45th Polish Mathematical Olympiad Problems 1994

45th Polish Mathematical Olympiad Problems 1994

A1.  Find all triples (x,y,z) of positive rationals such that x + y + z, 1/x + 1/y + 1/z and xyz are all integers.

A2.  L, L' are parallel lines. C is a circle that does not intersect L. A is a variable point on L. The two tangents to C from A meet L' in two points with midpoint M. Show that the line AM passes through a fixed point (as A varies).

A3.  k is a fixed positive integer. Let a_n be the number of maps f from the subsets of {1, 2, ... , n} to {1, 2, ... , k} such that for all subsets A, B of {1, 2, ... , n} we have f(A ∩ B) = min(f(A), f(B)). Find lim_n→∞ a_n^1/n.

B1.  m, n are relatively prime. We have three jugs which contain m, n and m+n liters. Initially the largest jug is full of water. Show that for any k in {1, 2, ... , m+n} we can get exactly k liters into one of the jugs.

B2.  A parallelepiped has vertices A₁, A₂, ... , A₈ and center O. Show that 4 ∑ |OA_i|² ≤ (∑|OA_i|)².

B3.  The distinct reals x₁, x₂, ... , x_n (n > 3) satisfy ∑ x_i = 0, &sum x_i² = 1. Show that four of the numbers a, b, c, d must satisfy a + b + c + nabc ≤ ∑ x_i³ ≤ a + b + d + nabd.