Difference between revisions of "2012 USAJMO Problems"
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[[2012 USAJMO Problems/Problem 6|Solution]] | [[2012 USAJMO Problems/Problem 6|Solution]] | ||
− | == See | + | == See Also == |
− | {{USAJMO box|year=2012|before=[[2011 USAJMO Problems | + | *[[USAJMO Problems and Solutions]] |
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+ | {{USAJMO box|year=2012|before=[[2011 USAJMO Problems]]|after=[[2013 USAJMO Problems]}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 15:41, 5 August 2023
Contents
[hide]Day 1
Problem 1
Given a triangle , let and be points on segments and , respectively, such that . Let and be distinct points on segment such that lies between and , , and . Prove that , , , are concyclic (in other words, these four points lie on a circle).
Problem 2
Find all integers such that among any positive real numbers , , , with there exist three that are the side lengths of an acute triangle.
Problem 3
Let , , be positive real numbers. Prove that
Day 2
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 .
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 .
Problem 6
Let be a point in the plane of triangle , and a line passing through . Let , , be the points where the reflections of lines , , with respect to intersect lines , , , respectively. Prove that , , are collinear.
See Also
{{USAJMO box|year=2012|before=2011 USAJMO Problems|after=[[2013 USAJMO Problems]}} The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.