IMO Shortlist 1969 problem 24


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2. travnja 2012.
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(GBR 1) The polynomial P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k, where a_0,\cdots, a_k are integers, is said to be divisible by an integer m if P(x) is a multiple of m for every integral value of x. Show that if P(x) is divisible by m, then a_0 \cdot k! is a multiple of m. Also prove that if a, k,m are positive integers such that ak! is a multiple of m, then a polynomial P(x) with leading term ax^kcan be found that is divisible by m.
Izvor: Međunarodna matematička olimpijada, shortlist 1969