Problem Set Workbook
Access the downloadable workbook for 2008 USAMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2008 USAMO Day 1 math contest by visiting [Random Math USAMO Day 1 2008 Forum](https://forums.randommath.com/c/tournaments/MAA/2008-usamo-day 1)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2008 USAMO Day 1 problems, please refer below:
Problem 1: Prove that for each positive integer , there are pairwise relatively prime integers , , all strictly greater than 1 , such that is the product of two consecutive integers.
Solution:
Problem 2: Let be an acute, scalene triangle, and let , and be the midpoints of , and , respectively. Let the perpendicular bisectors of and intersect ray in points and respectively, and let lines and intersect in point , inside of triangle . Prove that points , and all lie on one circle.
Solution:
Problem 3: Let be a positive integer. Denote by the set of points with integer coordinates such that
A path is a sequence of distinct points in such that, for
, the distance between and is 1 (in other words, the points and are neighbors in the lattice of points with integer coordinates).
Prove that the points in cannot be partitioned into fewer than paths (a partition of into paths is a set of nonempty paths such that each point in appears in exactly one of the paths in ).
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions