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.
of positive integers such that 
is an integer.
has two isogonal conjugate points 
and 
. The circle 
intersects circle 
at 
, and the circle 
intersects circle 
at 
. Prove that 
and 
are isogonal conjugates in triangle . is the circle with diameter 
, Circle is the circle with diameter 
.
+1 w
that satisfy the functional equation:![\[
f(f(x)(1 + y)) = f(x) + f(xy), \quad \forall x, y > 0.
\]](http://latex.artofproblemsolving.com/9/f/c/9fcb6b114b87842f23f31a97ddae6c722fe71611.png)
. Determine all functions 
such that![\[ xf(y) + yf(x) = (x+y) f(x^2+y^2) \]](http://latex.artofproblemsolving.com/0/c/3/0c3b10a82eb78d0dac62560b6736e755ecefe350.png)
and 
in 
.
a function satifying 
for all 
.
, then what is 
, the answer is in terms of 
.
such that for all 
,![\[
mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).
\]](http://latex.artofproblemsolving.com/c/0/c/c0cdf1bcb8053f5dc26e9df6355207b5e479f861.png)
and the recursive relation
for all natural number 
. Find the general form of 
.
of the set 
, define its colorfulness as the greatest natural number 
such that:
, 
, if 
, then 
.
? Determine how many such permutations have maximal colorfulness.
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
is the midpoint of base 
of an isosceles trapezoid 
with circumcircle 
. The angle bisector of 
intersects at 
. Line 
meets again at 
. From point 
, tangents 
are drawn to 
. Prove that 
are concurrent.
such that 
.![\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]](http://latex.artofproblemsolving.com/b/3/2/b32e0bcba967ca847e878ac186c3168cd6c5fc66.png)
be real numbers such that 
.
and 
Prove that
.
lies on 
by Reim. Also, note that 
are isogonal wrt 
.
is a kite and therefore 
.
,
,
lies on 
,