30th USAMO 2001


A1. What is the smallest number of colors needed to color 8 boxes of 6 balls (one color for each ball), so that the balls in each box are all different colors and any pair of colors occurs in at most one box.
A2. The incircle of the triangle PBC touches BC at U and PC at V. The point S on BC is such that BS = CU. PS meets the incircle at two points. The nearer to P is Q. Take W on PC such that PW = CV. Let BW and PS meet at R. Show that PQ = RS.
A3. Non-negative reals x, y, z satisfy x² + y² + z² + xyz = 4. Show that xyz ≤ xy + yz + zx <= xyz + 2.
B1. ABC is a triangle and X is a point in the same plane. The three lengths XA, XB, XC can be used to form an obtuse-angled triangle. Show that if XA is the longest length, then ∠BAC is acute.
B2. A set of integers is such that if a and b belong to it, then so do a² - a, and a² - b. Also, there are two members a, b whose greatest common divisor is 1 and such that a - 2 and b - 2 also have greatest common divisor 1. Show that the set contains all the integers.
B3. Every point in the plane is assigned a real number, so that for any three points which are not collinear, the number assigned to the incenter is the mean of the numbers assigned to the three points. Show that the same number is assigned to every point.

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