically opposite to $A$. Let $\\Delta$ be a line passing through $P$ and intersecting $\\Gamma$ at points $C$ a... |
[
{
"content": "If $\\varepsilon _{k}, k=1,2,...,2013$ are all $2013^{th}$ roots of unity, evaluate the product: \\[ \\prod_{k=1}^{2013}(2-\\varepsilon _{k}^{2}). \\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>\nHere's a thinking pro... |
[
{
"content": "Arrange the arithmetic operation signs and parentheses in the expression consisting of the three numbers $\\frac{1}{8} \\ldots \\frac{1}{9} \\ldots \\frac{1}{28}$ so that the result of the calculations is equal to $\\frac{1}{2016}$. Let's think step by step and output the final answer within \\box... |
[
{
"content": "Let \\( p \\in \\mathbb{N} \\) and \\( f \\in \\mathbb{Z}[X] \\) such that \\( f(p) = 6 \\), \\( f(p+1) = 18 \\), and \\( f(p+2) = 54 \\). How many solutions does the equation \\( f(x) = 2 \\) have over \\( \\mathbb{Z} \\)? Let's think step by step and output the final answer within \\boxed{}.",
... |
[{"content":"Consider the function \\( f: \\mathbb{N} \\setminus \\{0,1\\} \\to \\mathbb{R} \\), def(...TRUNCATED) |
[{"content":"Find all $n \\geq 10$ such that $\\binom{n}{10} \\binom{n+1}{10}$ is a perfect square. (...TRUNCATED) |
[{"content":"Calculate the integral \\[ \\int \\frac{dx}{(x+a)^{m}(x+b)^{n}} \\quad \\text{where } m(...TRUNCATED) |
[{"content":"Determine the general term of the sequence $(a_n)_{n\\ge 1}$ given by $a_0=0$, $a_1=b$,(...TRUNCATED) |
[{"content":"Find the functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that for all \\(...TRUNCATED) |
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