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
Let 
be an acute-angled triangle with 
and let 
be the foot of its altitude from
points 
and 
lie on the rays 
and 
respectively , so that points 
and are collinear and points 
and lie on one circle with center 
Prove that if 
is the midpoint of 
and 
is the orthocenter of 
then 
is a parallelogram.
be an acute-angled triangle with 
and let 
be the foot of its altitude from
points 
and 
lie on the rays 
and 
respectively , so that points 
and are collinear and points 
and lie on one circle with center 
Prove that if 
is the midpoint of 
and 
is the orthocenter of 
then 
is a parallelogram.This post has been edited 1 time. Last edited by sqing, Sep 3, 2018, 12:52 AM
cut 
at 
.
hence
or 
.
in the middle perpendicular to 
is a parallelogram.Suppose that a circle with center 
and radius 
intersects 
at 
and 
.
be the reflections of 
onto 
.
is on the circumcircle of 
.
which is easily seen to be equivalent to the claim.
