Problem

Source: Iranian TST 2019, third exam day 1, problem 1

Tags: algebra, polynomial



Find all polynomials $P(x,y)$ with real coefficients such that for all real numbers $x,y$ and $z$: $$P(x,2yz)+P(y,2zx)+P(z,2xy)=P(x+y+z,xy+yz+zx).$$ Proposed by Sina Saleh