Question 79
Consider an odd prime $p$ and a positive integer $N \lt 50p$. Let $a_1, a_2, \dots, a_N$ be a list of positive integers less than $p$ such that any specific value occurs at most $\frac{51}{100}N$ times and $a_1 + a_2 + \dots + a_N$ is not divisible by $p$. Prove that there exists a permutation $b_1, b_2, \dots, b_N$ of the $a_i$ such that, for all $k = 1, 2, \dots, N$, the sum $b_1 + b_2 + \dots + b_k$ is not divisible by $p$.