Maths Olympiad Exam

Let ABC be a triangle with circum circle. Let M be a point in the interior of triangle ABC which is also on the bisector of A. Let AM, BM, CM meet in A1, B1, C1 respectively. Suppose P is the point of intersection of A1C1 with AB; and Q is the point of intersection of A1B1 with AC. Prove that PQ is parallel to BC.

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Maths Olympiad Other Question
Let ABC be a triangle in which AB = AC and CAB = 90◦. Suppose M and
N are points on the hypotenuse BC such that BM2 + CN2 = MN2. Prove that
MAN = 45◦ Find all natural numbers n > 1 such that n2 does not divide (n - 2)!.