messages listlengths 1 1 | label stringlengths 1 4 | metadata dict | data_source stringclasses 1
value |
|---|---|---|---|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 mil... | 27 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 0,
"raw_problem": "Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many mi... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPositive real numbers $x$ and $y$ satisfy $y^3=x^2$ and $(y-x)^2=4y^2$. What is $x+y$?\n\nRemember to put your answer ... | 36 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 1,
"raw_problem": "Positive real numbers $x$ and $y$ satisfy $y^3=x^2$ and $(y-x)^2=4y^2$. What is $x+y$?",
"source_id": 1,
"split": null,
"url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Pro... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the degree measure of the acute angle formed by lines with slopes $2$ and $\\frac{1}{3}$?\n\nRemember to put y... | 45 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 2,
"raw_problem": "What is the degree measure of the acute angle formed by lines with slopes $2$ and $\\frac{1}{3}$?",
"source_id": 2,
"split": null,
"url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_P... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the value of\n\\[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \\dots + 18^3 - 17^3?\\]\n\nRemember to put your answer o... | 3159 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 3,
"raw_problem": "What is the value of\n\\[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \\dots + 18^3 - 17^3?\\]",
"source_id": 3,
"split": null,
"url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Prob... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn a table tennis tournament every participant played every other participant exactly once. Although there were twice ... | 36 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 4,
"raw_problem": "In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed pl... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many complex numbers satisfy the equation $z^5=\\overline{z}$, where $\\overline{z}$ is the conjugate of the compl... | 7 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 5,
"raw_problem": "How many complex numbers satisfy the equation $z^5=\\overline{z}$, where $\\overline{z}$ is the conjugate of the complex number $z$?",
"source_id": 5,
"split": null,
"url": "https://artofproblemsolvin... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nConsider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can... | 21 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 7,
"raw_problem": "Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\\tfrac{\\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positi... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFlora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent... | 3 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 8,
"raw_problem": "Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\\frac{1}{2^... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the product of all solutions to the equation\n\\[\\log_{7x}2023\\cdot \\log_{289x}2023=\\log_{2023x}2023\\]\n\... | 1 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 10,
"raw_problem": "What is the product of all solutions to the equation\n\\[\\log_{7x}2023\\cdot \\log_{289x}2023=\\log_{2023x}2023\\]",
"source_id": 10,
"split": null,
"url": "https://artofproblemsolving.com/wiki/inde... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe weight of $\\frac{1}{3}$ of a large pizza together with $3 \\frac{1}{2}$ cups of orange slices is the same as the ... | 4 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 11,
"raw_problem": "The weight of $\\frac{1}{3}$ of a large pizza together with $3 \\frac{1}{2}$ cups of orange slices is the same as the weight of $\\frac{3}{4}$ of a large pizza together with $\\frac{1}{2}$ cup of orange slices. ... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below.\n1\n1 1\n1 3 1\n1 5 5 1\n1 7 11 7 1\nEach ro... | 5 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 12,
"raw_problem": "Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below.\n1\n1 1\n1 3 1\n1 5 5 1\n1 7 11 7 1\nEach row after the first row is formed by placing a 1 at each end of the row, and each interior entr... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the pol... | 29 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 13,
"raw_problem": "If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\\overline{AB}$ i... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f$ be the unique function defined on the positive integers such that \\[\\sum_{d\\mid n}d\\cdot f\\left(\\frac{n}... | 96 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 14,
"raw_problem": "Let $f$ be the unique function defined on the positive integers such that \\[\\sum_{d\\mid n}d\\cdot f\\left(\\frac{n}{d}\\right)=1\\] for all positive integers $n$. What is $f(2023)$?",
"source_id": 14,
... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many ordered pairs of positive real numbers $(a,b)$ satisfy the equation\n\\[(1+2a)(2+2b)(2a+b) = 32ab?\\]\n\nReme... | 1 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 15,
"raw_problem": "How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation\n\\[(1+2a)(2+2b)(2a+b) = 32ab?\\]",
"source_id": 15,
"split": null,
"url": "https://artofproblemsolving.com/wiki/index.php... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $K$ be the number of sequences $A_1$, $A_2$, $\\dots$, $A_n$ such that $n$ is a positive integer less than or equa... | 5 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 16,
"raw_problem": "Let $K$ be the number of sequences $A_1$, $A_2$, $\\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\\{1, 2, 3, \\dots, 10\\}$, and $A_{i-1}$ is a subset o... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere is a unique sequence of integers $a_1, a_2, \\cdots a_{2023}$ such that\n\\[\\tan2023x = \\frac{a_1 \\tan x + a_... | -1 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 17,
"raw_problem": "There is a unique sequence of integers $a_1, a_2, \\cdots a_{2023}$ such that\n\\[\\tan2023x = \\frac{a_1 \\tan x + a_3 \\tan^3 x + a_5 \\tan^5 x + \\cdots + a_{2023} \\tan^{2023} x}{1 + a_2 \\tan^2 x + a_4 \\ta... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many positive perfect squares less than $2023$ are divisible by $5$?\n\nRemember to put your answer on its own lin... | 8 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 18,
"raw_problem": "How many positive perfect squares less than $2023$ are divisible by $5$?",
"source_id": 18,
"split": null,
"url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_3"
}
... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many digits are in the base-ten representation of $8^5 \\cdot 5^{10} \\cdot 15^5$?\n\nRemember to put your answer ... | 18 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 19,
"raw_problem": "How many digits are in the base-ten representation of $8^5 \\cdot 5^{10} \\cdot 15^5$?",
"source_id": 19,
"split": null,
"url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/P... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJanet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. What is the probabi... | 265 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 20,
"raw_problem": "Janet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3$? The final answer can be written in... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPoints $A$ and $B$ lie on the graph of $y=\\log_{2}x$. The midpoint of $\\overline{AB}$ is $(6, 2)$. What is the posit... | 9 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 21,
"raw_problem": "Points $A$ and $B$ lie on the graph of $y=\\log_{2}x$. The midpoint of $\\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$? The final answer can be written in... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-dig... | 9 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 22,
"raw_problem": "A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is ... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nMaureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her... | 7 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 23,
"raw_problem": "Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nMrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses co... | 7 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 25,
"raw_problem": "Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only $\\frac{1}{3}$ full. What fracti... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origi... | 7 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 26,
"raw_problem": "In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-a... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCalculate the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the oth... | 13 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 27,
"raw_problem": "Calculate the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other. The final answer can be written in the form $\\frac{m}{n}$, where $m$ and $n$ are relativ... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor complex number $u = a+bi$ and $v = c+di$ (where $i=\\sqrt{-1}$), define the binary operation\n$u \\otimes v = ac +... | 50 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 28,
"raw_problem": "For complex number $u = a+bi$ and $v = c+di$ (where $i=\\sqrt{-1}$), define the binary operation\n$u \\otimes v = ac + bdi$\nSuppose $z$ is a complex number such that $z\\otimes z = z^{2}+40$. What is $|z|^2$?",... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA rectangular box $P$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $P$ is ... | 13 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 29,
"raw_problem": "A rectangular box $P$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $P$ is $13$, the areas of all $6$ faces of $P$ is $\\frac{11}{2}$, and the volume of $P$ is $\\frac{... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?\n\... | 5 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 30,
"raw_problem": "For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?",
"source_id": 30,
"split": null,
"url": "https://artofproblemsolving.com/wiki/inde... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the state of Coinland, coins have values $6,10,$ and $15$ cents. Suppose $x$ is the value in cents of the most expe... | 11 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 32,
"raw_problem": "In the state of Coinland, coins have values $6,10,$ and $15$ cents. Suppose $x$ is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is ... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTriangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has a... | 18 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 33,
"raw_problem": "Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\\circ,$ Find the area of $ABC$. The final answer can be simplified in the fo... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCarlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\\%$ on ever... | 50 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 36,
"raw_problem": "Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\\%$ sales tax on the discounted p... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhen $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. Wh... | 11 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 40,
"raw_problem": "When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$?",
"source_id": 40,
"split": null,
"url": "https://artofproblemsolving.... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.\n\\[abcd=2^6\\cdot... | 3 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 41,
"raw_problem": "Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.\n\\[abcd=2^6\\cdot 3^9\\cdot 5^7\\]\n\\[\\text{lcm}(a,b)=2^3\\cdot 3^2\\cdot 5^3\\]\n\\[\\text{lcm}(a,c)=2^3\\c... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. Find t... | 194 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 43,
"raw_problem": "A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. Find the ratio of the area of circle $A$ to the area of circle $B$. The final answer can be written... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip ... | 1625 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 44,
"raw_problem": "Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint?",
... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nYou are playing a game. A $2 \\times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertica... | 4 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 45,
"raw_problem": "You are playing a game. A $2 \\times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \\times 3$ grid of squares, but you are not told which two squares are covered. ... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhen the roots of the polynomial \n\\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \\cdot \\cdot \\cdot (x-10)^{10}\\]\nare removed... | 6 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 46,
"raw_problem": "When the roots of the polynomial \n\\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \\cdot \\cdot \\cdot (x-10)^{10}\\]\nare removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor how many integers $n$ does the expression\\[\\sqrt{\\frac{\\log (n^2) - (\\log n)^2}{\\log n - 3}}\\]represent a r... | 901 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 47,
"raw_problem": "For how many integers $n$ does the expression\\[\\sqrt{\\frac{\\log (n^2) - (\\log n)^2}{\\log n - 3}}\\]represent a real number, where log denotes the base $10$ logarithm?",
"source_id": 47,
"split": nu... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many nonempty subsets $B$ of ${0, 1, 2, 3, \\cdots, 12}$ have the property that the number of elements in $B$ is e... | 144 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 48,
"raw_problem": "How many nonempty subsets $B$ of ${0, 1, 2, 3, \\cdots, 12}$ have the property that the number of elements in $B$ is equal to the least element of $B$? For example, $B = {4, 6, 8, 11}$ satisfies the condition.",... | aime-amc23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the area of the region in the coordinate plane defined by\n$| | x | - 1 | + | | y | - 1 | \\le 1$?\n\nRemember... | 8 | {
"ability": "MATH",
"data_source": "aime-amc23",
"extra_info": {
"index": 49,
"raw_problem": "What is the area of the region in the coordinate plane defined by\n$| | x | - 1 | + | | y | - 1 | \\le 1$?",
"source_id": 49,
"split": null,
"url": "https://artofproblemsolving.com/wiki/index.php/202... | aime-amc23 |
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