Problem Set Workbook
Access the downloadable workbook for 2010 USAJMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2010 USAJMO Day 1 math contest by visiting Random Math USAJMO Day 1 2010 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2010 USAJMO Day 1 problems, please refer below:
Problem 1: A permutation of the set of positive integers is a sequence such that each element of appears precisely one time as a term of the sequence. For example, is a permutation of [5]. Let be the number of permutations of for which is a perfect square for all . Find with proof the smallest such that is a multiple of 2010 .
Solution:
Problem 2: Let be an integer. Find, with proof, all sequences of positive integers with the following three properties:
(1) ;
(2) for all ;
(3) given any two indices and (not necessarily distinct) for which , there is an index such that .
Solution:
Problem 3: Let be a convex pentagon inscribed in a semicircle of diameter . Denote by the feet of the perpendiculars from onto lines , respectively. Prove that the acute angle formed by lines and is half the size of , where is the midpoint of segment .
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions