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.
be a fixed positive real number. Find the minimum possible value 
such that for any positive reals 
, 
, 
, 
, we have
and 
with no common divisor greater than 1 such that 
divides 
. (Professor Yongjin Song)
and 
. Prove that there always exists a positive integer 
such that 
. is a power of a prime number, but for arbitrary then well.....
, 
, 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 
. )
, , with , . Prove that there exists such that for every (For and , the notation represents . )
be the circumcircle of acute triangle 
. Points 
and 
are on segments 
and 
respectively such that 
. The perpendicular bisectors of 
and 
intersect minor arcs and of at points 
and 
respectively. Prove that lines 
and 
are either parallel or they are the same line., , with , . Prove that there exists such that for every (For and , the notation represents . )
is a positive real number, find the minimum of the following expression
. The angle bisectors of the angles and 
intersect the sides 
and 
at the points 
and 
, and intersect each other at the point 
. The line 
intersects the circumcircle of triangle at the points 
and 
. Prove that the circumradius of triangle 
is twice as long as the circumradius of triangle .
such that 
is also a prime number.
be a cyclic quadrilateral. Let 
, 
, 
be the feet of the perpendiculars from to the lines 
, 
, , respectively. Show that 
if and only if the bisectors of 
and 
are concurrent with .
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)
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
and 
so 
so we have that 
is cylic.
be the intersection of the line through parallel to with 
.
so 
is cylic which implies belongs to the circumcircle of 
. Hence passes through the intersection of the circumcircle of and the line through parallel to which is fixed.
![\[
\frac{5^{2000} - 1}{4}
\]](http://latex.artofproblemsolving.com/8/b/a/8ba5d1ed8bafd6653f9e6f351558a9976c43a57a.png)
is divided by 
.
circuncírculos 
and cut again in 
.
since the angles MNC