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Existence problem about quadratic trinomial

Source: 2019 Belarus Team Selection Test 2.1

September 2, 2019

quadraticsalgebrapolynomial

Problem Statement

Given a quadratic trinomial

p(x)

with integer coefficients such that

p(x)

is not divisible by

3

for all integers

x

. Prove that there exist polynomials

f(x)

and

h(x)

with integer coefficients such that

p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1.

(I. Gorodnin)

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