Maths Olympiad Exam

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.

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Maths Olympiad Other Question

If a, b, c are positive real numbers such that abc = 1, prove that
ab+c bc+a ca+b ≤ 1. Let ABC be a triangle and D be the mid-point of side BC. Suppose DAB = BCA
and DAC = 15◦. Show that ADC is obtuse. Further, if O is the circumcentre of
ADC, prove that triangle AOD is equilateral.