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"content": "In a mathematical competition, six teams are participating, and each team has four members. The teams are arranged in two rows of three teams each, with each team facing another team in the opposite row. The sum of the ages of the members of each team is written on a card, and ... |
[
{
"role": "user",
"content": "Let A=(2,3,4,6,7,10,12,13) and (p,q,r,s,t,u,v,w) be distinct elements in A. What is the minimum possible value of pqrs+tuwu?\nPlease reason step by step, and put your final answer within \\boxed{{}}.\n"
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"role": "assistant",
"content": "<think>\nOkay, so I need ... |
[
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"role": "user",
"content": "The number $x$ can be expressed as a three-place decimal $0.abc,$ where $a, b,$ and $c$ represent digits, any of which could be zero. It is desired to approximate $x$ by a fraction whose numerator is 1 or 3 and whose denominator is an integer. The closest such fraction to $x$ i... |
[{"role":"user","content":"In the following grid, each square is to be colored red, white, or blue, (...TRUNCATED) |
[{"role":"user","content":"Find all values of $x$ such that $x + \\lfloor x \\rfloor^2 = 22.5$.\n\n((...TRUNCATED) |
[{"role":"user","content":"In pentagon ABCDE, AB = BC = CD = DE = 2 and ∠B = ∠C = ∠D = 90°. F(...TRUNCATED) |
[{"role":"user","content":"A bookshelf contains 20 books, each written by a different author. If you(...TRUNCATED) |
[{"role":"user","content":"Quadrilateral $ABCD$ has $AB=9$ and $\\frac{BC}{AD}=\\frac{40}{41}$. What(...TRUNCATED) |
[{"role":"user","content":"Find the largest positive integer that is not the sum of a positive integ(...TRUNCATED) |
[{"role":"user","content":"There are three boxes containing $n$ balls each. Each ball is one of thre(...TRUNCATED) |
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