[/quote]
,
and 
intersect at points 
and 
.
is tangent to 
and 
,
closer to point 
passes through 
and intersecting 
.
,
are collinear, 
,
,
.
.
such that 
for all 
.
.
amount of people stand on coordinates 
where 
.
.
got 
people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups.
litres of water.
be positive real numbers such that 
.
has centroid 
.
passing through 
and 
.
is chosen on 
at 
.
,
,
,
(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 
,
:![\[
\begin{cases}
f(n+1) > f(n) \\
f(f(n)) = 2n + 2
\end{cases}
\]](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.
Y by TheOneYouWant, Tawan, skys, Adventure10, Mango247
Let the nervous waiting commence!
No talking about about how to solve the problems.
I've done ~4, and fully locked down 2.
How is everyone else doing?
No talking about about how to solve the problems.
I've done ~4, and fully locked down 2.
How is everyone else doing?
This post has been edited 1 time. Last edited by xPythagoras, Feb 15, 2016, 10:10 PM
Reason: No one seemed to like the other 'rules'
Reason: No one seemed to like the other 'rules'
skills
