Problem Set Workbook
Access the downloadable workbook for 2015 USAJMO Day 1 problems here.
Discussion Forum
Engage in discussion about the 2015 USAJMO Day 1 math contest by visiting Random Math USAJMO Day 1 2015 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2015 USAJMO Day 1 problems, please refer below:
Problem 1: Given a sequence of real numbers, a move consists of choosing two terms and replacing each by their arithmetic mean. Show that there exists a sequence of 2015 distinct real numbers such that after one initial move is applied to the sequence - no matter what move - there is always a way to continue with a finite sequence of moves so as to obtain in the end a constant sequence.
Solution:
Problem 2: Solve in integers the equation
Solution:
Problem 3: Quadrilateral is inscribed in circle with and . Let be a variable point on segment . Line meets again at (other than ). Point lies on arc of such that is perpendicular to . Let denote the midpoint of chord . As varies on segment , show that moves along a circle.
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Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions