46th Polish Mathematical Olympiad Problems 1995

46th Polish Mathematical Olympiad Problems 1995

 
A1.  How many subsets of {1, 2, ... , 2n} do not contain two numbers with sum 2n+1?
A2.  The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area?

A3.  p ≥ 5 is prime. The sequence a₀, a₁, a₂, ... is defined by a₀ = 1, a₁ = 1, ... , a_p-1 = p-1 and a_n = a_n-1 + a_n-p for n ≥ p. Find a_p³ mod p.

B1.  The positive reals x₁, x₂, ... , x_n have harmonic mean 1. Find the smallest possible value of x₁ + x₂²/2 + x₃³/3 + ... + x_nⁿ/n.

B2.  An urn contains n balls labeled 1, 2, ... , n. We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by k. Find all k such that the expected number of balls removed is k.

B3.  PA, PB, PC are three rays in space. Show that there is just one pair of points B', C' with B' on the ray PB and C' on the ray PC such that PC' + B'C' = PA + AB' and PB' + B'C' = PA + AC'.