data_source stringclasses 1
value | answer stringlengths 1 204 | source_dataset stringclasses 4
values | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|---|
numina_math | 2851 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the largest prime factor of $45^{5}-1$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2851",
"style": "rule"
} | {
"index": 1600,
"split": "test"
} |
numina_math | \frac{1}{2026} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_{n}$ be a recursively defined sequence with $a_{0}=2024$ and $a_{n+1}=a_{n}^{3}+5 a_{n}^{2}+10 a_{n}+6$ for $n \\geq 0$. Determine the value of\n\n$$\n\\sum_{n=0}^{\\infty... | math | {
"ground_truth": "\\frac{1}{2026}",
"style": "rule"
} | {
"index": 1601,
"split": "test"
} |
numina_math | \frac{1}{6} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Robert has five beads in his hand, with the letters C, M, I, M, and C, and he wants to make a circular bracelet spelling \"CMIMC.\" However, the power went out, so Robert can no ... | math | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": 1602,
"split": "test"
} |
numina_math | \frac{683}{1024} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Every day, Pinky the flamingo eats either $1$ or $2$ shrimp, each with equal probability. Once Pinky has consumed $10$ or more shrimp in total, its skin will turn pink. Once Pink... | math | {
"ground_truth": "\\frac{683}{1024}",
"style": "rule"
} | {
"index": 1603,
"split": "test"
} |
numina_math | \frac{2}{17} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are $34$ friends are sitting in a circle playing the following game. Every round, four of them are chosen at random, and have a rap battle. The winner of the rap battle sta... | math | {
"ground_truth": "\\frac{2}{17}",
"style": "rule"
} | {
"index": 1604,
"split": "test"
} |
numina_math | 1088 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ and $k$ be positive integers, with $k \\leq n$. Define a (simple, undirected) graph $G_{n, k}$ as follows: its vertices are all of the binary strings of length $n$, and t... | math | {
"ground_truth": "1088",
"style": "rule"
} | {
"index": 1605,
"split": "test"
} |
numina_math | \frac{138}{55} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a $12$-card deck containing all four suits of $2,3$, and $4$. A double is defined as two cards directly next to each other in the deck, with the same value. Suppose we s... | math | {
"ground_truth": "\\frac{138}{55}",
"style": "rule"
} | {
"index": 1606,
"split": "test"
} |
numina_math | 256 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a $4 \\times 4$ grid of squares. We place coins in some of the grid squares so that no two coins are orthogonally adjacent, and each $2 \\times 2$ square in the grid has... | math | {
"ground_truth": "256",
"style": "rule"
} | {
"index": 1607,
"split": "test"
} |
numina_math | \frac{1507^{2}}{360^{2}} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alan is bored one day and decides to write down all the divisors of $1260^{2}$ on a wall. After writing down all of them, he realizes he wrote them on the wrong wall and needs to... | math | {
"ground_truth": "\\frac{1507^{2}}{360^{2}}",
"style": "rule"
} | {
"index": 1608,
"split": "test"
} |
numina_math | 2240 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Divide a regular 8960-gon into non-overlapping parallelograms. Suppose that $R$ of these parallelograms are rectangles. What is the minimum possible value of $R$ ? Let's think st... | math | {
"ground_truth": "2240",
"style": "rule"
} | {
"index": 1609,
"split": "test"
} |
numina_math | \frac{9349}{327680} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $p(k)$ be the probability that if we choose a uniformly random subset $S$ of $\\{1,2, \\ldots, 18\\}$, then $|S| \\equiv k$ $(\\bmod 5)$.\nEvaluate\n\n$$\n\\sum_{k=0}^{4}\\le... | math | {
"ground_truth": "\\frac{9349}{327680}",
"style": "rule"
} | {
"index": 1610,
"split": "test"
} |
numina_math | 4102652 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_{n}$ be the number of ways to express $n$ as an ordered sum of powers of $3$. For example, $a_{4}=3$, since\n\n$$\n4=1+1+1+1=1+3=3+1 .\n$$\n\nLet $b_{n}$ denote the remain... | math | {
"ground_truth": "4102652",
"style": "rule"
} | {
"index": 1611,
"split": "test"
} |
numina_math | 14\sqrt{3} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "I'm given a square of side length $7$, and I want to make a regular tetrahedron from it. Specifically, my strategy is to cut out a net. If I cut out a parallelogram-shaped net th... | math | {
"ground_truth": "14\\sqrt{3}",
"style": "rule"
} | {
"index": 1612,
"split": "test"
} |
numina_math | \sqrt{3}-1 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a cube of side length $4$, place eight spheres of radius $1$ inside the cube so that each sphere is externally tangent to three others. What is the radius of the largest sp... | math | {
"ground_truth": "\\sqrt{3}-1",
"style": "rule"
} | {
"index": 1613,
"split": "test"
} |
numina_math | \frac{11\sqrt{3}}{4} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A B$ be a segment of length $1$. Let $\\odot A, \\odot B$ be circles with radius $\\overline{A B}$ centered at $A, B$. Denote their intersection points $C, D$. Draw circles ... | math | {
"ground_truth": "\\frac{11\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": 1614,
"split": "test"
} |
numina_math | \frac{2\sqrt{39}}{13} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A B C D E F$ be a regular hexagon with side length $1$, and let $G$ be the midpoint of side $\\overline{C D}$, and define $H$ to be the unique point on side $\\overline{D E}... | math | {
"ground_truth": "\\frac{2\\sqrt{39}}{13}",
"style": "rule"
} | {
"index": 1615,
"split": "test"
} |
numina_math | \frac{12\sqrt{3}+4}{13} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle A B C$ be an equilateral triangle. Let $E_{A B}$ be the ellipse with foci $A, B$ passing through $C$, and in the parallel manner define $E_{B C}, E_{A C}$. Let $\... | math | {
"ground_truth": "\\frac{12\\sqrt{3}+4}{13}",
"style": "rule"
} | {
"index": 1616,
"split": "test"
} |
numina_math | \frac{31}{7} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Points $A, B, C, D, E$, and $F$ lie on a sphere with center $O$ and radius $R$ such that $\\overline{A B}, \\overline{C D}$, and $\\overline{E F}$ are pairwise perpendicular and ... | math | {
"ground_truth": "\\frac{31}{7}",
"style": "rule"
} | {
"index": 1617,
"split": "test"
} |
numina_math | \frac{7\sqrt{3}}{3} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A B C$ be a triangle with altitude $\\overline{A F}$. Let $A B=5, A C=8, B C=7$. Let $P$ be on $\\overline{A F}$ such that it lies between $A$ and $F$. Let $\\omega_{1}, \\o... | math | {
"ground_truth": "\\frac{7\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": 1618,
"split": "test"
} |
numina_math | \frac{97-3\sqrt{1001}}{2} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\omega$ be a circle with diameter $\\overline{A B}$, center $O$, and cyclic quadrilateral $A B C D$ inscribed in it, with $C$ and $D$ on the same side of $\\overline{A B}$.... | math | {
"ground_truth": "\\frac{97-3\\sqrt{1001}}{2}",
"style": "rule"
} | {
"index": 1619,
"split": "test"
} |
numina_math | \frac{11}{3} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Define the ratio of an ellipse to be the length of the major axis divided by the length of the minor axis.\n\nGiven a trapezoid $A B C D$ with $A B \\| D C$ and that $\\angle A D... | math | {
"ground_truth": "\\frac{11}{3}",
"style": "rule"
} | {
"index": 1620,
"split": "test"
} |
numina_math | 9.94 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle A B C$ exist such that $A B=6, B C=8, A C=10$. Let $P$ lie on the circumcircle of $A B C, \\omega$, such that $P$ lies strictly on the arc in between $B$ and $C$ ... | math | {
"ground_truth": "9.94",
"style": "rule"
} | {
"index": 1621,
"split": "test"
} |
numina_math | 13 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "I define a \"good day\" as a day when both the day and the month evenly divide the concatenation of the two. For example, today (March 15) is a good day since $3$ and $15$ both d... | math | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 1622,
"split": "test"
} |
numina_math | \frac{11}{4} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "We are searching for the number $7$ in the following binary tree:\n\n 4\n / \\\n 2 6\n / \\ / \\\n1 3 5 7\n\n- **Level 1 (Root):** 4\n- **Level 2:** 2 (left), ... | math | {
"ground_truth": "\\frac{11}{4}",
"style": "rule"
} | {
"index": 1623,
"split": "test"
} |
numina_math | 512 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(x)=x^{4}-4 x^{2}+2$. Find the smallest natural $n \\in \\mathbb{N}$ such that there exists $k, c \\in \\mathbb{N}$ with\n\n$$\n\\left|f^{k}\\left(\\frac{n^{2}+1}{n}\\right... | math | {
"ground_truth": "512",
"style": "rule"
} | {
"index": 1624,
"split": "test"
} |
numina_math | 44 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A non-self intersecting hexagon $R A N D O M$ is formed by assigning the labels $R, A, N, D, O, M$ in some order to the points\n$$\n(0,0),(10,0),(10,10),(0,10),(3,4),(6,2) .\n$$\... | math | {
"ground_truth": "44",
"style": "rule"
} | {
"index": 1625,
"split": "test"
} |
numina_math | 2496 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose we have a uniformly random function from $\\{1,2,3, \\ldots, 25\\}$ to itself. Find the expected value of\n\n$$\n\\sum_{x=1}^{25}(f(f(x))-x)^{2}\n$$ Let's think step by s... | math | {
"ground_truth": "2496",
"style": "rule"
} | {
"index": 1626,
"split": "test"
} |
numina_math | 3+\sqrt{2}+\sqrt{3}+\sqrt{6} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose we have a regular $24$-gon labeled $A_{1} \\cdots A_{24}$. We will draw $2$ smaller regular $24$-gons within $A_{1} \\cdots A_{24}$. For the sake of this problem, make $A... | math | {
"ground_truth": "3+\\sqrt{2}+\\sqrt{3}+\\sqrt{6}",
"style": "rule"
} | {
"index": 1627,
"split": "test"
} |
numina_math | 648 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The binomial coefficient $\\binom{n}{k}$ can be defined as the coefficient of $x^{k}$ in the expansion of $(1+x)^{n}$. Similarly, define the trinomial coefficient $\\binom{n}{k}_... | math | {
"ground_truth": "648",
"style": "rule"
} | {
"index": 1628,
"split": "test"
} |
numina_math | 10+\frac{40\pi}{3} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $U$ be the set of all possible complex numbers $m$ so that the $4$ roots of $\\left(x^{2}+2 x+5\\right)\\left(x^{2}-\\right.$ $2 m x+25)=0$ are concyclic in the complex plane... | math | {
"ground_truth": "10+\\frac{40\\pi}{3}",
"style": "rule"
} | {
"index": 1629,
"split": "test"
} |
numina_math | 8222 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a triangle, $A B=78, B C=50, A C=112$, construct squares $A B X Y, B C P Q, A C M N$ outside the triangle. Let $L_{1}, L_{2}, L_{3}$ be the midpoints of $\\overline{M P}, \... | math | {
"ground_truth": "8222",
"style": "rule"
} | {
"index": 1630,
"split": "test"
} |
numina_math | 2024^{4046} | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a $2024 \\times 2024$ grid of squares, each square is colored either black or white. An ant starts at some black square in the grid and starts walking parallel to the sides of... | math | {
"ground_truth": "2024^{4046}",
"style": "rule"
} | {
"index": 1631,
"split": "test"
} |
numina_math | 27.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 $A$... | math | {
"ground_truth": "27.0",
"style": "rule"
} | {
"index": 1632,
"split": "test"
} |
numina_math | 36.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Positive real numbers $x$ and $y$ satisfy $y^3=x^2$ and $(y-x)^2=4y^2$. What is $x+y$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"... | math | {
"ground_truth": "36.0",
"style": "rule"
} | {
"index": 1633,
"split": "test"
} |
numina_math | 45.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the degree measure of the acute angle formed by lines with slopes $2$ and $\\frac{1}{3}$? Let's think step by step and output the final answer within \\boxed{}.",
"ro... | math | {
"ground_truth": "45.0",
"style": "rule"
} | {
"index": 1634,
"split": "test"
} |
numina_math | 3159.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the value of\n\\[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \\dots + 18^3 - 17^3?\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... | math | {
"ground_truth": "3159.0",
"style": "rule"
} | {
"index": 1635,
"split": "test"
} |
numina_math | 36.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 num... | math | {
"ground_truth": "36.0",
"style": "rule"
} | {
"index": 1636,
"split": "test"
} |
numina_math | 7.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many complex numbers satisfy the equation $z^5=\\overline{z}$, where $\\overline{z}$ is the conjugate of the complex number $z$? Let's think step by step and output the final... | math | {
"ground_truth": "7.0",
"style": "rule"
} | {
"index": 1637,
"split": "test"
} |
numina_math | 21.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 $m$ a... | math | {
"ground_truth": "21.0",
"style": "rule"
} | {
"index": 1638,
"split": "test"
} |
numina_math | 3.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "3.0",
"style": "rule"
} | {
"index": 1639,
"split": "test"
} |
numina_math | 1.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the product of all solutions to the equation\n\\[\\log_{7x}2023\\cdot \\log_{289x}2023=\\log_{2023x}2023\\] Let's think step by step and output the final answer within \\... | math | {
"ground_truth": "1.0",
"style": "rule"
} | {
"index": 1640,
"split": "test"
} |
numina_math | 4.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 $\\f... | math | {
"ground_truth": "4.0",
"style": "rule"
} | {
"index": 1641,
"split": "test"
} |
numina_math | 5.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 o... | math | {
"ground_truth": "5.0",
"style": "rule"
} | {
"index": 1642,
"split": "test"
} |
numina_math | 29.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 $B$. F... | math | {
"ground_truth": "29.0",
"style": "rule"
} | {
"index": 1643,
"split": "test"
} |
numina_math | 96.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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(2... | math | {
"ground_truth": "96.0",
"style": "rule"
} | {
"index": 1644,
"split": "test"
} |
numina_math | 1.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation\n\\[(1+2a)(2+2b)(2a+b) = 32ab?\\] Let's think step by step and output the final answer within \\boxed... | math | {
"ground_truth": "1.0",
"style": "rule"
} | {
"index": 1645,
"split": "test"
} |
numina_math | 5.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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... | math | {
"ground_truth": "5.0",
"style": "rule"
} | {
"index": 1646,
"split": "test"
} |
numina_math | -1.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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... | math | {
"ground_truth": "-1.0",
"style": "rule"
} | {
"index": 1647,
"split": "test"
} |
numina_math | 8.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many positive perfect squares less than $2023$ are divisible by $5$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "8.0",
"style": "rule"
} | {
"index": 1648,
"split": "test"
} |
numina_math | 18.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many digits are in the base-ten representation of $8^5 \\cdot 5^{10} \\cdot 15^5$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"... | math | {
"ground_truth": "18.0",
"style": "rule"
} | {
"index": 1649,
"split": "test"
} |
numina_math | 265.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 $3$? T... | math | {
"ground_truth": "265.0",
"style": "rule"
} | {
"index": 1650,
"split": "test"
} |
numina_math | 9.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 $B$? T... | math | {
"ground_truth": "9.0",
"style": "rule"
} | {
"index": 1651,
"split": "test"
} |
numina_math | 9.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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... | math | {
"ground_truth": "9.0",
"style": "rule"
} | {
"index": 1652,
"split": "test"
} |
numina_math | 7.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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... | math | {
"ground_truth": "7.0",
"style": "rule"
} | {
"index": 1653,
"split": "test"
} |
numina_math | 7.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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... | math | {
"ground_truth": "7.0",
"style": "rule"
} | {
"index": 1654,
"split": "test"
} |
numina_math | 7.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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... | math | {
"ground_truth": "7.0",
"style": "rule"
} | {
"index": 1655,
"split": "test"
} |
numina_math | 13.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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... | math | {
"ground_truth": "13.0",
"style": "rule"
} | {
"index": 1656,
"split": "test"
} |
numina_math | 50.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "50.0",
"style": "rule"
} | {
"index": 1657,
"split": "test"
} |
numina_math | 13.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 $\\frac{11}{2}$, ... | math | {
"ground_truth": "13.0",
"style": "rule"
} | {
"index": 1658,
"split": "test"
} |
numina_math | 5.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots? Let's think step by step and output the final answer within \\... | math | {
"ground_truth": "5.0",
"style": "rule"
} | {
"index": 1659,
"split": "test"
} |
numina_math | 11.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "11.0",
"style": "rule"
} | {
"index": 1660,
"split": "test"
} |
numina_math | 18.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 a... | math | {
"ground_truth": "18.0",
"style": "rule"
} | {
"index": 1661,
"split": "test"
} |
numina_math | 50.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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\... | math | {
"ground_truth": "50.0",
"style": "rule"
} | {
"index": 1662,
"split": "test"
} |
numina_math | 11.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$? Let's think step by step and output the final ans... | math | {
"ground_truth": "11.0",
"style": "rule"
} | {
"index": 1663,
"split": "test"
} |
numina_math | 3.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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... | math | {
"ground_truth": "3.0",
"style": "rule"
} | {
"index": 1664,
"split": "test"
} |
numina_math | 194.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 $B$... | math | {
"ground_truth": "194.0",
"style": "rule"
} | {
"index": 1665,
"split": "test"
} |
numina_math | 1625.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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... | math | {
"ground_truth": "1625.0",
"style": "rule"
} | {
"index": 1666,
"split": "test"
} |
numina_math | 4.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 tol... | math | {
"ground_truth": "4.0",
"style": "rule"
} | {
"index": 1667,
"split": "test"
} |
numina_math | 6.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 $11$ dis... | math | {
"ground_truth": "6.0",
"style": "rule"
} | {
"index": 1668,
"split": "test"
} |
numina_math | 901.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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? Let's... | math | {
"ground_truth": "901.0",
"style": "rule"
} | {
"index": 1669,
"split": "test"
} |
numina_math | 144.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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, $B = {4, 6, 8... | math | {
"ground_truth": "144.0",
"style": "rule"
} | {
"index": 1670,
"split": "test"
} |
numina_math | 8.0 | weqweasdas/aime_hmmt_brumo_cmimc_amc23 | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the area of the region in the coordinate plane defined by\n$| | x | - 1 | + | | y | - 1 | \\le 1$? Let's think step by step and output the final answer within \\boxed{}."... | math | {
"ground_truth": "8.0",
"style": "rule"
} | {
"index": 1671,
"split": "test"
} |
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