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 Adventure10, Mango247
Circle 
with center 
is touched from inside by two circles in points 
and 
respectively. Let those two circles intersect at points 
and 
, such that is closer to line 
. Prove that 
and 
are perpendicular iff , and 
are collinear
with center 
is touched from inside by two circles in points 
and 
respectively. Let those two circles intersect at points 
and 
, such that is closer to line 
. Prove that 
and 
are perpendicular iff , and 
are collinear
