show that![\[\frac{a_1}{a_1+a_2}+\frac{a_2}{a_2+a_3}+\cdots +\frac{a_n}{a_n+a_1}\ge\frac{a_1+a_2+\cdots +a_n}{a_1+a_2+\cdots +a_n-(n-2)\sqrt[n]{a_1a_2\cdots a_n}}>1.\]](http://latex.artofproblemsolving.com/c/e/2/ce286b2083f9fcbb7a976b511eef0a223a464a77.png)
be a positive integer. Paul has a 
rectangular strip consisting of 
unit squares, where the 
square is labelled with 
for all 
.
square satisfying the following property: if the unit square in the 
column is labelled with 
,
is divisible by 
.
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
and 
A 
between 
,
,
and 
for every 
;
A 
between 
A 
between 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of 
be integer numbers 
satisfy that
.
is a perfect square number
can’t be any power 
of a positive integer number
Y by Adventure10, Mango247
The picture below shows the way Juan wants to divide a square field in three regions, so that all three of them share a well at vertex 
. If the side length of the field is 
meters, and each one of the three regions has the same area, how far must the points 
and 
be from 
?
Note: the area of each region includes the area the well occupies.
![[asy]
pair A=(0,0),B=(60,0),C=(60,-60),D=(0,-60),M=(0,-40),N=(20,-60);
pathpen=black;
D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle);
D(B--MP("M",M,W));
D(B--MP("N",N,S));
D(CR(B,3));[/asy]](//latex.artofproblemsolving.com/f/7/8/f7822683c73fc810675415e8466a0810e99f3063.png)
. If the side length of the field is 
meters, and each one of the three regions has the same area, how far must the points 
and 
be from 
?Note: the area of each region includes the area the well occupies.
![[asy]
pair A=(0,0),B=(60,0),C=(60,-60),D=(0,-60),M=(0,-40),N=(20,-60);
pathpen=black;
D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle);
D(B--MP("M",M,W));
D(B--MP("N",N,S));
D(CR(B,3));[/asy]](http://latex.artofproblemsolving.com/f/7/8/f7822683c73fc810675415e8466a0810e99f3063.png)
This post has been edited 1 time. Last edited by Leicich, Sep 20, 2014, 9:32 PM
,
is the desired length![\begin{align*}
[ABM]&=1200\\
[ABM]&=\frac{60\times d}{2}\\
60d&=2400\\
d&=40\\
x&=\boxed{20\text{\;meters}}
\end{align*}](http://latex.artofproblemsolving.com/d/4/e/d4ea1fdf7d33c0e4d639325a3a548ef43ae0da38.png)
