USA Team Selection Test 2014

1 Let be an acute triangle, and let be a variable interior point on the minor arc of its circumcircle. Let and be the feet of the perpendiculars from to lines and , respectively. Let be the intersection of line and the perpendicular from to . Let be the line through parallel to . Prove that as varies along minor arc , the line always passes through a fixed point. (Specifically: prove that there is a point , determined by triangle , such that no matter where is on arc , line passes through .)

Robert Simson et al.

 2 Let be a sequence of integers, with the property that every consecutive group of 's averages to a perfect square. More precisely, for every positive integers and , the quantity

is always the square of an integer. Prove that the sequence must be constant (all are equal to the same perfect square).

Evan O'Dorney and Victor Wang 

3 Let be an even positive integer, and let be an -vertex graph with exactly edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices is said to be amicable if they have a common neighbor (there is a vertex such that and are both edges). Prove that has at least pairs of vertices which are amicable.

Po-Shen Loh

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