[/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 a point on side 
be a point on side 
are parallel. Let 
be an interior point of 
.
and 
meet side 
,
and 
intersect at a point 
.
,
are collinear.
.
Y by Abdollahpour, Adventure10, Mango247, and 2 other users
The incircle of triangle 
touches its sides 
, 
, 
at the points 
, 
, 
respectively. A segment 
intersects the incircle at the point 
. A line 
through 
is parallel to . Lines 
and 
intersect at the points 
and 
respectively. Prove that 
.
A. Polyansky
touches its sides 
, 
, 
at the points 
, 
, 
respectively. A segment 
intersects the incircle at the point 
. A line 
through 
is parallel to . Lines 
and 
intersect at the points 
and 
respectively. Prove that 
.A. Polyansky
so the quadrilateral 
is cyclic and therefore we get:
Q.E.D.
be the antipode of 
and 
are collinear.
then 
are collinear (by considering the radical axes of the following circles : incircle, Euler's circle of 
and the circumcircle 
where 
).
is cyclic and thus: ![\[\angle{PQR}=\angle{PTR}=\angle{B_{1}QC_{1}}.\]](http://latex.artofproblemsolving.com/7/7/c/77c56496eeeb7fdc4e12faa6a6229107a67003ed.png)
because 
is cyclic.
,
,
is cyclic, and similar the quadrilateral 
is cyclic, so:
,
the midpoints of 
respectively. Prove that 
lie on the same circle.
lie on circle but how can we proof 
all lie on the nine point circle of 
be the reflection of 
over 
is cyclic. Also, 
and, 
is median and 
h
is cyclic 
.
concylic.Hence 
.
are cyclic and therefore 
onto 
.
,
,
is the foot of the altitude from 
.
and so 
.
to 
respectively.
,
is cyclic. Likewise, 
is cyclic. Then
which finishes.
