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 
.
Equality holds when 
or 
and 
,
cookies from one of the piles, of which he eats 
and puts 
for which Marica has a winning strategy.
Y by
Inside the acute-angled triangle 
, mark the point 
so that 
, a point 
on the side 
such that 
, and a point 
on the segment 
such that 
. Let 
be the point of intersection of the lines 
and 
, and let 
be a point on the side 
that such 
. Prove that the lines 
and 
intersect at one point.
, mark the point 
so that 
, a point 
on the side 
such that 
, and a point 
on the segment 
such that 
. Let 
be the point of intersection of the lines 
and 
, and let 
be a point on the side 
that such 
. Prove that the lines 
and 
intersect at one point.
