,
,
be real numbers, such that 
and 
.
.
be a positive integer such that ![\[
1 + \sum_{i=1}^{n-1} i^{n-1}.
\]](http://latex.artofproblemsolving.com/e/2/d/e2daf7800da3f984cd761ed1fecf10083444db25.png)
Prove that 
.
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.
,
,
.![\[\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}\]](http://latex.artofproblemsolving.com/2/6/4/264d110e2a40c7f787c650aa8b8dc30af5d28263.png)
Y by Adventure10, Mango247
In a circle 
with centre 
and radius 
, let 
, 
be two circles with centres 
, 
and radii 
, 
respectively, so that each circle 
is internally tangent to at 
and so that , are externally tangent to each other at 
.
Prove that the three lines
, 
, and 
are concurrent.
with centre 
and radius 
, let 
, 
be two circles with centres 
, 
and radii 
, 
respectively, so that each circle 
is internally tangent to at 
and so that , are externally tangent to each other at 
.Prove that the three lines
, 
, and 
are concurrent.
to 
,
.
to 
balance at 
to 
,
balance at 
at 
.
now balance at 
,
in the triangle 
are concurrent.
,
,
in 
,
so 
are on same line 
a
are on same line.
is perpendicular to line that tangent to circl
is perpendicular to line that tangent to circl
are on same line 
!
and 
and 
AND 
we must proof:
a
so the problem is proof!
.![\[\frac{OA_1}{A_1O_1} \cdot \frac{O_1A}{AO_2} \cdot \frac{O_2A_2}{A_2O} = 1.\]](http://latex.artofproblemsolving.com/5/8/f/58fe3d80b59713a430f2bccd0d008e182f16eb4c.png)
![\[\frac{r}{r_1} \cdot \frac{r_1}{r_2} \cdot \frac{r_2}{r} = 1,\]](http://latex.artofproblemsolving.com/6/d/9/6d92534ae3bab68d714a138855a202469eb1b2e7.png)
