problem string | answer string | tokenized_prompt string | responses list |
|---|---|---|---|
Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$. | 70 | <|im_start|>user
Return your final response within \boxed{}. Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$.<|im_end|>
<|im_start|>assistant
| [
"We are given a problem involving numbers in different bases.\n\nWe are to **find the sum of all integer bases $ b > 9 $** for which $ 17_b $ divides $ 97_b $.\n\n---\n\n### Step 1: Convert $ 17_b $ and $ 97_b $ to base 10\n\n- $ 17_b = 1 \\cdot b + 7 = b + 7 $\n- $ 97_b = 9 \\cdot b + 7 = 9b + 7 $\n\nWe are told t... |
"On $\\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\\overline{AB}$ with $AD=4, DE=(...TRUNCATED) | 588 | "<|im_start|>user\nReturn your final response within \\boxed{}. On $\\triangle ABC$ points $A,D,E$, (...TRUNCATED) | ["We are given a geometric configuration involving a triangle $ \\triangle ABC $, with points on sid(...TRUNCATED) |
"The 9 members of a baseball team went to an ice cream parlor after their game. Each player had a si(...TRUNCATED) | 16 | "<|im_start|>user\nReturn your final response within \\boxed{}. The 9 members of a baseball team wen(...TRUNCATED) | ["We are given:\n\n- 9 players.\n- Each player chooses one of three flavors: chocolate (C), vanilla (...TRUNCATED) |
"Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $1(...TRUNCATED) | 117 | "<|im_start|>user\nReturn your final response within \\boxed{}. Find the number of ordered pairs $(x(...TRUNCATED) | ["We are given the Diophantine equation:\n\n$$\n12x^2 - xy - 6y^2 = 0\n$$\n\nWe are to find the numb(...TRUNCATED) |
"There are $8!=40320$ eight-digit positive integers that use each of the digits $1,2,3,4,5,6,7,8$ ex(...TRUNCATED) | 279 | "<|im_start|>user\nReturn your final response within \\boxed{}. There are $8!=40320$ eight-digit pos(...TRUNCATED) | ["We are given that there are $ 8! = 40320 $ eight-digit positive integers formed by using each of t(...TRUNCATED) |
"An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the(...TRUNCATED) | 504 | "<|im_start|>user\nReturn your final response within \\boxed{}. An isosceles trapezoid has an inscri(...TRUNCATED) | ["We are given an **isosceles trapezoid** with an **inscribed circle** (a tangential quadrilateral),(...TRUNCATED) |
"The twelve letters $A,B,C,D,E,F,G,H,I,J,K$, and $L$ are randomly grouped into six pairs of letters.(...TRUNCATED) | 821 | "<|im_start|>user\nReturn your final response within \\boxed{}. The twelve letters $A,B,C,D,E,F,G,H,(...TRUNCATED) | ["We are given 12 distinct letters: \n$ A, B, C, D, E, F, G, H, I, J, K, L $ \nThese are randomly (...TRUNCATED) |
"Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has exactly one (...TRUNCATED) | 77 | "<|im_start|>user\nReturn your final response within \\boxed{}. Let $k$ be real numbers such that th(...TRUNCATED) | ["We are given a system of two equations involving a complex number $ z $ and a real number $ k $:\n(...TRUNCATED) |
"The parabola with equation $y=x^{2}-4$ is rotated $60^{\\circ}$ counterclockwise around the origin.(...TRUNCATED) | 62 | "<|im_start|>user\nReturn your final response within \\boxed{}. The parabola with equation $y=x^{2}-(...TRUNCATED) | ["We are given the parabola:\n\n$$\ny = x^2 - 4\n$$\n\nThis parabola is rotated $60^\\circ$ **counte(...TRUNCATED) |
"The 27 cells of a $3\\times9$ grid are filled in using the numbers 1 through 9 so that each row con(...TRUNCATED) | 81 | "<|im_start|>user\nReturn your final response within \\boxed{}. The 27 cells of a $3\\times9$ grid a(...TRUNCATED) | ["We are given a $3 \\times 9$ grid filled with numbers 1 through 9, such that:\n\n- Each **row** co(...TRUNCATED) |
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