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China South East Mathematical Olympiad 2021 Grade10 Q3

Source: China Nanchang, Jul 28,2021

July 28, 2021
algebranumber theoryChina

Problem Statement

Let pp be an odd prime and {ui}i≥0\{u_i\}_{i\ge 0}be an integer sequence. Let vn=∑i=0nCnipiuiv_n=\sum_{i=0}^{n} C_{n}^{i} p^iu_i where CniC_n^i denotes the binomial coefficients. If vn=0v_n=0 holds for infinitely many nn , prove that it holds for every positive integer nn.
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