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original polynomial

Source: 2023 Viet Nam math olympiad for high school students D2 P2

March 25, 2023
algebra

Problem Statement

a) Given a prime number pp and 22 polynomialsP(x)=anxn+...+a1x+a0;Q(x)=bmxm+...+b1x+b0.P(x)=a_nx^n+...+a_1x+a_0; Q(x)=b_mx^m+...+b_1x+b_0. We know that the product P(x)Q(x)P(x)Q(x) is a polynomial whose coefficents are all divisible by p.p. Prove that: at least 11 in 22 polynomials P(x),Q(x)P(x),Q(x) has all coefficents are all divisible by p.p.
b) Prove that the product of 22 original polynomials is a original polynomial.
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