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 circumcircle of the 
is circumcircle of the 
is midline of the 
if 
then 
is isosceles trapezoid and the 
is isosceles. Similarly, 
is midline of the if 
then 
is isosceles trapezoid and 
is isosceles. Therefore, 
is at least a kite, let 
be its diagonal intersection. Assuming 
are both cyclic, 
implies 
(see http://www.mathlinks.ro/viewtopic.php?t=204352) and is a rhombus. Power of 
to is
and reducing by 
we get the following equation for the angle 
:
and 
only the 1st one acceptable.
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
and 
be the midpoints of sides 
and 
of the convex quadrangle 
Is it possible that points 
lie on the same circle and points 
lie on the same circle?
