[/quote]
,
be positive real numbers such that 
.
has centroid 
.
passing through 
.
and 
.
is chosen on 
at 
and 
.
,
,
,
(in this order, from left to right). Determine whether the set of all beautiful integers is finite.
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.
.
where 
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 
intersects 
at 
,
intersects 
at 
.
intersects 
and 
,
and 
intersect at 
,
and 
intersect at 
.
. Prove that
Y by
There are coins in denominations of 
and 
doubloons, where and are given mutually prime natural numbers, with 
. A non-negative integer 
is called lucky if the sum in doubloons can be scored with using no more than 
coins. Find the number of lucky numbers.
From the folklore
and 
doubloons, where and are given mutually prime natural numbers, with 
. A non-negative integer 
is called lucky if the sum in doubloons can be scored with using no more than 
coins. Find the number of lucky numbers.From the folklore
