AoPS Academy
in the array can be changed into either 
or 
so that the row-sums and column-sums remain unchanged. (Note that is the least integer greater than or equal to , while is the greatest integer less than or equal to .)
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 , i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex 
a non-negative real number 
, for 
, such that 
. Bob then selects a sequence of distinct vertices 
such that 
and 
are connected by an edge for every 
. (Note that the length 
is not fixed and the first selected vertex always has to be 
.) Bob wins if![\[
\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;
\]](http://latex.artofproblemsolving.com/1/7/9/1795db8e4a509dd465f6ff462093ae75b04de2b4.png)
otherwise, Alice wins. Depending on 
, determine the largest possible value of 
for which Bobby has a winning strategy.
, 
, 
, 
, 
, 
be positive integers and let 
.
divides 
and 
. Prove that is composite.
the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
, and to the lines 
and 
at 
and 
, respectively. The lines 
and 
meet at 
, and the lines 
and 
meet at 
Let 
be the point of intersection of the lines 
and , and let 
be the point of intersection of the lines 
and 
Prove that is the midpoint of 
opposite the vertex 
is the circle that is tangent to the line segment , to the ray beyond 
, and to the ray beyond 
.)
and 
are defined in the following manner:
then 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:
where 
.
such that for every 
we have 
. is irrational then for every 
there exists 
such that![\[u<P(n)-\lfloor P(n)\rfloor <v.\]](http://latex.artofproblemsolving.com/9/4/a/94a8cb1f064e1dea37e8894fb4fd4e34c4c6c845.png)
Generalize the problem!
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 
