IMO 1974 LL CUB7

Let P be a prime number and n a natural number. Prove that

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IMO 1974 LL CUB7

Origin: CUB

Problem

Let P be a prime number and n a natural number. Prove that the product N = pn2 2n−1

i=1; 2∤i ((p −1)i)! p2i pi  is a natural number that is not divisible by p.