Problem Set Workbook
Access the downloadable workbook for 2007 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2007 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2007 Forum](https://forums.randommath.com/c/tournaments/MAA/2007-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2007 USAMO Day 1 problems, please refer below:
Problem 1: Let be a positive integer. Define a sequence by setting and, for each , letting be the unique integer in the range for which is divisible by . For instance, when the obtained sequence is . Prove that for any the sequence eventually becomes constant.
Solution:
Problem 2: A square grid on the Euclidean plane consists of all points , where and are integers. Is it possible to cover all grid points by an infinite family of discs with nonoverlapping interiors if each disc in the family has radius at least ?
Solution:
Problem 3: Let be a set containing elements, for some positive integer . Suppose that the -element subsets of are partitioned into two classes. Prove that there are at least pairwise disjoint sets in the same class.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions