41st Polish Mathematical Olympiad Problems 1990

41st Polish Mathematical Olympiad Problems 1990

A1.  Find all real-valued functions f on the reals such that (x-y)f(x+y) - (x+y)f(x-y) = 4xy(x²-y²) for all x, y.

A2.  For n > 1 and positive reals x1, x2, ... , xn, show that x₁²/(x₁²+x²x₃) + x₂²/(x₂²+x³x₄) + ... + x_n²/(x_n²+x¹x₂) ≤ n-1.

A3.  In a tournament there are n players. Each pair of players play each other just once. There are no draws. Show that either (1) one can divide the players into two groups A and B, such that every player in A beat every player in B, or (2) we can label the players P₁, P₂, ... , P_n such that P_i beat P_i+1 for i = 1, 2, ... n (where we use cyclic subscripts, so that P_n+1 means P₁).

B1.  A triangle with each side length at least 1 lies inside a square side 1. Show that the center of the square lies inside the triangle.

B2.  a₁, a₂, a₃, ... is a sequence of positive integers such that lim_n→∞ n/a_n = 0. Show that we can find k such that there are at least 1990 squares between a₁ + a₂ + ... + a_k and a₁ + a₂ + ... + a_k+1.

B3.  Show that ∑_k=0^[n/3] (-1)^k nC3k is a multiple of 3 for n > 2. (nCm is the binomial coefficient)