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2(P(d))^2 + 2P(ab + bc + ca) = (P(a + b + c))^2 , when a^2 + b^2 + c^2 = 2d^2

Source: Indian Postal Coaching 2009 set 2 p5

May 26, 2020
algebrapolynomial

Problem Statement

Find all real polynomials P(x)P(x) such that for every four distinct natural numbers a,b,c,da, b, c, d such that a2+b2+c2=2d2a^2 + b^2 + c^2 = 2d^2 with gcd(a,b,c,d)=1gcd(a, b, c, d) = 1 the following equality holds: 2(P(d))2+2P(ab+bc+ca)=(P(a+b+c))22(P(d))^2 + 2P(ab + bc + ca) = (P(a + b + c))^2 .
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