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 
be an isosceles triangle (
) with incenter 
. Circle 
passes through 
and and is tangent to 
. intersects 
and circumcircle of at 
and 
, respectively. Let 
be the midpoint of 
and 
be the midpoint of 
. Prove that 
, 
and 
are concurrent.
+1 w
, 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
with sidelengths 
is given. It appears that it is impossible to construct a triangle from three segments whose lengths are equal to the altitudes of . Prove that 
.
such that:
and 
: Prove that :
,
are the altitudes of 
Suppose 
,
,