[/quote]
,
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.
be non-negative real numbers such that 
Prove that
be the circumcenter, symmedian point of 
.
at 
at 
,
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
and 
be a point on side 
such that lines 
are parallel. Let 
.
and 
meet side 
and 
,
and 
intersect at a point 
.
,
.
Y by
Given an acute triangle 
. Let 
be a point symmetric to point 
wrt 
. The lines 
and 
intersect the circumscribed circle 
of the triangle for the second time at points 
and 
respectively. Assume that the points and lie inside the segments and respectively. Prove that the center of the circumscribed circle of triangle 
lies on the circle .
(Danilo Hilko)
. Let 
be a point symmetric to point 
wrt 
. The lines 
and 
intersect the circumscribed circle 
of the triangle for the second time at points 
and 
respectively. Assume that the points and lie inside the segments and respectively. Prove that the center of the circumscribed circle of triangle 
lies on the circle .(Danilo Hilko)
