Qwen3-30B-A3B-Instruct-2507
Qwen3-4B-Instruct-2507
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Qwen3-32B
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aime_2024
aime_2025
amc23
gaokao_math_cloze
gpqa_diamond
olympiad
Instruction 1
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for which there exists a strategy for Bob that guarantees that Bob will win the game regardless of Alice's play. Please reason step by step, and put your final answer within \boxed{}.
Instruction 2
Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$. Find the number of intersections of the graphs of \[y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).\] Please reason step by step, and put your final answer within \boxed{}.
Instruction 3
Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p^{2}$. Find the least positive integer $m$ such that $m^{4}+1$ is divisible by $p^{2}$. Please reason step by step, and put your final answer within \boxed{}.
Instruction 4
Let $\omega\neq 1$ be a 13th root of unity. Find the remainder when \[\prod_{k=0}^{12}(2-2\omega^k+\omega^{2k})\] is divided by 1000. Please reason step by step, and put your final answer within \boxed{}.
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