number int64 | base_question string | final_answer string | dag string | ques_node string | dag_node string | ques_edge string | dag_edge string | success bool |
|---|---|---|---|---|---|---|---|---|
1 | Patrick started walking at a constant speed along a straight road from his school to the park. One hour after Patrick left, Tanya started running at a constant speed of $2$ miles per hour faster than Patrick walked, following the same straight road from the school to the park. One hour after Tanya left, Jose started bi... | 277 | {"nodes": [{"id": "n1", "type": "given", "value": 1, "label": "patrick_head_start", "description": "Patrick left 1 hour before Tanya"}, {"id": "n2", "type": "given", "value": 1, "label": "tanya_head_start", "description": "Tanya left 1 hour before Jose"}, {"id": "n3", "type": "given", "value": 2, "label": "tanya_speed_... | Patrick started walking at a constant speed along a straight road from his school to the park. One hour after Patrick left, Tanya started running at a constant speed of 2 miles per hour faster than Patrick walked, following the same straight road from the school to the park. One hour after Tanya left, Jose started bicy... | {"nodes": [{"id": "n1", "type": "given", "value": 1, "label": "patrick_head_start", "description": "Patrick left 1 hour before Tanya"}, {"id": "n2", "type": "given", "value": 1, "label": "tanya_head_start", "description": "Tanya left 1 hour before Jose"}, {"id": "n3", "type": "given", "value": 2, "label": "tanya_speed_... | Patrick started walking at a constant speed along a straight road from his school to the park. One hour after Patrick left, Tanya started running at a constant speed of 2 miles per hour faster than Patrick walked, following the same straight road from the school to the park. One hour after Tanya left, Jose started bicy... | {"nodes": [{"id": "n1", "type": "given", "value": 1, "label": "patrick_head_start", "description": "Patrick left 1 hour before Tanya"}, {"id": "n2", "type": "given", "value": 1, "label": "tanya_head_start", "description": "Tanya left 1 hour before Jose"}, {"id": "n3", "type": "given", "value": 2, "label": "tanya_speed_... | true |
2 | Find the number of positive integer palindromes written in base $10$, with no zero digits, and whose digits add up to $13$. For example, $42124$ has these properties. Recall that a palindrome is a number whose representation reads the same from left to right as from right to left. | 62 | {"nodes": [{"id": "n1", "type": "given", "value": 13, "label": "target_digit_sum", "description": "The digits of the palindrome must add up to 13"}, {"id": "n2", "type": "given", "value": 10, "label": "base", "description": "Numbers are written in base 10"}, {"id": "n3", "type": "given", "value": "no_zeros", "label": "... | Find the number of positive integer palindromes written in base $10$, with no zero digits, and whose digits add up to some target value. Recall that a palindrome is a number whose representation reads the same from left to right as from right to left. | {"nodes": [{"id": "n2", "type": "given", "value": 10, "label": "base", "description": "Numbers are written in base 10"}, {"id": "n3", "type": "given", "value": "no_zeros", "label": "no_zero_digits_constraint", "description": "The palindrome must contain no zero digits (each digit is 1-9)"}, {"id": "n4", "type": "given"... | Find the number of positive integer palindromes written in base 10 whose digits add up to some specified total. Recall that a palindrome is a number whose representation reads the same from left to right as from right to left. | {"nodes": [{"id": "n1", "type": "unknown", "value": null, "label": "target_digit_sum", "description": "The digits of the palindrome must add up to some unspecified value"}, {"id": "n2", "type": "given", "value": 10, "label": "base", "description": "Numbers are written in base 10"}, {"id": "n3", "type": "unknown", "valu... | true |
3 | A hemisphere with radius $200$ sits on top of a horizontal circular disk with radius $200$, and the hemisphere and disk have the same center. Let $\mathcal T$ be the region of points $P$ in the disk such that a sphere of radius 42 can be placed on top of the disk at $P$ and lie completely inside the hemisphere. The are... | 079 | {"nodes": [{"id": "n1", "type": "given", "value": 200, "label": "R_hemisphere", "description": "Radius of the hemisphere (and disk)"}, {"id": "n2", "type": "given", "value": 200, "label": "R_disk", "description": "Radius of the horizontal circular disk"}, {"id": "n3", "type": "given", "value": 42, "label": "r_sphere", ... | A hemisphere with radius $200$ sits on top of a horizontal circular disk with radius $200$, and the hemisphere and disk have the same center. A sphere of some unknown radius is to be placed on top of the disk. Let $\mathcal T$ be the region of points $P$ in the disk such that a sphere of this radius can be placed on to... | {"nodes": [{"id": "n1", "type": "given", "value": 200, "label": "R_hemisphere", "description": "Radius of the hemisphere (and disk)"}, {"id": "n2", "type": "given", "value": 200, "label": "R_disk", "description": "Radius of the horizontal circular disk"}, {"id": "n4", "type": "relationship", "value": null, "label": "sh... | A hemisphere with radius $200$ sits on top of a horizontal circular disk with radius $200$, and the hemisphere and disk have the same center. Let $\mathcal T$ be the region of points $P$ in the disk such that a sphere can be placed on top of the disk at $P$ and lie completely inside the hemisphere. The area of $\mathca... | {"nodes": [{"id": "n1", "type": "given", "value": 200, "label": "R_hemisphere", "description": "Radius of the hemisphere (and disk)"}, {"id": "n2", "type": "given", "value": 200, "label": "R_disk", "description": "Radius of the horizontal circular disk"}, {"id": "n3", "type": "unknown", "value": null, "label": "r_spher... | true |
4 | Find the number of integers less than or equal to $100$ that are equal to $a + b + ab$ for some choice of distinct positive integers $a$ and $b$. | 070 | {"nodes": [{"id": "n1", "type": "given", "value": 100, "label": "upper_bound", "description": "Upper bound: integers less than or equal to 100"}, {"id": "n2", "type": "given", "value": null, "label": "a", "description": "First distinct positive integer a"}, {"id": "n3", "type": "given", "value": null, "label": "b", "de... | Find the number of integers less than or equal to some unspecified upper bound $N$ that are equal to $a + b + ab$ for some choice of distinct positive integers $a$ and $b$. | {"nodes": [{"id": "n2", "type": "given", "value": null, "label": "a", "description": "First distinct positive integer a"}, {"id": "n3", "type": "given", "value": null, "label": "b", "description": "Second distinct positive integer b"}, {"id": "n4", "type": "constraint", "value": null, "label": "distinct_positive", "des... | Find the number of integers less than or equal to $N$ that are equal to $a + b + ab$ for some choice of distinct positive integers $a$ and $b$. | {"nodes": [{"id": "n1", "type": "unknown", "value": null, "label": "upper_bound", "description": "Upper bound N: integers less than or equal to N (value unspecified)"}, {"id": "n2", "type": "given", "value": null, "label": "a", "description": "First distinct positive integer a"}, {"id": "n3", "type": "given", "value": ... | true |
5 | A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\theta$ around point $B$ to a point $A \prime$. Point $B$ is rotated across a angle of $\theta$ around point $A \prime$ clockwise to a point $B \prime$. $A B \prime = \frac {4}{3}$. If $\cos \th... | 065 | {"nodes": [{"id": "n1", "type": "given", "value": 1, "label": "AB", "description": "Distance from A to B equals 1"}, {"id": "n2", "type": "given", "value": "acute", "label": "theta_type", "description": "Angle theta is acute"}, {"id": "n3", "type": "given", "value": "4/3", "label": "AB_prime", "description": "Distance ... | A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\theta$ around point $B$ to a point $A'$. Point $B$ is rotated across an angle of $\theta$ around point $A'$ clockwise to a point $B'$. If $\cos \theta = \frac{m}{n}$ where $m$ and $n$ are relat... | {"nodes": [{"id": "n1", "type": "given", "value": 1, "label": "AB", "description": "Distance from A to B equals 1"}, {"id": "n2", "type": "given", "value": "acute", "label": "theta_type", "description": "Angle theta is acute"}, {"id": "n4", "type": "unknown", "value": null, "label": "theta", "description": "The acute r... | A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\theta$ around point $B$ to a point $A'$. Point $B$ is rotated through an angle of $\theta$ around point $A'$ clockwise to a point $B'$. If $\cos \theta = \frac{m}{n}$ where $m$ and $n$ are rela... | {"nodes": [{"id": "n1", "type": "given", "value": 1, "label": "AB", "description": "Distance from A to B equals 1"}, {"id": "n2", "type": "given", "value": "acute", "label": "theta_type", "description": "Angle theta is acute"}, {"id": "n3", "type": "unknown", "value": null, "label": "AB_prime", "description": "Distance... | true |
6 | The product of all positive real numbers $x$ satisfying the equation\[\sqrt[20]{x^{\log_{2026} x}} = 26x\]is an integer $P$. Find the number of positive integer divisors of $P$. | 441 | {"nodes": [{"id": "n1", "type": "given", "value": 20, "label": "root_degree", "description": "The 20th root in the equation"}, {"id": "n2", "type": "given", "value": 2026, "label": "log_base", "description": "Base of the logarithm in the exponent"}, {"id": "n3", "type": "given", "value": 26, "label": "rhs_coefficient",... | The product of all positive real numbers $x$ satisfying the equation
\[\sqrt[k]{x^{\log_{2026} x}} = 26x\]
for some positive integer $k$ is an integer $P$. Find the number of positive integer divisors of $P$. | {"nodes": [{"id": "n2", "type": "given", "value": 2026, "label": "log_base", "description": "Base of the logarithm in the exponent"}, {"id": "n3", "type": "given", "value": 26, "label": "rhs_coefficient", "description": "Coefficient 26 on right-hand side of equation"}, {"id": "n4", "type": "unknown", "value": null, "la... | The product of all positive real numbers $x$ satisfying the equation
\[\sqrt[k]{x^{\log_{2026} x}} = 26x\]
is an integer $P$. Find the number of positive integer divisors of $P$. | {"nodes": [{"id": "n1", "type": "unknown", "value": null, "label": "root_degree", "description": "The k-th root in the equation \u2014 value unspecified"}, {"id": "n2", "type": "given", "value": 2026, "label": "log_base", "description": "Base of the logarithm in the exponent"}, {"id": "n3", "type": "given", "value": 26... | true |
7 | Find the number of functions $\pi$ mapping the set $A=\{1,2,3,4,5,6\}$ onto $A$ such that for every $a\in A$,\[\pi(\pi(\pi(\pi(\pi(\pi(a))))))=a.\] | 396 | {"nodes": [{"id": "n1", "type": "given", "value": 6, "label": "set_size", "description": "Size of set A = {1,2,3,4,5,6}"}, {"id": "n2", "type": "given", "value": [1, 2, 3, 4, 5, 6], "label": "set_A", "description": "The domain and codomain set A = {1,2,3,4,5,6}"}, {"id": "n3", "type": "given", "value": 6, "label": "com... | Find the number of functions $\pi$ mapping the set $A=\{1,2,3,4,5,6\}$ onto $A$ such that for every $a\in A$, applying $\pi$ some number of times to $a$ returns $a$. | {"nodes": [{"id": "n1", "type": "given", "value": 6, "label": "set_size", "description": "Size of set A = {1,2,3,4,5,6}"}, {"id": "n2", "type": "given", "value": [1, 2, 3, 4, 5, 6], "label": "set_A", "description": "The domain and codomain set A = {1,2,3,4,5,6}"}, {"id": "n4", "type": "constraint", "value": null, "labe... | Find the number of functions $\pi$ mapping the set $A=\{1,2,3,4,5,6\}$ onto $A$ such that for every $a\in A$, applying $\pi$ some number of times to $a$ returns $a$. | {"nodes": [{"id": "n1", "type": "given", "value": 6, "label": "set_size", "description": "Size of set A = {1,2,3,4,5,6}"}, {"id": "n2", "type": "given", "value": [1, 2, 3, 4, 5, 6], "label": "set_A", "description": "The domain and codomain set A = {1,2,3,4,5,6}"}, {"id": "n3", "type": "unknown", "value": null, "label":... | true |
8 | Let $N$ be the number of positive integer divisors of $17017^{17}$ that leave a remainder of $5$ upon division by $12$. Find the remainder when $N$ is divided by $1000$. | 244 | {"nodes": [{"id": "n1", "type": "given", "value": 17017, "label": "base_number", "description": "The base number 17017"}, {"id": "n2", "type": "given", "value": 17, "label": "exponent", "description": "The exponent 17 in 17017^17"}, {"id": "n3", "type": "given", "value": 5, "label": "remainder_target", "description": "... | Let $N$ be the number of positive integer divisors of some number raised to the $17$th power that leave a remainder of $5$ upon division by $12$. The base number has four distinct prime factors. Find the remainder when $N$ is divided by $1000$. | {"nodes": [{"id": "n2", "type": "given", "value": 17, "label": "exponent", "description": "The exponent 17"}, {"id": "n3", "type": "given", "value": 5, "label": "remainder_target", "description": "Required remainder upon division by 12"}, {"id": "n4", "type": "given", "value": 12, "label": "divisor_mod", "description":... | Let $N$ be the number of positive integer divisors of $17017^{17}$ that leave some remainder upon division by $12$. Find the remainder when $N$ is divided by $1000$. | {"nodes": [{"id": "n1", "type": "given", "value": 17017, "label": "base_number", "description": "The base number 17017"}, {"id": "n2", "type": "given", "value": 17, "label": "exponent", "description": "The exponent 17 in 17017^17"}, {"id": "n3", "type": "unknown", "value": null, "label": "remainder_target", "descriptio... | true |
9 | Joanne has a blank fair six-sided die and six stickers each displaying a different integer from $1$ to $6$. Joanne rolls the die and then places the sticker labeled $1$ on the top face of the die. She then rolls the die again, places the sticker labeled $2$ on the top face, and continues this process to place the rest ... | 029 | {"nodes": [{"id": "n1", "type": "given", "value": 6, "label": "num_faces", "description": "A fair six-sided die with 6 faces"}, {"id": "n2", "type": "given", "value": 6, "label": "num_stickers", "description": "Six stickers labeled with integers 1 through 6"}, {"id": "n3", "type": "given", "value": [1, 2, 3, 4, 5, 6], ... | Joanne has a blank fair six-sided die and some number of stickers each displaying a different integer. Joanne rolls the die and then places the sticker labeled $1$ on the top face of the die. She then rolls the die again, places the next sticker on the top face, and continues this process to place the rest of the stick... | {"nodes": [{"id": "n1", "type": "given", "value": 6, "label": "num_faces", "description": "A fair six-sided die with 6 faces"}, {"id": "n2", "type": "given", "value": 6, "label": "num_stickers", "description": "Six stickers labeled with integers 1 through 6"}, {"id": "n3", "type": "given", "value": [1, 2, 3, 4, 5, 6], ... | Joanne has a blank fair six-sided die and six stickers each displaying a different integer from $1$ to $6$. Joanne rolls the die and then places the sticker labeled $1$ on the top face of the die. She then rolls the die again, places the sticker labeled $2$ on the top face, and continues this process to place the rest ... | {"nodes": [{"id": "n1", "type": "given", "value": 6, "label": "num_faces", "description": "A fair six-sided die with 6 faces"}, {"id": "n2", "type": "given", "value": 6, "label": "num_stickers", "description": "Six stickers labeled with integers 1 through 6"}, {"id": "n3", "type": "given", "value": [1, 2, 3, 4, 5, 6], ... | true |
10 | Let $\triangle ABC$ have side lengths $AB=13$, $BC=14$, and $CA=15$. Triangle $\triangle A'B'C'$ is obtained by rotating $\triangle ABC$ about its circumcenter so that $\overline{A'C'}$ is perpendicular to $\overline{BC}$, with $A'$ and $B$ not on the same side of line $B'C'$. Find the integer closest to the area of he... | 156 | {"nodes": [{"id": "n1", "type": "given", "value": 13, "label": "AB", "description": "Side length AB = 13"}, {"id": "n2", "type": "given", "value": 14, "label": "BC", "description": "Side length BC = 14"}, {"id": "n3", "type": "given", "value": 15, "label": "CA", "description": "Side length CA = 15"}, {"id": "n4", "type... | Let $\triangle ABC$ have side length $CA=15$, with $AB$ and $BC$ unspecified. Triangle $\triangle A'B'C'$ is obtained by rotating $\triangle ABC$ about its circumcenter so that $\overline{A'C'}$ is perpendicular to $\overline{BC}$, with $A'$ and $B$ not on the same side of line $B'C'$. Find the integer closest to the a... | {"nodes": [{"id": "n3", "type": "given", "value": 15, "label": "CA", "description": "Side length CA = 15"}, {"id": "n4", "type": "derived_info", "value": null, "label": "triangle_ABC", "description": "Original triangle ABC with only CA specified"}, {"id": "n5", "type": "derived_info", "value": null, "label": "circumcen... | Let $\triangle ABC$ have side lengths $AB=13$, $BC=14$, and $CA=15$. Triangle $\triangle A'B'C'$ is obtained by rotating $\triangle ABC$ about its circumcenter by some angle $\theta$, with $A'$ and $B$ not on the same side of line $B'C'$. Find the integer closest to the area of hexagon $AA'CC'BB'$. | {"nodes": [{"id": "n1", "type": "given", "value": 13, "label": "AB", "description": "Side length AB = 13"}, {"id": "n2", "type": "given", "value": 14, "label": "BC", "description": "Side length BC = 14"}, {"id": "n3", "type": "given", "value": 15, "label": "CA", "description": "Side length CA = 15"}, {"id": "n4", "type... | true |
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