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yoonholee

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completions_AIME2024_qwen3-8b-hint-rpo-0.1-0.1-5e-6init_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 rewrite any logarithm with a (...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>When a point lies on a line segment, its coordinates can be exp(...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>Use complementary counting to find the probability of “at lea(...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>Model the game as a directed graph where each state (number of (...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 sequence of equal-radius circles is tangent along a stra(...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>Use the definition of the absolute value function to break a pi(...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 probability depends on an outcome’s symmetry (e.g. rot(...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>Use the multinomial coefficient to count the number of nonnegat(...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 convert logarithms from one b(...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 a set of positive integers is partitioned by a chosen elem(...TRUNCATED)