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[ { "content": "Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$", "role": "user" } ]
70
math_aime_2025
MathArena/aime_2025
1
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$
70
[ "Number Theory" ]
[ "To find the sum of all integer bases \\(b > 9\\) for which \\(17_b\\) is a divisor of \\(97_b\\), we need to convert the numbers from base \\(b\\) to base \\(10\\) and then determine when one is a divisor of the other.\n\nFirst, let's convert \\(17_b\\) and \\(97_b\\) to base \\(10\\):\n\n\\(17_b\\) converts to:\n...
7
7/32
32
Qwen/Qwen3-4B-Base
qwen_instruct_user_boxed_math
1
1
8,192
train
[{"content":"On $\\triangle ABC$ points $A, D, E$, and $B$ lie in that order on side $\\overline{AB}(...TRUNCATED)
588
math_aime_2025
MathArena/aime_2025
2
"On $\\triangle ABC$ points $A, D, E$, and $B$ lie in that order on side $\\overline{AB}$ with $AD =(...TRUNCATED)
588
[ "Geometry" ]
["To find the area of the heptagon \\(AFNBCEM\\), we start by analyzing the given points and their p(...TRUNCATED)
0
0/32
32
Qwen/Qwen3-4B-Base
qwen_instruct_user_boxed_math
1
1
8,192
train
[{"content":"The 9 members of a baseball team went to an ice-cream parlor after their game. Each pla(...TRUNCATED)
16
math_aime_2025
MathArena/aime_2025
3
"The 9 members of a baseball team went to an ice-cream parlor after their game. Each player had a si(...TRUNCATED)
16
[ "Combinatorics" ]
["To find the number of different assignments of ice cream flavors to the baseball players, we start(...TRUNCATED)
2
2/32
32
Qwen/Qwen3-4B-Base
qwen_instruct_user_boxed_math
1
1
8,192
train
[{"content":"Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $(...TRUNCATED)
117
math_aime_2025
MathArena/aime_2025
4
"Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $1(...TRUNCATED)
117
[ "Algebra" ]
["To find the number of ordered pairs \\((x, y)\\) such that both \\(x\\) and \\(y\\) are integers b(...TRUNCATED)
0
0/32
32
Qwen/Qwen3-4B-Base
qwen_instruct_user_boxed_math
1
1
8,192
train
[{"content":"There are $8!= 40320$ eight-digit positive integers that use each of the digits 1, 2, 3(...TRUNCATED)
279
math_aime_2025
MathArena/aime_2025
5
"There are $8!= 40320$ eight-digit positive integers that use each of the digits 1, 2, 3, 4, 5, 6, 7(...TRUNCATED)
279
[ "Combinatorics", "Number Theory" ]
["To find the number of eight-digit integers using each of the digits 1, 2, 3, 4, 5, 6, 7, 8 exactly(...TRUNCATED)
0
0/32
32
Qwen/Qwen3-4B-Base
qwen_instruct_user_boxed_math
1
1
8,192
train
[{"content":"An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The r(...TRUNCATED)
504
math_aime_2025
MathArena/aime_2025
6
"An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the(...TRUNCATED)
504
[ "Geometry" ]
["To find \\( r^2 + s^2 \\) for the given isosceles trapezoid with an inscribed circle, we start by (...TRUNCATED)
3
3/32
32
Qwen/Qwen3-4B-Base
qwen_instruct_user_boxed_math
1
1
8,192
train
[{"content":"The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly gr(...TRUNCATED)
821
math_aime_2025
MathArena/aime_2025
7
"The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into s(...TRUNCATED)
821
[ "Combinatorics" ]
["To find the probability that the last word listed contains the letter $G$, let's break down the pr(...TRUNCATED)
0
0/32
32
Qwen/Qwen3-4B-Base
qwen_instruct_user_boxed_math
1
1
8,192
train
[{"content":"Let $k$ be a real number such that the system\n\\begin{align*} |25+20i-z|&=5\\\\ |z-4-k(...TRUNCATED)
77
math_aime_2025
MathArena/aime_2025
8
"Let $k$ be a real number such that the system\n\\begin{align*} |25+20i-z|&=5\\\\ |z-4-k|&=|z-3i-k| (...TRUNCATED)
77
[ "Algebra" ]
["To find the sum of all possible values of \\( k \\) such that the given system has exactly one com(...TRUNCATED)
0
0/32
32
Qwen/Qwen3-4B-Base
qwen_instruct_user_boxed_math
1
1
8,192
train
[{"content":"The parabola with equation $y = x^2 - 4$ is rotated $60^\\circ$ counterclockwise around(...TRUNCATED)
62
math_aime_2025
MathArena/aime_2025
9
"The parabola with equation $y = x^2 - 4$ is rotated $60^\\circ$ counterclockwise around the origin.(...TRUNCATED)
62
[ "Algebra" ]
["To find the unique point in the fourth quadrant where the original parabola \\( y = x^2 - 4 \\) an(...TRUNCATED)
2
2/32
32
Qwen/Qwen3-4B-Base
qwen_instruct_user_boxed_math
1
1
8,192
train
[{"content":"The $27$ cells of a $3 \\times 9$ grid are filled in using the numbers $1$ through $9$ (...TRUNCATED)
81
math_aime_2025
MathArena/aime_2025
10
"The $27$ cells of a $3 \\times 9$ grid are filled in using the numbers $1$ through $9$ so that each(...TRUNCATED)
81
[ "Combinatorics" ]
["To find the number of different ways to fill a $3 \\times 9$ grid under the given constraints, we (...TRUNCATED)
0
0/32
32
Qwen/Qwen3-4B-Base
qwen_instruct_user_boxed_math
1
1
8,192
train
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