uuid stringlengths 5 7 | question stringlengths 37 2.66k | teacher_reasoning stringlengths 1 10.5k | answer stringclasses 1
value | rewritten_reasoning stringlengths 386 149k | rewrite_model stringclasses 1
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row-0 | Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$. | In base \(b\)
\[
17_b = 1\cdot b+7 = b+7,
\qquad
97_b = 9\cdot b+7 = 9b+7 .
\]
We need \((b+7)\mid (9b+7)\).
Write \(b+7=m\) (\(m>16\) because \(b>9\)).
Then \(b=m-7\) and
\[
9b+7 = 9(m-7)+7 = 9m-56 .
\]
Hence
\[
9b+7 \equiv -56 \pmod{m}\quad\Longrightarrow\quad
m\mid 56 .
\]
So \(m=b+7\) must be a divisor of ... | To find the sum of all bases \(b>9\) where \(17_b\) divides \(97_b\):
1. In base \(b\),
\[
17_b = b + 7,
\quad 97_b = 9b + 7.
\]
2. We need \(b+7\) to divide \(9b+7\). Let \(b+7 = m\) where \(m > 16\).
3. Then \(b = m - 7\) and
\[
9b + 7 = 9(m - 7) + 7 = 9m - 63 + 7 = 9m - 56.
\]
4. So, \(9m - 56 ... | Qwen/Qwen2.5-7B-Instruct | vllm | |
row-1 | On $\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\overline{AB}$ with $AD=4, DE=16$, and $EB=8$. Points $A,F,G$, and $C$ lie in that order on side $\overline{AC}$ with $AF=13, FG=52$, and $GC=26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. Quadrilate... | Let
\[
AB=AD+DE+EB=4+16+8=28,\qquad AC=AF+FG+GC=13+52+26=91 .
\]
Place \(A\) at \((0,0)\) and \(B\) at \((28,0)\).
Let \(C=(c,42)\); we will determine \(c\) from the given area of \(DEGF\).
--------------------------------------------------------------------
### 1. Coordinates of the points on \(AB\) and \(AC\)... | Let's find the lengths and coordinates step-by-step:
1. Calculate \(AB\) and \(AC\):
\[
AB = 4 + 16 + 8 = 28, \quad AC = 13 + 52 + 26 = 91.
\]
2. Place \(A\) at \((0,0)\) and \(B\) at \((28,0)\). Let \(C = (c, 42)\).
3. Find coordinates of points \(D, E, F, G\):
\[
D = (4,0), \quad E = (20,0), \quad F... | Qwen/Qwen2.5-7B-Instruct | vllm | |
row-2 | The 9 members of a baseball team went to an ice cream parlor after their game. Each player had a singlescoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was great... | The counts of chocolate (\(c\)), vanilla (\(v\)), and strawberry (\(s\)) must satisfy
\[
c+v+s=9,\qquad c>v>s\ge 1 .
\]
---
### Find all possible \((c,v,s)\)
Let \(s=k\).
Then \(v\ge k+1\) and \(c\ge v+1\).
Write \(v=k+1+t\) with \(t\ge 0\).
Then
\[
c=9-(k+1+t)-k = 8-2k-t .
\]
We need
\[
c\ge v+1 \;\L... | To solve the problem, we need to find the number of ways to assign ice cream flavors to 9 players such that each flavor is chosen at least once and the number of players choosing chocolate is greater than those choosing vanilla, which is greater than those choosing strawberry.
First, let's determine the possible count... | Qwen/Qwen2.5-7B-Instruct | vllm | |
row-3 | Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$, inclusive, such that $12x^{2}-xy-6y^{2}=0$. | The equation is
\[
12x^2-xy-6y^2=0 .
\]
Treat it as a quadratic in \(y\):
\[
-6y^2-xy+12x^2=0 \quad\Longrightarrow\quad 6y^2+xy-12x^2=0 .
\]
Using the quadratic formula for \(y\),
\[
y=\frac{-x\pm\sqrt{x^2-4(6)(-12x^2)}}{12}
=\frac{-x\pm\sqrt{x^2+288x^2}}{12}
=\frac{-x\pm 17|x|}{12}.
\]
For any integer \(x\... | The equation is \(12x^2 - xy - 6y^2 = 0\). Treat it as a quadratic in \(y\):
\[
6y^2 + xy - 12x^2 = 0.
\]
Using the quadratic formula for \(y\):
\[
y = \frac{-x \pm \sqrt{x^2 + 288x^2}}{12} = \frac{-x \pm 17|x|}{12}.
\]
If \(x \ge 0\), \(y = \frac{4x}{12} = \frac{x}{3}\) or \(y = -\frac{3x}{2}\).
If \(x \le 0\), t... | Qwen/Qwen2.5-7B-Instruct | vllm | |
row-4 | There are $8!=40320$ eight-digit positive integers that use each of the digits $1,2,3,4,5,6,7,8$ exactly once. Let $N$ be the number of these integers that are divisible by 22. Find the difference between $N$ and 2025. | First, a number formed with the digits \(1,2,\dots ,8\) is divisible by \(22\) iff it is divisible by both \(2\) and \(11\).
---
### 1. Divisibility by \(2\)
The last (8th) digit must be even.
The even digits available are \(2,4,6,8\).
---
### 2. Divisibility by \(11\)
For an 8‑digit number \(d_1d_2d_3d_4d_5d... | To find numbers divisible by 22, they must end in an even digit and have an alternating sum divisible by 11.
1. **Divisibility by 2**: The last digit must be even. There are 4 choices: 2, 4, 6, 8.
2. **Divisibility by 11**: Sum of digits in odd positions must equal the sum in even positions, both 18. Possible sets o... | Qwen/Qwen2.5-7B-Instruct | vllm | |
row-5 | "An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the(...TRUNCATED) | "For a tangential quadrilateral the sums of lengths of opposite sides are equal. \nLet the parallel(...TRUNCATED) | "Given an isosceles trapezoid with an inscribed circle, the sum of the lengths of the parallel sides(...TRUNCATED) | Qwen/Qwen2.5-7B-Instruct | vllm | |
row-6 | "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) | "Let the letters be the numbers \\(1,2,\\dots ,12\\) with \\(G=7\\).\n\n---\n\n### 1. Pairing by a (...TRUNCATED) | "To find the probability that the last word contains \\(G\\), first note that \\(G\\) pairs with ano(...TRUNCATED) | Qwen/Qwen2.5-7B-Instruct | vllm | |
row-7 | "Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has exactly one (...TRUNCATED) | "The first equation \n\n\\[\n|25+20i-z|=5\n\\]\n\ndescribes a circle with centre \\(C=(25,20)\\) an(...TRUNCATED) | "To find \\(k\\) such that the system has exactly one complex solution \\(z\\), start by understandi(...TRUNCATED) | Qwen/Qwen2.5-7B-Instruct | vllm | |
row-8 | "The parabola with equation $y=x^{2}-4$ is rotated $60^{\\circ}$ counterclockwise around the origin.(...TRUNCATED) | "The parabola \n\n\\[\ny=x^{2}-4\n\\]\n\nis rotated \\(60^\\circ\\) counter‑clockwise about the o(...TRUNCATED) | "The parabola $y = x^2 - 4$ is rotated $60^\\circ$ counterclockwise around the origin. Let the origi(...TRUNCATED) | Qwen/Qwen2.5-7B-Instruct | vllm | |
row-9 | "The 27 cells of a $3\\times9$ grid are filled in using the numbers 1 through 9 so that each row con(...TRUNCATED) | "The grid consists of three \\(3\\times 3\\) blocks. \nLet the rows of the grid be \\(R_1,R_2,R_3\\(...TRUNCATED) | "The 27 cells of a $3\\times9$ grid are filled using numbers 1 through 9. Each row and each $3\\time(...TRUNCATED) | Qwen/Qwen2.5-7B-Instruct | vllm |
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