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yoonholee

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completions_AIME2024_qwen3-8b-hint-rpo-0.1-0.1-1e-5init_Qwen3-1.7B

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"Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: \n\\[\\l(...TRUNCATED)

"<notes>\n <note>\n <description>Use the change-of-base formula to convert any logarithm to a co(...TRUNCATED)

"Let $O(0,0), A(\\tfrac{1}{2}, 0),$ and $B(0, \\tfrac{\\sqrt{3}}{2})$ be points in the coordinate pl(...TRUNCATED)

"<notes>\n <note>\n <description>Any segment of fixed length between two axes can be described b(...TRUNCATED)

"Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers(...TRUNCATED)

"<notes>\n <note>\n <description>When an event E can occur in multiple ways, compute its probabi(...TRUNCATED)

"Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take tur(...TRUNCATED)

"<notes>\n <note>\n <description>When a game involves removing a fixed number of tokens (or piec(...TRUNCATED)

"Eight circles of radius $34$ are sequentially tangent, and two of the circles are tangent to $AB$ a(...TRUNCATED)

"<notes>\n <note>\n <description>When a circle of radius r is tangent to a line, its center lies(...TRUNCATED)

"Define $f(x)=|| x|-\\tfrac{1}{2}|$ and $g(x)=|| x|-\\tfrac{1}{4}|$. Find the number of intersection(...TRUNCATED)

"<notes>\n <note>\n <description>When a function involves an absolute value, split the domain in(...TRUNCATED)

"Each vertex of a regular octagon is independently colored either red or blue with equal probability(...TRUNCATED)

"<notes>\n <note>\n <description>When a geometric figure is invariant under a group of symmetrie(...TRUNCATED)

"Find the number of triples of nonnegative integers $(a,b,c)$ satisfying $a + b + c = 300$ and \\[a^(...TRUNCATED)

"<notes>\n <note>\n <description>When a symmetric polynomial in several variables is given, intr(...TRUNCATED)

"There exist real numbers $x$ and $y$, both greater than 1, such that $\\log_x\\left(y^x\\right)=\\l(...TRUNCATED)

"<notes>\n <note>\n <description>Use the change-of-base formula to rewrite logarithms with diffe(...TRUNCATED)

"Alice chooses a set $A$ of positive integers. Then Bob lists all finite nonempty sets $B$ of positi(...TRUNCATED)

"<notes>\n <note>\n <description>When each element of a set can be chosen independently, the tot(...TRUNCATED)