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Bertrand's Theorem in Polynomials!!

Source: Iran 3rd round 2013 - Algebra Exam - Problem 1

September 11, 2013
algebrapolynomialalgebra proposed

Problem Statement

Let a0,a1,…,an∈Na_0,a_1,\dots,a_n \in \mathbb N. Prove that there exist positive integers b0,b1,…,bnb_0,b_1,\dots,b_n such that for 0≤i≤n:ai≤bi≤2ai0 \leq i \leq n : a_i \leq b_i \leq 2a_i and polynomial P(x)=b0+b1x+⋯+bnxnP(x) = b_0 + b_1 x + \dots + b_n x^n is irreducible over Q[x]\mathbb Q[x]. (10 points)
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