Question 3
Let $p>2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$. Suppose that for every integer $r$ not divisible by $p$,
\[\left\{\frac{ra}{p}\right\}+\left\{\frac{rb}{p}\right\}+\left\{\frac{rc}{p}\right\}+\left\{\frac{rd}{p}\right\}=2,\]
where $\{t\}=t-\lfloor t\rfloor$ is the fractional part. Prove that at least two of $a+b,a+c,a+d,b+c,b+d,c+d$ are divisible by $p$.
\[\left\{\frac{ra}{p}\right\}+\left\{\frac{rb}{p}\right\}+\left\{\frac{rc}{p}\right\}+\left\{\frac{rd}{p}\right\}=2,\]
where $\{t\}=t-\lfloor t\rfloor$ is the fractional part. Prove that at least two of $a+b,a+c,a+d,b+c,b+d,c+d$ are divisible by $p$.