Indian National Mathematics Olympiad 2001 Problems

Indian National Mathematics Olympiad 2001 Problems

1.  ABC is a triangle which is not right-angled. P is a point in the plane. A', B', C' are the reflections of P in BC, CA, AB. Show that [incomplete].

2.  Show that a² + b² + c² = (a-b)(b-c)(c-a) has infinitely many integral solutions.

3.  a, b, c are positive reals with product 1. Show that a^b+cb^c+ac^a+b ≤ 1.

4.  Show that given any nine integers, we can find four, a, b, c, d such that a + b - c - d is divisible by 20. Show that this is not always true for eight integers.

5.  ABC is a triangle. M is the midpoint of BC. ∠MAB = ∠C, and ∠MAC = 15 ^o. Show that ∠AMC is obtuse. If O is the circumcenter of ADC, show that AOD is equilateral.

6.  Find all real-valued functions f on the reals such that f(x+y) = f(x) f(y) f(xy) for all x, y.