[/quote]
,
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 
and 
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 
,
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 
.
such that:
and 
: Prove that :
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New Edition.
Download all AoPS competitions from 1959 to 2012.
Since it is difficult to read a long 1500 pages document, I managed to make a zip file for each country and put all the Olympiad problems of that country in the zip file. I've uploaded them separately so that you can choose each country you want and download all its problems. Also, I put all the documents in one compressed file for those who want it all at once!
Here you go: http://www.mediafire.com/?rid3p3qa87r6h
Enjoy your problem solving!
Download all AoPS competitions from 1959 to 2012.
Since it is difficult to read a long 1500 pages document, I managed to make a zip file for each country and put all the Olympiad problems of that country in the zip file. I've uploaded them separately so that you can choose each country you want and download all its problems. Also, I put all the documents in one compressed file for those who want it all at once!
Here you go: http://www.mediafire.com/?rid3p3qa87r6h
Enjoy your problem solving!
This post has been edited 70 times. Last edited by Amir Hossein, Feb 8, 2015, 8:13 AM
Reason: Updated
Reason: Updated
