[/quote]
,
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of 
be real numbers . Prove that : There exist positive integer 
such that 
Where 
be an isosceles triangle 
)
.
passes through 
and 
.
and circumcircle of 
and 
,
be the midpoint of 
be the midpoint of 
.
,
and 
are concurrent.
,
in the array can be changed into either 
or 
so that the row-sums and column-sums remain unchanged. (Note that 
be 
given points in the plane, and let 
be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points 
a non-negative real number 
,
,
.
such that 
and 
are connected by an edge for every 
.
is not fixed and the first selected vertex always has to be 
.![\[
\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;
\]](http://latex.artofproblemsolving.com/1/7/9/1795db8e4a509dd465f6ff462093ae75b04de2b4.png)
otherwise, Alice wins. Depending on 
,
for which Bobby has a winning strategy.
,
,
,
,
,
be positive integers and let 
.
divides 
and 
.
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
,
and 
meet at 
and 
meet at 
Let 
be the point of intersection of the lines 
and 
be the point of intersection of the lines 
and 
Prove that 
is the circle that is tangent to the line segment 
,
be positive reals such that 
.
Y by Adventure10, Mango247, Rounak_iitr, and 1 other user
Let 
be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at 
. The midpoints of 
and 
are 
and 
respectively, and 
is the line through parallel to . The feet of the perpendiculars from E onto the lines and are 
and 
, respectively. Prove that the lines 
and 
are perpendicular.
be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at 
. The midpoints of 
and 
are 
and 
respectively, and 
is the line through parallel to . The feet of the perpendiculars from E onto the lines and are 
and 
, respectively. Prove that the lines 
and 
are perpendicular.
WRT 
it follows that 
is the E-circumdiameter of 
Let 
cuts 
From the cyclic quadrilateral 
(due to its right angles <EKG and <EHG) we deduce that 
Since 
are isogonals WRT 
then 
are also isogonals WRT 
From 
we have then 
are collinear, i.e. 
.
.
.
.
and 
are similar and the angles 
are these that are formed by the median and the height to these similar triangles so they are equal and we are done.
,
.
,
,
.
The condition can be written as 
,
are cocyclic, so 
.
,
so 
Denote 
.
Denote 
,
,
,
,
(
.
.
.
.
,
.
THen 
is cyclic.
and 
,
as desired.
perpendicular to 
by Similarity
is parallel to 
so 
!
wrt 
and 
and 
and it is so easy that we can proof by 
so 
and 
problem proof! 
bring the symmedian of triangle 
.
and 
![\[\frac{sin \ \alpha}{sin \ \beta}=\frac{sin \ \lambda}{sin \ (\alpha+\beta+\lambda)} \ (\star)\]](http://latex.artofproblemsolving.com/3/1/b/31b500f686afdc58684e270964507a6d173a2ea7.png)
and similarly in triangle ![\[\frac{sin \ \alpha}{sin \ \beta}=\frac{sin \ \phi}{sin \ (\alpha +\beta + \phi)} \ (\star\star)\]](http://latex.artofproblemsolving.com/7/d/d/7dd993247d44cfcb55a1270e4a9003c53961a214.png)
Combinig 
and 
we get ![\[\frac{sin \ \phi}{sin \ (\alpha+\beta+\phi)} = \frac{sin \ \lambda}{sin \ (\alpha +\beta + \lambda)}\]](http://latex.artofproblemsolving.com/4/e/f/4ef1a24f91b539ee6f54faa8ca113628880ae217.png)
Solving the equation we get 
(Which is impossible) then 
By angle Chasing we get 
Where 
and since 
which follows 
and let 
and 
we have 
Since 
we get 
which means that 
be the intersection of 
and let 
be the intersection of 
with 
since the quadrilateral 
and, 
(since, the triangles 
and 
are simillar). hence, the triangles 
and 
are simillar and, 
now, we see that the quadrilateral 
is Cyclic since, 
therefore, 
and thus, the quadrilateral 
is Cyclic. hence, 
since, 
.
