39th Polish Mathematical Olympiad Problems 1988

39th Polish Mathematical Olympiad Problems 1988

A1.  The real numbers x₁, x₂, ... , x_n belong to the interval (0,1) and satisfy x₁ + x₂ + ... + x_n = m + r, where m is an integer and r ∈ [0,1). Show that x₁² + x₂² + ... + x_n² ≤ m + r².

A2.  For a permutation P = (p₁, p₂, ... , p_n) of (1, 2, ... , n) define X(P) as the number of j such that p_i < p_j for every i < j. What is the expected value of X(P) if each permutation is equally likely?

A3.  W is a polygon. W has a center of symmetry S such that if P belongs to W, then so does P', where S is the midpoint of PP'. Show that there is a parallelogram V containing W such that the midpoint of each side of V lies on the border of W.

B1.  d is a positive integer and f : [0,d] → R is a continuous function with f(0) = f(d). Show that there exists x ∈ [0,d-1] such that f(x) = f(x+1).

B2.  The sequence a₁, a₂, a₃, ... is defined by a₁ = a₂ = a₃ = 1, a_n+3 = a_n+2a_n+1 + a_n. Show that for any positive integer r we can find s such that a_s is a multiple of r.

B3.  Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius 1.