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).
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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