RMM 2015 Problem 5

Let $p \geq 5$ be a prime number. For a positive integer $k$ we denote by $R(k)$ the remainder of $k$ when divided by $p$. Determine all positive integers $a<p$ such that

$$m+R(m a)>a$$

for every $m=1,2, \ldots, p-1$. (Bulgaria) Alexander Ivanov