34th Polish Mathematical Olympiad Problems 1983

34th Polish Mathematical Olympiad Problems 1983

A1.  The angle bisectors of the angles A, B, C in the triangle ABC meet the circumcircle again at K, L, M. Show that |AK| + |BL| + |CM| > |AB| + |BC| + |CA|.

A2.  For given n, we choose k and m at random subject to 0 ≤ k ≤ m ≤ 2ⁿ. Let p_n be the probability that the binomial coefficient mCk is even. Find lim_n→∞ p_n.

A3.  Q is a point inside the n-gon P₁P₂...P_n which does not lie on any of the diagonals. Show that if n is even, then Q must lie inside an even number of triangles P_iP_jP_k.

B1.  Given a real numbers x ∈ (0,1) and a positive integer N, prove that there exist positive integers a, b, c, d such that (1) a/b < x < c/d, (2) c/d - a/b < 1/n, and (3) qr - ps = 1.

B2.  There is a piece in each square of an m x n rectangle on an infinite chessboard. An allowed move is to remove two pieces which are adjacent horizontally or vertically and to place a piece in an empty square adjacent to the two removed and in line with them (as shown below)

X X . to . . X, or  . to X
                    X    .
                    X    .

Show that if mn is a multiple of 3, then it is not possible to end up with only one piece after a sequence of moves.

B3.  Show that if the positive integers a, b, c, d satisfy ab = cd, then we have gcd(a,c) gcd(a,d) = a gcd(a,b,c,d).