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 
we define:
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
Let 
be the center of mass of a triangle 
and the points 
on the segments 
respectively, 
(excluding the extremities) such that
are the centers of mass of the triangles 
respectively, 
Pove that:
a) The centers of mas of
and 
are the same.
b) For any planar point
the inequality
holds.
be the center of mass of a triangle 
and the points 
on the segments 
respectively, 
(excluding the extremities) such that
are the centers of mass of the triangles 
respectively, 
Pove that:a) The centers of mas of
and 
are the same.b) For any planar point
the inequality
holds.
