RMO–1998

1. Let ABCD be a convex quadrilateral in which ∠BAC = 50⁰, ∠CAD = 60⁰, ∠CBD = 30⁰, and ∠BDC = 25⁰. If E is the point of intersection of AC and BD, find ∠AEB.

2. Let n be a positive integer and p¹, p², ·pⁿ be n prime numbers all larger than 5 such that 6 divides p²₁ + p²₂ + ·p²_n
. Prove that 6 divides n.

3. Prove the following inequality for every natural number

4. Let ABC be a triangle with AB = BC and ∠ BAC = 30⁰. 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 (lcm) 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.

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