Problem Set Workbook
Access the downloadable workbook for 2018 USAJMO Day 2 problems here.
Discussion Forum
Engage in discussion about the 2018 USAJMO Day 2 math contest by visiting Random Math USAJMO Day 2 2018 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2018 USAJMO Day 2 problems, please refer below:
Problem 4: Triangle is inscribed in a circle of radius 2 with , and is a real number satisfying the equation , where . Find all possible values of .
Solution:
Problem 5: Let be a prime, and let be integers. Show that there exists an integer such that the numbers
produce at least distinct remainders upon division by .
Solution:
Problem 6: Karl starts with cards labeled lined up in a random order on his desk. He calls a pair of these cards swapped if and the card labeled is to the left of the card labeled . For instance, in the sequence of cards , there are three swapped pairs of cards, , and .
He picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had cards to its left, then it now has cards to its right. He then picks up the card labeled 2 and reinserts it in the same manner, and so on until he has picked up and put back each of the cards exactly once in that order. (For example, the process starting at would be .)
Show that, no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions