Problem Set Workbook
Access the downloadable workbook for 2012 USAJMO Day 2 problems here.
Discussion Forum
Engage in discussion about the 2012 USAJMO Day 2 math contest by visiting Random Math USAJMO Day 2 2012 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2012 USAJMO Day 2 problems, please refer below:
Problem 4:Let be an irrational number with , and draw a circle in the plane whose circumference has length 1 . Given any integer , define a sequence of points , as follows. First select any point on the circle, and for define as the point on the circle for which the length of arc is , when travelling counterclockwise around the circle from to . Suppose that and are the nearest adjacent points on either side of . Prove that .
Solution:
Problem 5: For distinct positive integers , define to be the number of integers with such that the remainder when divided by 2012 is greater than that of divided by 2012. Let be the minimum value of , where and range over all pairs of distinct positive integers less than 2012. Determine .
Solution:
Problem 6: Let be a point in the plane of , and a line passing through . Let be the points where the reflections of lines with respect to intersect lines , respectively. Prove that are collinear.
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Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions