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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 ... |
[
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"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... |
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"content": "Let $f(x) = x^3 + ax^2 + bx + c$ be a cubic polynomial with rational coefficients, and let $\\alpha$, $\\beta$, and $\\gamma$ be its roots. If $f(x)$ has a local maximum at $\\alpha$, and $\\beta$ and $\\gamma$ are positive integers, find the minimum possible value of $\\beta +... |
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"content": "Find all values of $x$ such that $x + \\lfloor x \\rfloor^2 = 22.5$.\n\n(Note: $\\lfloor x \\rfloor$ represents the greatest integer less than or equal to $x$)\nPlease reason step by step, and put your final answer within \\boxed{{}}.\n"
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[{"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":"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) |
[{"role":"user","content":"A nonzero polynomial with rational coefficients has all of the numbers \n(...TRUNCATED) |
[{"role":"user","content":"Consider a trapezoid $ABCD$ with $AB$ parallel to $CD$, $AB = 7$, $BC = 5(...TRUNCATED) |
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