Question 36
Given a sequence of integers $a_1, a_2, a_3, \dots$ such that
$$
0 \le a_k \le k-1 \quad \text{and} \quad a_1 + \dots + a_k \equiv 0 \pmod{k}
$$
for all $k \gt 1$. Prove that the sequence is constant from some point on. For example, when $a_1 = 9$ the sequence is $9, 1, 2, 0, 3, 3, 3, \dots$.
$$
0 \le a_k \le k-1 \quad \text{and} \quad a_1 + \dots + a_k \equiv 0 \pmod{k}
$$
for all $k \gt 1$. Prove that the sequence is constant from some point on. For example, when $a_1 = 9$ the sequence is $9, 1, 2, 0, 3, 3, 3, \dots$.