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Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree n with real coefficients can be written as the average of two monic polynomials of degree n with n real roots.

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This is USAMO 2002 problem 3.

The result is trivial for $n=1$, so assume $n>1$. Let $g(x)=(x-2)(x-4)...(x-2n+2)$. We claim that $q(x)=x^n-kg(x),r(x)=2p(x)-q(x)$ works for sufficiently large $k$.

The values $g(1),g(3), ... , g(2n-1)$ alternate in sign and have magnitude at least 1. Take $k$ so that $x^n<k/3$ and $|p(x)-x^n|<k/3$ for all $1\le x\le 2n-1$.

Then if follows that $q(1),q(3),...,q(2n-1)$ also alternate in sign, and similarly $r(1),r(3),...,r(2n-1)$ alternate in sign. So $q(x),r(x)$ each have at least $n-1$ real roots. But polynomials with real coefficients must have an even number of non-real roots, so $q(x),r(x)$ each have $n$ real roots.

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