MathDB

3

Part of 2007 Baltic Way

Problems(1)

3 conditions on the polynomials F,G,H

Source: Baltic Way 2007

11/30/2010
Suppose that F,G,HF,G,H are polynomials of degree at most 2n+12n+1 with real coefficients such that: i) For all real xx we have F(x)≤G(x)≤H(x)F(x)\le G(x)\le H(x). ii) There exist distinct real numbers x1,x2,…,xnx_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 x0x_0 different from x1,x2,…,xnx_1,x_2,\ldots ,x_n such that F(x0)+H(x0)=2G(x0)F(x_0)+H(x_0)=2G(x_0). Prove that F(x)+H(x)=2G(x)F(x)+H(x)=2G(x) for all real numbers xx.
algebrapolynomialalgebra proposed