28th USAMO 1999


A1. Certain squares of an n x n board are colored black and the rest white. Every white square shares a side with a black square. Every pair of black squares can be joined by chain of black squares, so that consecutive members of the chain share a side. Show that there are at least (n² - 2)/3 black squares.
A2. For each pair of opposite sides of a cyclic quadrilateral take the larger length less the smaller length. Show that the sum of the two resulting differences is at least twice the difference in length of the diagonals.
A3. p is an odd prime. The integers a, b, c, d are not multiples of p and for any integer n not a multiple of p, we have {na/p} + {nb/p} + {nc/p} + {nd/p} = 2, where { } denotes the fractional part. Show that we can find at least two pairs from a, b, c, d whose sum is divisible by p.
B1. A set of n > 3 real numbers has sum at least n and the sum of the squares of the numbers is at least n². Show that the largest positive number is at least 2.
B2. Two players play a game on a line of 2000 squares. Each player in turn puts either S or O into an empty square. The game stops when three adjacent squares contain S, O, S in that order and the last player wins. If all the squares are filled without getting S, O, S, then the game is drawn. Show that the second player can always win.
B3. I is the incenter of the triangle ABC. The point D outside the triangle is such DA is parallel to BC and DB = AC, but ABCD is not a parallelogram. The angle bisector of BDC meets the line through I perpendicular to BC at X. The circumcircle of CDX meets the line BC again at Y. Show that DXY is isosceles.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.