problem string | hint string | completions sequence | corrects sequence | acc float64 | answer string |
|---|---|---|---|---|---|
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <notes>
<note>
<description>Convert any number from base b to base 10 by expressing it as a polynomial in b: for a k-digit number with digits d₁…dₖ, its value is ∑_{i=1}^k d_i·b^{k–i}. This lets you work with familiar arithmetic in base 10.</description>
<example>For the 3-digit base-b number d₁d₂d₃, its base... | [
"<think>\nOkay, let's try to solve this problem. The question is asking for the sum of all integer bases b > 9 where 17_b is a divisor of 97_b. \n\nFirst, I need to understand what 17_b and 97_b represent in base b. In any base b, a number like 17_b is equal to 1*b^1 + 7*b^0, which simplifies to b + 7. Similarly, 9... | [
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] | 1 | 70 |
"In $\\triangle ABC$ points $D$ and $E$ lie on $\\overline{AB}$ so that $AD < AE < AB$, while points(...TRUNCATED) | "<notes>\n <note>\n <description>When two points are reflected across a line, the segment joinin(...TRUNCATED) | ["<think>\nOkay, let's try to tackle this geometry problem. Hmm, so we have triangle ABC with points(...TRUNCATED) | [
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] | 0.5625 | 588 |
"The $9$ members of a baseball team went to an ice-cream parlor after their game. Each player had a (...TRUNCATED) | "<notes>\n <note>\n <description>When assigning distinct labels to objects with no further restr(...TRUNCATED) | ["<think>\nOkay, let's tackle this problem step by step. So, we have 9 players, each choosing one of(...TRUNCATED) | [
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] | 1 | 16 |
"Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $1(...TRUNCATED) | "<notes>\n <note>\n <description>When a binary quadratic Diophantine equation has integer coeffi(...TRUNCATED) | ["<think>\nOkay, let's see. I need to find the number of ordered pairs (x, y) where both x and y are(...TRUNCATED) | [
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] | 0.625 | 117 |
"There are $8!= 40320$ eight-digit positive integers that use each of the digits $1, 2, 3, 4, 5, 6, (...TRUNCATED) | "<notes>\n <note>\n <description>When a divisibility rule involves a product of two coprime fact(...TRUNCATED) | ["<think>\nOkay, let's tackle this problem step by step. The question is about finding the number of(...TRUNCATED) | [
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] | 0.3125 | 279 |
"An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the(...TRUNCATED) | "<notes>\n <note>\n <description>When a circle is inscribed in a quadrilateral, the sum of the l(...TRUNCATED) | ["<think>\nOkay, let's see. The problem is about an isosceles trapezoid with an inscribed circle. Th(...TRUNCATED) | [
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] | 0.875 | 504 |
"The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into s(...TRUNCATED) | "<notes>\n <note>\n <description>When arranging n distinct items into ordered pairs, the total n(...TRUNCATED) | ["<think>\nOkay, let's tackle this probability problem step by step. First, I need to understand wha(...TRUNCATED) | [
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] | 0 | 821 |
"Let $k$ be a real number such that the system \\begin{align*} &|25 + 20i - z| = 5 \\ &|z - 4 - k| =(...TRUNCATED) | "<notes>\n <note>\n <description>Interpret the equation |z – a| = r geometrically as a circle (...TRUNCATED) | ["<think>\nOkay, let's try to solve this problem step by step. So, we have a system of two equations(...TRUNCATED) | [
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] | 0.1875 | 77 |
"The parabola with equation $y = x^2 - 4$ is rotated $60^\\circ$ counterclockwise around the origin.(...TRUNCATED) | "<notes>\n <note>\n <description>Rotation of a point (x,y) by an angle θ around the origin can (...TRUNCATED) | ["<think>\nOkay, let's tackle this problem step by step. The question is about rotating a parabola a(...TRUNCATED) | [
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] | 0.0625 | 62 |
"The $27$ cells of a $3 \\times 9$ grid are filled in using the numbers $1$ through $9$ so that each(...TRUNCATED) | "<notes>\n <note>\n <description>When arranging n distinct symbols into n slots with no repetiti(...TRUNCATED) | ["<think>\nOkay, let's try to tackle this problem step by step. So, we have a 3x9 grid filled with n(...TRUNCATED) | [
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] | 0 | 81 |
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