53rd Polish Mathematical Olympiad Problems 2002

53rd Polish Mathematical Olympiad Problems 2002

A1.  Find all triples of positive integers (a, b, c) such that a² + 1 and b² + 1 are prime and (a² + 1)(b² + 1) = c² + 1.

A2.  ABC is an acute-angled triangle. BCKL, ACPQ are rectangles on the outside of two of the sides and have equal area. Show that the midpoint of PK lies on the line through C and the circumcenter.

A3.  Three non-negative integers are written on a blackboard. A move is to replace two of the integers by their sum and (non-negative) difference. Can we always get two zeros by a sequence of moves?

B1.  Given any finite sequence x1, x2, ... , xn of at least 3 positive integers, show that either ∑₁ⁿ x_i/(x_i+1 + x_i+2) ≥ n/2 or ∑ ₁ⁿ x_i/(x_i-1 + x_i-2) ≥ n/2. (We use the cyclic subscript convention, so that x_n+1 means x₁ and x_-1 means x_n-1 etc).

B2.  ABC is a triangle. A sphere does not intersect the plane of ABC. There are 4 points K, L, M, P on the sphere such that AK, BL, CM are tangent to the sphere and AK/AP = BL/BP = CM/CP. Show that the sphere touches the circumsphere of ABCP.

B3.  k is a positive integer. The sequence a₁, a₂, a₃, ... is defined by a₁ = k+1, a_n+1 = a_n² - ka_n + k. Show that a_m and a_n are coprime (for m ≠ n).