28th Spanish Mathematical Olympiad Problems 1992

A1.  Find the smallest positive integer N which is a multiple of 83 and is such that N² has exactly 63 positive divisors.

A2.  Given two circles (neither inside the other) with different radii, a line L, and k > 0, show how to construct a line L' parallel to L so that L intersects the two circles in chords with total length k.

A3.  a, b, c, d are positive integers such that (a+b)² + 2a + b = (c+d)² + 2c + d. Show that a = c and b = d. Show that the same is true if a, b, c, d satisfy (a+b)² + 3a + b = (c+d)² + 3c + d. But show that there exist a, b, c, d such that (a+b)² + 4a + b = (c+d)² + 4c + d, but a ≠ c and b ≠ d.

B1.  Show that there are infinitely many primes in the arithmetic progression 3, 7, 11, 15, ... .

B2.  Given the triangle ABC, show how to find geometrically the point P such that ∠PAB = ∠PBC = ∠PCA. Express this angle in terms of ∠A, ∠B, ∠C using trigonometric functions.

B3.  For each positive integer n let S(n) be the set of complex numbers z such that |z| = 1 and (z + 1/z)ⁿ = 2^n-1(zⁿ + 1/zⁿ). Find S(2), S(3), S(4). Find an upper bound for |S(n)| for n ≥ 5.