Maths Olympiad Exam

If a, b, c are positive real numbers such that abc = 1, prove that
ab+c bc+a ca+b ≤ 1.

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Maths Olympiad Other Question

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. Given any nine integers show that it is possible to choose, from among them, four
integers a, b, c, d such that a + b − c − d is divisible by 20. Further show that such a
selection is not possible if we start with eight integers instead of nine.