[/quote]
,
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 
.
.
,
,
or 
.
,
and 
Prove that
of the set 
,
such that:
,
,
,
.
?
,
,
,
.
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 
represents 
.
Prove that
Let 
Prove that
.
measures 40 units. Let 
be the orthocenter of triangle 
its circumcenter. Let 
be the foot of the altitude from 
and 
the foot of the altitude from 
.
.
passes through point 
.
be real number such that 
Prove that
Let 
be real number . Prove that
be a triangle. The points 
and 
lie on the segments 
and 
respectively, such that the lines 
and 
intersect in a common point. Prove that it is possible to choose two of the triangles 
and 
whose inradii sum up to at least the inradius of the triangle 
integers is 
.
to each of the first four and subtracting 
from each of the last seven, what is the new mean?
Y by Adventure10 and 1 other user
I am currently in the process of making a Mock Mathcounts Sprint and Target Examination. The difficulty is about National difficulty, if not more difficult. PM if you are interested, and I will PM (and post) when my rounds are completed and for more details. PLEASE be patient, as it may take a while for me to finish. 
But if you are interesting please PM and/or (but at least PM me to officially sign up) if you are interested.
Also note that I may not even finish it.
*Also you may need Word 2007 or higher to view some of the problems
*CHECK MY REVISED SAMPLE PROBLEMS OUT (if you scroll down)
But if you are interesting please PM and/or (but at least PM me to officially sign up) if you are interested.
Also note that I may not even finish it.
*Also you may need Word 2007 or higher to view some of the problems
*CHECK MY REVISED SAMPLE PROBLEMS OUT (if you scroll down)
This post has been edited 2 times. Last edited by luppleAOPS, Mar 1, 2011, 2:52 AM
and 
.
? Would greatly appreciate it.
