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.
, 
, 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)
-by- board of unit squares for some odd positive integer . We say that a collection 
of identical dominoes is a maximal grid-aligned configuration on the board if consists of 
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let 
be the number of distinct maximal grid-aligned configurations obtainable from by repeatedly sliding dominoes. Find the maximum value of as a function of .
that satisfy the equation
![\[
\frac{5^{2000} - 1}{4}
\]](http://latex.artofproblemsolving.com/8/b/a/8ba5d1ed8bafd6653f9e6f351558a9976c43a57a.png)
is divided by 
.
