VMO 1982 Problem 5

n is a positive integer. x and y are reals such that 0 ≤ x ≤ 1 and x $^{n+1}$ ≤ y ≤ 1. Show that the absolute value of (y - x)(y - $x^2$)(y - $x^3$) ... (y - $x^n$ )(1 + x)(1 + $x^2$) ... (1 + $x^n$ ) is at most (y + x)(y + $x^2$) ... (y + $x^n$ )(1 - x)(1 - $x^2$) ... (1 - $x^n$ ).