Maths Olympiad Exam

Show that the equation
x^2 + y^2 + z^2 = (x - y)(y - z)(z - x)
has infinitely many solutions in integers x, y, z.

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Maths Olympiad Other Question

Let ABC be a triangle in which no angle is 90◦. For any point P in the plane
of the triangle, let A1,B1,C1 denote the reflections of P in the sides BC,CA,AB
respectively. Prove the following statements:
(a) If P is the incentre or an excentre of ABC, then P is the circumcentre of A1B1C1;
(b) If P is the circumcentre of ABC, then P is the orthocentre of A1B1C1;
(c) If P is the orthocentre of ABC, then P is either the incentre or an excentre of
A1B1C1. If a, b, c are positive real numbers such that abc = 1, prove that
ab+c bc+a ca+b ≤ 1.