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p^2|binomial sum of p choose k, k<2p/3

Source: VJIMC 2001 1.2

July 23, 2021
number theorySummationbinomial coefficients

Problem Statement

Prove that for any prime p≥5p\ge5, the number ∑0<k<2p3(pk)\sum_{0<k<\frac{2p}3}\binom pkis divisible by p2p^2.
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