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 
.
the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
,
at 
and 
,
and 
meet at 
and 
meet at 
Let 
and 
be the point of intersection of the lines 
and 
Prove that 
is the circle that is tangent to the line segment 
and 
are defined in the following manner:
then find the value of 
then find the value of 
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
Y by Adventure10
Let 
and 
be two circles intersecting at 
and 
. Let 
and 
be the centres of 
and , respectively. Let 
be a point on the segment 
such that 
and 
. We draw a line perpendicular to 
through and denote by 
and 
the points at which this line intersects . We likewise draw a line perpendicular to 
through and denote by 
and F the points at which this line intersects . Show that 
and 
are the vertices of a rectangle.
and 
be two circles intersecting at 
and 
. Let 
and 
be the centres of 
and , respectively. Let 
be a point on the segment 
such that 
and 
. We draw a line perpendicular to 
through and denote by 
and 
the points at which this line intersects . We likewise draw a line perpendicular to 
through and denote by 
and F the points at which this line intersects . Show that 
and 
are the vertices of a rectangle.
is a chord of the circle 
and 
,
.
are the vertices of a parallelogram.
.
