[/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 
be a point on side 
and 
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
The 
is a cyclic quadrilateral with no parallel sides inscribed in circle 
. Let 
be the intersection of two diagonals and the angle bisector of 
cut the lines 
at 
respectively .
a) Prove that the circles
are passing through a point. Call that point 
.
b) Denote
. Prove that 
.
is a cyclic quadrilateral with no parallel sides inscribed in circle 
. Let 
be the intersection of two diagonals and the angle bisector of 
cut the lines 
at 
respectively .a) Prove that the circles
are passing through a point. Call that point 
.b) Denote
. Prove that 
.Attachments:
;
)
;
)
;
)
)
;
)
;
)
;
)
)
;
)
;
;
)
;
)
= 
+ 1 = 
+ 1 = 
+ 1 = 
+ 1 = 
;
)
)
;
)
)
)
)
)
)
. 
= 
. 
= 
= 
= 
= 
(With 
is distance from point 
)
is diameter of 
;
)
and 
.
.
Let 
is cyclic 
Hence Done!
