BALTIC-WAY 2007 Problem 3

Suppose that $F,G,H$ are polynomials of degree at most $2n+1$ with real coefficients such that: i) For all real $x$ we have $F(x)\le G(x)\le H(x)$ . ii) There exist distinct real numbers $x_1,x_2,\ldots ,x_n$ such that $F(x_i)=H(x_i) \text{for}\ i=1,2,3,\ldots ,n$ . iii) There exists a real number $x_0$ different from $x_1,x_2,\ldots ,x_n$ such that $F(x_0)+H(x_0)=2G(x_0)$ . Prove that $F(x)+H(x)=2G(x)$ for all real numbers $x$ .