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Generaliztion of 1999 USAMO
EthanWYX2009   5
N an hour ago by kingu
Let $p > 2$ be a prime and let $a_1,a_2,\ldots,a_{2n}$ be integers not divisible by $p,$ such that
\[ \left\{ \dfrac{ra_1}{p} \right\} + \left\{ \dfrac{ra_2}{p} \right\} + \cdots+ \left\{ \dfrac{ra_{2n}}{p} \right\} = n  \]for any integer $r$ not divisible by $p$. Prove that there exists $2\le j\le 2n$ such that $a_1+a_j$ is divisible by $p$.

Created by Haojia Shi
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EthanWYX2009
Nov 14, 2024
kingu
an hour ago
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