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
There are three cones on the table, touching each other. The heights of the cones are the same, and the radii of their bases are equal to 
and 
. A ball of radius 
was placed on the table, touching all the cones. It turned out that the center of the ball is equidistant from all points of contact of the cones. Find 
.
and 
. A ball of radius 
was placed on the table, touching all the cones. It turned out that the center of the ball is equidistant from all points of contact of the cones. Find 
.This post has been edited 1 time. Last edited by parmenides51, Feb 14, 2021, 5:52 PM
