AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be real numbers . Prove that : There exist positive integer 
such that 
Where 
, for which
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 
.)
be positive reals such that 
. Find the minimum and maximum value of
, and 
are all polynomials such that ![\[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\]](http://latex.artofproblemsolving.com/1/6/a/16a71abc110ece558d427a07d7be2d1f72148024.png)
prove that 
is a factor of 
.
, let 
denote the sum of squares of digits of . Prove that the sequence 
is eventually periodic.
we have that 
. But when 
, then 
, so from some point all elements in this sequence will be smaller than 
. For each 
sequence will be unique, which means that if some value in our sequence repeat, then whole sequence will have to start to repeat. We know that from some points all elements are in interval ![$[1,1000]$](http://latex.artofproblemsolving.com/a/b/b/abb6453846b59c5e5c88a4dcce4f6a03e915c347.png)
, so problem follows.
such that:
and 
: Prove that :
