RMO–1998
1. Let ABCD be a convex quadrilateral in which angle BAC = 50 ^o, angle CAD = 60 ^o,
    angle CBD = 30 ^o,and angle BDC = 25^o . If E is the point of intersection of AC and
    BD, find angle AEB.

2. Let n be a positive integer and p₁, p₂,....p_n be n prime numbers all larger than 5 such
    that 6 divides p₁²+ p₂²+..p_n². Prove that 6 divides n.

3. Prove the following inequality for every natural number n:
    {1/(n + 1)}(1 +1/3+1/5+ ....1/(2n − 1)) >(1/n)(1/2+1/4+1/6+.......1/2n).
4. Let ABC be a triangle with AB = BC and angle BAC = 30^o . Let A' be the reflection
    of A in the line BC; B' be the reflection of B in the line CA; C' be the reflection of C in
    the line AB.Show that A', B', C' form the vertices of an equilateral triangle.

5. Find the minimum possible least common multiple of twenty (not necessarily distinct)
    natural numbers whose sum is 801.

6. Given the 7-element set A = {a, b, c, d, e, f, g}, find a collection T of 3-element subsets
    of A such that each pair of elements from A occurs exactly in one of the subsets of T.