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China South East Mathematical Olympiad 2021 Grade11 P2

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July 28, 2021
number theoryprime numbers

Problem Statement

Let p≥5p\geq 5 be a prime number, and set M={1,2,⋯ ,p−1}.M=\{1,2,\cdots,p-1\}. Define T={(n,xn):p∣nxn−1 and n,xn∈M}.T=\{(n,x_n):p|nx_n-1\ \textup{and}\ n,x_n\in M\}. If ∑(n,xn)∈Tn[nxnp]≡k(modp),\sum_{(n,x_n)\in T}n\left[\dfrac{nx_n}{p}\right]\equiv k \pmod {p}, with 0≤k≤p−1,0\leq k\leq p-1, where [α]\left[\alpha\right] denotes the largest integer that does not exceed α,\alpha, determine the value of k.k.
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