[/quote]
,
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
as the set of all numbers of the form 
for some nonnegative 
and 
.
such that 
and 
.
,
. Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.
- Greatest common divisor of numbers 
and 
board, there is initially a tree of height 
.
.
such that the gardener can ensure there are eventually 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that the equation
be a positive integer and 
be a polynomial with integer coefficients such that 
.
is divisible by 
.
such that for every positive real 
,
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
be a positive integer and let 
be a strictly increasing sequence of 
be a subset of 
such that the value of![\[
\left|1-\sum_{i \in X} a_{i}\right|
\]](http://latex.artofproblemsolving.com/2/5/1/2515e074de1782f38453e1fce8ab7c2ec7cfe4c0.png)
is minimised. Prove that there exists a strictly increasing sequence of 
with sum equal to 2 such that![\[
\sum_{i \in X} b_{i}=1.
\]](http://latex.artofproblemsolving.com/3/7/4/37410820fc4c546f11d57212cfce53a7ec4941e3.png)
be a scalene triangle with circumcircle 
,
at 
.
meets 
and 
.
intersects the 
-
,
,
.
passes through either 
be positive real numbers such that : 
. Prove that : 
![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
be relatively prime nonconstant polynomials. Show that there can be at most three real numbers 
such that 
is the square of a polynomial.
be real numbers less than 2.Then prove that 
Y by
Two circles 
and 
with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.
Thanks.
and 
with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.Thanks.
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