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 CZRorz, Adventure10, Mango247, lian_the_noob12
Let 
be an acute angled triangle whose shortest side is 
. Consider a variable point 
on the side , and let 
and 
be points on 
and 
, respectively, such that 
and 
. Prove that, as traces , the circumcircle of the triangle 
passes through a fixed point.
be an acute angled triangle whose shortest side is 
. Consider a variable point 
on the side , and let 
and 
be points on 
and 
, respectively, such that 
and 
. Prove that, as traces , the circumcircle of the triangle 
passes through a fixed point.
.
be the bisector of 
with 
on 
Clearly, 
is the circumcenter of 
are kites. 
bisects the 
respectively. 
is the incenter.
,
.
is isosceles and 
is 
angle bisector, then 
so 
and 
Similarly, 
and 
Hence, 
so 
Note that 
and 
Hence, 
.
so 
is cyclic so all 
goes through 
,
are concyclic
and 
are kite which implies 
.
is the 
a
.
which is fixed therefore by using 
.
wrt ![\[ f(\bullet)=P(\bullet, (ADE))-P(\bullet, (ABC)) \]](http://latex.artofproblemsolving.com/5/f/1/5f1b4debfedc233c68d96966a28a06b603fe139e.png)
It is known that 
is linear over the plane. We know 
.
is fixed. By the linearity we have
which is fixed.
The fixed point is 
.
, 
because 
. Likewise 
.
.
and 
.
the problem will end. 
because 
.
because it is the central angle 
