AoPS Academy
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)
and 
. Prove that
Equality holds when 
or 
and 
, Ivica and Marica play the following game: Given two piles of cookies, on each turn a player takes 
cookies from one of the piles, of which he eats 
and puts of is arbitrary in every move. Players take turns alternatively, with Ivica going
for which Marica has a winning strategy.
be an acute triangle and let 
be its orthocenter. Let 
be a segment through with 
on side 
, 
on side 
and such that 
. Finally, let 
be the intersection between the segment 
and the bisector of the angle 
. Prove that 
.
be an acute triangle and let be its orthocenter. Let be a segment through with on side , on side and such that . Finally, let be the intersection between the segment and the bisector of the angle . Prove that .
means 
(since 
). Hence, 
. Also, 
is the angle bisector of 
. Hence, is the perpendicular bisector of ![$[PQ]$](http://latex.artofproblemsolving.com/2/1/c/21ca08816cf8b23ddf756ce9ae098ad327f2443d.png)
. Thus, 
, as desired.
