[/quote]
,
such that for all positive integers 
,![\[
f(mn) = \mathrm{lcm}(m, n) \cdot \gcd(f(m), f(n)),
\]](http://latex.artofproblemsolving.com/6/0/7/6071692e2716901e88cc1cfc19ed60388c10701d.png)
where 
and 
denote the least common multiple and the greatest common divisor of 
and 
,
such that 
for all 
.
such that 
coprime, 
and 
.
:![\[
\begin{cases}
f(n+1) > f(n) \\
f(f(n)) = 2n + 2
\end{cases}
\]](http://latex.artofproblemsolving.com/0/4/7/047e28c9d899d7de443ae40ac2c756fda851fafb.png)
Find the value of 
.
differents subjects, each consist of 
students. Given that for any 
different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least 
different subjects.
has 
.
.
where 
and 
are integers and 
.
with real coefficients such that for all reals 
we have the following relations![\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]](http://latex.artofproblemsolving.com/4/4/9/4491a53f9c6f96c18e42eb984b8e83566140bfdb.png)
,
may have in the range 
, as 
.
be its incenter. A point 
is chosen on line 
,
intersects 
at 
,
intersects 
at 
.
intersects 
and 
,
and 
intersect at 
,
and 
intersect at 
.
.
. Prove that
,
intersect at 
,
intersect at 
,
intersect at 
.
intersects line 
at 
and 
intersects line 
and 
are collinear in that order. Prove that if lines 
and 
intersect at 
.
and 
are written on a blackboard, where 
and 
are relatively prime positive integers. At any point, Evan may pick two of the numbers 
and 
written on the board and write either their arithmetic mean 
or their harmonic mean 
on the board as well. Find all pairs 
such that Evan can write 1 on the board in finitely many steps.
Y by
9Poll:
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Hi everyone,
For an AIME score of 7-11, would you recommend MathPath or AMSP Level 2/3?
Thanks in advance!
Also people who have gone to them, please tell me more about the programs!
For an AIME score of 7-11, would you recommend MathPath or AMSP Level 2/3?
Thanks in advance!
Also people who have gone to them, please tell me more about the programs!
This post has been edited 1 time. Last edited by FuturePanda, Jan 30, 2025, 3:16 AM
