IMO Shortlist 1968 problem 16


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2. travnja 2012.
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A polynomial p(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k with integer coefficients is said to be divisible by an integer m if p(x) is divisible by m for all integers x. Prove that if p(x) is divisible by m, then k!a_0 is also divisible by m. Also prove that if a_0, k,m are non-negative integers for which k!a_0 is divisible by m, there exists a polynomial p(x) = a_0x^k+\cdots+ a_k divisible by m.
Izvor: Međunarodna matematička olimpijada, shortlist 1968