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if Q(i),Q(i+1),Q(i+2),Q(i+3) are integers for 1 integer i, then Q(n) is integer

Source: Irmo 2014 p2 q8

September 15, 2018
algebrapolynomialInteger

Problem Statement

(a) Let a0,a1,a2a_0, a_1,a_2 be real numbers and consider the polynomial P(x)=a0+a1x+a2x2P(x) = a_0 + a_1x + a_2x^2 . Assume that P(−1),P(0)P(-1), P(0) and P(1)P(1) are integers. Prove that P(n)P(n) is an integer for all integers nn. (b) Let a0,a1,a2,a3a_0,a_1, a_2, a_3 be real numbers and consider the polynomial Q(x)=a0+a1x+a2x2+a3x3Q(x) = a0 + a_1x + a_2x^2 + a_3x^3 . Assume that there exists an integer ii such that Q(i),Q(i+1),Q(i+2)Q(i),Q(i+1),Q(i+2) and Q(i+3)Q(i+3) are integers. Prove that Q(n)Q(n) is an integer for all integers nn.
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