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of positive integers such that 
is an integer.
be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through 
to 
such that 
. Define 
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
be an acute triangle with 
, its orthocenter and 
it's circumcenter. Let be a point on the parallel through to such that 
and and are on different sides of . Denote by 
the intersection of the circumcircle of and 
, where 
is the reflection of over , 
the midpoint of 
, the intersection of 
and the parallel through to 
, the intersection of 
and the perpendicular through to 
and 
a point on such that 
, where 
is the midpoint of 
. Prove that 
lie on a line.
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
, where 
. Determine the measure of the acute angle formed by the lines of edges 
and .
