[/quote]
,
and 
.
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
and 
be two sequences of integers such that 
and 
for all 
.
divides 
for an infinity of integers 
such that 
for all 
are coprime. Consider the sequence 
given by 
.
such that 
is even and 
is an integer. Prove that the numerator in its simplest form of 
is an odd integer greater than 
.
be an integer and let 
be 
distinct subsets of 
.
such that the n subsets 
are also disjoint.
s
.
satisfies 
.
,
.
such that for all irrational numbers 
and 
,
where 
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
with incenter 
,
touches sides 
and 
at points 
.
meets 
at 
.
is parallel to 
.
with incircle 
.
and 
Prove that
Let 
Prove that
Let 
Prove that
Y by Adventure10, Mango247
The circle 
is inside the circle 
, and the circles touch each other at 
. A line through intersects also at 
, and also at 
. The tangent to at intersects at 
and 
. The tangents of passing thorugh touch at 
and 
. Prove that , , and are concyclic.
is inside the circle 
, and the circles touch each other at 
. A line through intersects also at 
, and also at 
. The tangent to at intersects at 
and 
. The tangents of passing thorugh touch at 
and 
. Prove that , , and are concyclic.
.
,
into a line which must be tangent to 
.
are fixed by the inversion (since they lie both on 
,
all lie on a circle centered at 
to 
at 
and 
and from here 
and we are done.
