[/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.
be an acute triangle. Let 
and 
be a point on side 
such that lines 
and 
are parallel. Let 
be an interior point of 
.
and 
meet side 
and 
,
and 
intersect at a point 
.
,
are collinear.
.
Y by Adventure10, Mango247, Mango247
Let 
be a trapezoid, where 
, 
, and 
. A circle 
is inscribed in 
and a circle 
is an excircle for 
which is tangent to 
(opposite to 
). Prove that the tangent line to through 
, different than 
, is parallel to the tangent line to through 
, different than 
.
be a trapezoid, where 
, 
, and 
. A circle 
is inscribed in 
and a circle 
is an excircle for 
which is tangent to 
(opposite to 
). Prove that the tangent line to through 
, different than 
, is parallel to the tangent line to through 
, different than 
.
be the incenter of 
,
be the excenter of 
;
and the tangent from 
at 
.
.
and 
,
.
is the reflection of 
,
is cyclic, which is true as 
,
,
,
,
- hence the angle between this tangent and 
.
,
.
is an angle bisector of 
,
,
.
from the angle bisector, the latter is equivalent to 
.
,
,
.
and 
,
,
