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

We see that $\{\frac{ra+rb+rc+rd}{p}\}=0$ means that $p|r(a+b+c+d)$ . Now, since $p$ does nto divide $r$ and $p$ is prime, their GCD is 1 so $p|a+b+c+d$ .

Since \{ \frac{ra}p \}+\{ \frac{rb}p \}+\{ \frac{rc}p \}+\{ \frac{rd}p \} =2 $, then we see that they have to represent mods$ \mod p $, and thus, our possible values of$ p $are all such that$ k^4 \equiv 1 \pmod(p) $for all$ k $. This happens when$ p=2, 3 $or$ 5$.

When$ (Error compiling LaTeX. Unknown error_msg)p=2 $then$ r $is odd, meaning$ ra $,$ rb $,$ rc $and$ rd$are all 1 mod 2, or the sum wouldn't be 2. Any pairwise sum is 2.

When$ (Error compiling LaTeX. Unknown error_msg)p=3 $then$ r $is not divisible by 3, thus two are$ 1 $, and the other two are$ 2$. Thus, four pairwise sums sum to 3.

When$ (Error compiling LaTeX. Unknown error_msg)p=5 $then$ r $is not divisible by 5 so$ a, b, c, d $are$ 1, 2, 3 $and$ 4$, so two pairwise sums sum to 5.

All three possible cases work so we are done.

1999 USAMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6
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1999 USAMO Problems/Problem 3

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