Find all polynomials with real coefficients such that $$xp(x) + yp(y)\geq2p(xy)$$ for all $x, y\in\mathbb{R}$.

My attempt:

I noticed the similarity between the given expression and the square of the sum $$(x-y)^2 = x^2 + y^2 - 2xy\geq0$$ So I guessed that $p$ would be of the form $$p(x)=ax$$ and I found that $a>0$. I tried to do $p(x)=ax\cdot q(x) + r$ to proof that $p(x)=ax$ would be the only solution, but I didn't finish this.

Any help would be appreciated!

Thanks for attention.



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