messages listlengths 2 2 |
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[
{
"role": "user",
"content": "A linear function $f(x) = ax + b$ is defined on the interval $[-10, 10]$ such that the sum of the absolute values of $f(x)$ at $100$ evenly spaced points in the interval is equal to $1000$. Let $M$ be the maximum possible value of $|f(0)|$ and let $m$ be the minimum possible va... |
[
{
"role": "user",
"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": "Find the maximum value of $k$ such that the graph of the function $f(x) = x^3 - kx^2 + 2x - 1$ is above the $x$-axis for all $x$ in the interval $[0, 2]$.\nPlease reason step by step, and put your final answer within \\boxed{{}}.\n"
},
{
"role": "assistant",
"conten... |
[
{
"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"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need ... |
[{"role":"user","content":"The number $x$ can be expressed as a three-place decimal $0.abc,$ where $(...TRUNCATED) |
[{"role":"user","content":"Let $f(x) = x^3 + ax^2 + bx + c$ be a cubic polynomial with rational coef(...TRUNCATED) |
[{"role":"user","content":"Find the maximum value of \n\\[abc + def\\]\nwhere $a, b, c, d, e, f$ are(...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 the triangle $MNO$ with $\\angle M = 45^\\circ$, the side opposite to (...TRUNCATED) |
[{"role":"user","content":"In pentagon ABCDE, AB = BC = CD = DE = 2 and ∠B = ∠C = ∠D = 90°. F(...TRUNCATED) |
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