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exists plynomial with two conditions

Source: 4-th Taiwanese Mathematical Olympiad 1995

January 16, 2007
algebrapolynomialalgebra unsolved

Problem Statement

Let m1,m2,...,mnm_{1},m_{2},...,m_{n} be mutually distinct integers. Prove that there exists a f(x)∈Z[x]f(x)\in\mathbb{Z}[x] of degree nn satisfying the following two conditions: a)f(mi)=−1∀i=1,2,...,nf(m_{i})=-1\forall i=1,2,...,n. b)f(x)f(x) is irreducible.
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