Problem Set Workbook
Access the downloadable workbook for 2005 USAMO Day 2 problems here.
Discussion Forum
Engage in discussion about the 2005 USAMO Day 2 math contest by visiting [Random Math USAMO Day 2 2005 Forum](https://forums.randommath.com/c/tournaments/MAA/2005-usamo-day 2)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2005 USAMO Day 2 problems, please refer below:
Problem 1: Legs of a square table each have length , where is a positive integer. For how many ordered 4-tuples of nonnegative integers can we cut a piece of length from the end of leg and still have a stable table? (The table is stable if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)
Solution:
Problem 2: Let be an integer greater than 1. Suppose points are given in the plane, no three of which are collinear. Suppose of the given points are colored blue and the other colored red. A line in the plane is called a balancing line if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side. Prove that there exist at least two balancing lines.
Solution:
Problem 3: For a positive integer, let be the sum of the digits of . For , let be the minimal for which there exists a set of positive integers such that for any nonempty subset . Prove that there are constants with
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions