Problem Set Workbook
Access the downloadable workbook for 2013 USAJMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2013 USAJMO Day 1 math contest by visiting Random Math USAJMO Day 1 2013 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2013 USAJMO Day 1 problems, please refer below:
Problem 1: Are there integers and such that and are both perfect cubes of integers?
Solution:
Problem 2:Each cell of an board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent.) The filling is called a garden if it satisfies the following two conditions:
(1) The difference between any two adjacent numbers is either or .
(2) If a number is less than or equal to all of its adjacent numbers, then it is equal to . Determine the number of distinct gardens in terms of and .
Solution:
Problem 3: In triangle , points lie on sides , respectively. Let denote the circumcircles of triangles , respectively. Given the fact that segment intersects again at respectively, prove that .
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions