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 
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 a right-angled triangle in 
such that 
. Let 
be the midpoint of 
and let 
be the intersection of 
with the perpendicular line to which passes through . Let 
be the intersection point of the parallel line to which passes through with the perpendicular line to 
which passes through 
. Prove that triangles 
and 
are similar if and only if 
.
be a right-angled triangle in 
such that 
. Let 
be the midpoint of 
and let 
be the intersection of 
with the perpendicular line to which passes through . Let 
be the intersection point of the parallel line to which passes through with the perpendicular line to 
which passes through 
. Prove that triangles 
and 
are similar if and only if 
.
then it's very easy to prove that triangles AEM and MCA are similar. 
![$D\in [AC]$](http://latex.artofproblemsolving.com/c/5/a/c5acd77baad8d15d98170c2ac60f572eeeed5507.png)
s
so 
or 
so 
are similar 
