conversations listlengths 2 2 | source stringclasses 34 values | domain stringclasses 3 values | difficulty null |
|---|---|---|---|
[
{
"from": "human",
"value": "Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?"
},
{
"from": "gpt",
"value": "<think>\nThe ... | qq8933/AIME_1983_2024 | math | null |
[
{
"from": "human",
"value": "Let $ \\mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$ , and suppose that $ S$ is a set of points (not necessarily countable) in $ \\mathcal{H}$ such that the distance between any two distinct points in $ S$ is equal to $ d$ . Show that there is a poi... | AI-MO/NuminaMath-CoT/aops_forum | math | null |
[
{
"from": "human",
"value": "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$ ."
},
{
"from": "gpt",
"value": "<think>\nThe problem asks for the remainder when the product $P = 9 \\times 99 \\times 999 \\ti... | qq8933/AIME_1983_2024 | math | null |
[
{
"from": "human",
"value": "Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s"
},
{
"from": "gpt",
"value": "<think>Here's a thinking process to arrive at the solution:\n\n1. **Understand the Goal:** The request is to calculate the mean molecular speed of radon. This immedia... | TIGER-Lab/TheoremQA/float | math | null |
[
{
"from": "human",
"value": "Two capacitors with capacitance values $C_{1}=2000 \\pm 10 \\mathrm{pF}$ and $C_{2}=3000 \\pm 15 \\mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=5.00 \\pm 0.02 \\mathrm{~V}$. What is the percentage error in the calculation of the energy s... | daman1209arora/jeebench/phy | math | null |
[
{
"from": "human",
"value": "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let $x$ be the length of the segment joining the legs of the trapezoid that is paralle... | qq8933/AIME_1983_2024 | math | null |
[
{
"from": "human",
"value": "Let's say a language $L \\subseteq \\{0,1\\}^*$ is in $\\textbf{P}_{angel}$ if there exists a polynomial $p : \\mathbb{N} \\mapsto \\mathbb{N}$ , a sequence of strings $\\{\\alpha_n\\}_{n \\in \\mathbb{N}}$ with $\\alpha_n \\in \\{0,1\\}^{p(n)}$ , and a deterministic pol... | AI-MO/NuminaMath-CoT/aops_forum | math | null |
[
{
"from": "human",
"value": "In a mathematics test number of participants is $N < 40$ . The passmark is fixed at $65$ . The test results are\nthe following: \nThe average of all participants is $66$ , that of the promoted $71$ and that of the repeaters $56$ . \nHowever, due to an error in the wording ... | AI-MO/NuminaMath-CoT/aops_forum | math | null |
[
{
"from": "human",
"value": "Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRG... | qq8933/AIME_1983_2024 | math | null |
[
{
"from": "human",
"value": "Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$ , and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\\triangle GEM$ ."
},
{
"from": "gpt",
"value": "<think>... | qq8933/AIME_1983_2024 | math | null |
[
{
"from": "human",
"value": "Consider the following two person game. A number of pebbles are situated on the table. Two players make their moves alternately. A move consists of taking off the table $x$ pebbles where $x$ is the square of any positive integer. The player who is unable to make a move loses... | AI-MO/NuminaMath-CoT/aops_forum | math | null |
[
{
"from": "human",
"value": "Suppose $a,\\,b,$ and $c$ are three complex numbers with product $1$ . Assume that none of $a,\\,b,$ and $c$ are real or have absolute value $1$ . Define\n\\begin{tabular}{c c c} $p=(a+b+c)+\\left(\\dfrac 1a+\\dfrac 1b+\\dfrac 1c\\right)$ & \\text{and} & $q=\\dfrac a... | AI-MO/NuminaMath-CoT/aops_forum | math | null |
[
{
"from": "human",
"value": "Prove that the function \\[ f(\\nu)= \\int_1^{\\frac{1}{\\nu}} \\frac{dx}{\\sqrt{(x^2-1)(1-\\nu^2x^2)}}\\]\r\n(where the positive value of the square root is taken) is monotonically decreasing in the interval $ 0<\\nu<1$ . [P. Turan]"
},
{
"from": "gpt",
"value": "<... | AI-MO/NuminaMath-CoT/aops_forum | math | null |
[
{
"from": "human",
"value": "Joanie takes a $\\$6,\\!000$ loan to pay for her car. The annual interest rate on the loan is $12\\%$. She makes no payments for 4 years, but has to pay back all the money she owes at the end of 4 years. How much more money will she owe if the interest compounds quarterly than... | qfq/openaimath/Algebra | math | null |
[
{
"from": "human",
"value": "A function $f:[0,\\infty)\\to[0,\\infty)$ is integrable and $$ \\int_0^\\infty f(x)^2 dx<\\infty,\\quad \\int_0^\\infty xf(x) dx <\\infty $$ Prove the following inequality. $$ \\left(\\int_0^\\infty f(x) dx \\right)^3 \\leq 8\\left(\\int_0^\\infty f(x)^2 dx \\right) \\left... | AI-MO/NuminaMath-CoT/aops_forum | math | null |
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