Problem Set Workbook
Access the downloadable workbook for 2015 USAMO Day 2 problems here.
Discussion Forum
Engage in discussion about the 2015 USAMO Day 2 math contest by visiting [Random Math USAMO Day 2 2015 Forum](https://forums.randommath.com/c/tournaments/MAA/2015-usamo-day 2)
Individual Problems and Solutions
For problems and detailed solutions to each of the 2015 USAMO Day 2 problems, please refer below:
Problem 4: Steve is piling indistinguishable stones on the squares of an grid. Each square can have an arbitrarily high pile of stones. After he is finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions for some , such that and . A stone move consists of either removing one stone from each of and and moving them to and respectively, or removing one stone from each of and and moving them to and respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
How many different non-equivalent ways can Steve pile the stones on the grid?
Solution:
Problem 5: Let be distinct positive integers such that . Show that is a composite number.
Solution:
Problem 6: Consider , and let be a multiset of positive integers. Let . Assume that for every , the set contains at most numbers. Show that there are infinitely many for which the sum of the elements in is at most . (A multiset is a set-like collection of elements in which order is ignored, but repetition of elements is allowed and multiplicity of elements is significant. For example, multisets and are equivalent, but and differ.)
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions