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Source: China TST 2025 P20

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steven_zhang123

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steven_zhang123

#1 h Mar 29, 2025, 12:42 AM • 1 Y

Y by yofro

Let $n$ be an odd integer, $m = \frac{n+1}{2}$ . Consider $2m$ integers $a_1, a_2, \ldots, a_m, b_1, b_2, \ldots, b_m$ such that for any $1 \leq i < j \leq m$ , $a_i \not\equiv a_j \pmod{n}$ and $b_i \not\equiv b_j \pmod{n}$ . Prove that the number of $k \in \{0, 1, \ldots, n-1\}$ for which satisfy $a_i + b_j \equiv k \pmod{n}$ for some $i \neq j$ , $i, j \in \left \{ 1,2,\cdots,m \right \}$ is greater than $n - \sqrt{n} - \frac{1}{2}$ .

This post has been edited 1 time. Last edited by steven_zhang123, Mar 29, 2025, 12:54 AM

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