RMM 2024 Problem 2

Consider an odd prime $p$ and a positive integer $N<50 p$. Let $a_{1}, a_{2}, \ldots, a_{N}$ be a list of positive integers less than $p$ such that any specific value occurs at most $\frac{51}{100} N$ times and $a_{1}+a_{2}+\cdots+a_{N}$ is not divisible by $p$. Prove that there exists a permutation $b_{1}, b_{2}, \ldots, b_{N}$ of the $a_{i}$ such that, for all $k=1,2, \ldots, N$, the sum $b_{1}+b_{2}+\cdots+b_{k}$ is not divisible by $p$.