conversations listlengths 3 3 |
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[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$A$ and $B$ play a game where $A$ chooses a positive integer $n$ from the set $S = \\{1, 2, \\dots, 1001\\}$. $B$ must guess the value of $n$ by selecting subsets of $S$ an... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $AC$, and $DE$ and $BF$ are perpendicular to $AC$. If $AE=3$, $DE=5$, and $CE=7$, then... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$ABCD$ is a regular tetrahedron (right triangular pyramid). If $M$ is the midpoint of $\\overline{CD}$, then what is $\\cos \\angle AMB$?"
},
{
"from": "gpt",
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$ABCD$ is a square such that $AB$ lies on the line $y=x+4$ and points $C$ and $D$ lie on the graph of the parabola $y^2 = x$. Compute the sum of all possible areas of $ABCD... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$ABCDEF$ is an equiangular hexagon with side lengths $AB=2015$, $BC=2016$, $CD=2017$, and $DE=2018$. Find the perimeter of $ABCDEF$."
},
{
"from": "gpt",
"value... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$E$ is a point on side $AD$ of rectangle $ABCD$, so that $DE = 12$, while $DA = 16$, and $DC = 12$. If $CE$ extended meets the circumcircle of the rectangle at $F$, find th... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \\ne 0$ and $L \\ne 0$. Given that the sum $MATH + LMT$, where each letter represents a digit, equals... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$N$ visitors of the LycoReco cafe each give an integer rating from 1 to 10 on Takina Inoue's chocolate parfait. Given that the sum of all ratings given is 83 and the mean o... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$\\triangle ABC$ has all of its vertices on the parabola $y=x^{2}$. The slopes of $AB$ and $BC$ are $10$ and $-9$, respectively. If the $x$-coordinate of the triangle's cen... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$\\triangle ABC$ has all of its vertices on the parabola $y=x^{2}$. The slopes of $AB$ and $BC$ are $10$ and $-9$, respectively. If the x-coordinate of the triangle's centr... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$\\triangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute th... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$a, b, c, d$ are positive numbers with product 1. What is the minimum of $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$?"
},
{
"from": "gpt",
"value": "<think>\n<think>\nO... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$x = {1+\\frac{\\sqrt{2}}{1+\\frac{\\sqrt{2}}{1+...}}}$. Find $\\frac{1}{(x+1)(x-2)}$. When your answer is in the form $\\frac{A+\\sqrt{B}}{C}$, where $A$, $B$, and $C$ are... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "$x$ and $y$ are complex numbers such that $x^3 + y^3 = -16$ and $(x + y)^2 = xy$. What is the value of $|x + y|$?"
},
{
"from": "gpt",
"value": "<think>\n<think... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "(A) If \\(\\theta\\) is any quadrant 2 angle, what quadrant(s) could \\(2\\theta\\) lie in?\n(B) Given \\(\\sin \\theta = \\frac{8}{17}\\) where \\(\\frac{\\pi}{2} < \\thet... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "(b) Compute the remainder of $100!$ when divided by $101$."
},
{
"from": "gpt",
"value": "<think>\n<think>\nOkay, so I need to find the remainder when 100! is d... |
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