TUYMAADA 2009 S8

Determine the maximum number $h$ satisfying the following condition: for every $a\in [0,h]$ and every polynomial $P(x)$ of degree 99 such that $P(0)=P(1)=0$ , there exist $x_1,x_2\in [0,1]$ such that $P(x_1)=P(x_2)$ and $x_2-x_1=a$ .

Proposed by F. Petrov, D. Rostovsky, A. Khrabrov