[/quote]
,
Prove that 
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
.
board with a number written in each cell. Every two neighbour rows sum up to at least 
,
.
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
with no parallel sides is inscribed in a circle 
.
are inscribed in triangles 
,
and 
,
,
,
.
and the number of distinct prime divisors of 
.
be a positive integer. Prove that there exist infinitely many positive integers 
and 
does not divide 
for any positive integers 
satisfying 
.
such that for any 
:
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
such that 
for all 
be a triangle with circumcircle 
and let 
be the 
-
and 
be the intersection points of 
and 
be the projections of 
intersects the tangent line at 
at a point 
.
.
and 
: Prove that :
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Dear Mathlinkers,
The compilation of the Geometric Inequalities Marathon is ready and attached below.
The file may be distributed physically or electronically, in whole or in part, but for and only for non-commercial purposes. References to problems or solutions should credit the corresponding authors.
If you find any errors, please send me a PM.
Enjoy the document!
Samer Seraj (BigSams)
The compilation of the Geometric Inequalities Marathon is ready and attached below.
The file may be distributed physically or electronically, in whole or in part, but for and only for non-commercial purposes. References to problems or solutions should credit the corresponding authors.
If you find any errors, please send me a PM.
Enjoy the document!
Samer Seraj (BigSams)
This post has been edited 1 time. Last edited by BigSams, Jun 13, 2019, 2:41 PM
