1999 USAMO Problems/Problem 3

Problem

Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$ , such that $\left\{ \dfrac{ra}{p} \right\} + \left\{ \dfrac{rb}{p} \right\} + \left\{ \dfrac{rc}{p} \right\} + \left\{ \dfrac{rd}{p} \right\} = 2$ for any integer $r$ not divisible by $p$ . Prove that at least two of the numbers $a+b$ , $a+c$ , $a+d$ , $b+c$ , $b+d$ , $c+d$ are divisible by $p$ . (Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$ .)

Solution

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1999 USAMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
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1999 USAMO Problems/Problem 3

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