US-EGMO-TST 2026 Problem 2

Let $p$ be a prime and let $a$ and $b$ be positive integers less than $p$ . Show that

$\sum_{k=1}^{b} (-1)^{\lfloor (a-1)k/p \rfloor + \lfloor ak/p \rfloor} \geq 0.$

Official text USA EGMO Team Selection Test 2026 official paper, page 1. Quoted with source.

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Also US-TST 2026 Problem 2.

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