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Cubic with Horizontal Chord

Source: 1988 IrMO Paper 1 Problem 7

September 28, 2017
algebra

Problem Statement

A function ff, defined on the set of real numbers R\mathbb{R} is said to have a horizontal chord of length a>0a>0 if there is a real number xx such that f(a+x)=f(x)f(a+x)=f(x). Show that the cubic f(x)=x^3-x        (x\in \mathbb{R}) has a horizontal chord of length aa if, and only if, 0<a≤20<a\le 2.
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