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be a positive integer. Paul has a 
rectangular strip consisting of 
unit squares, where the 
square is labelled with 
for all 
. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or flipping) the pieces to obtain an 
square satisfying the following property: if the unit square in the row and 
column is labelled with 
, then 
is divisible by 
.
+1 w cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
and 
we define:
A 
between and as a sequence of distinct cities 
, 
, such that there are direct flights between 
and 
for every 
;
A 
between and as a path between and such that no other path between and has more cities;
A 
between and as a path between and such that no other path between and has fewer cities. and in the country, there exist a long path and a short path between them that have no cities in common (except and ). Let 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of , find all possible values 
be integer numbers 
satisfy that:
. Prove that: are even
is a perfect square number
can’t be any power 
of a positive integer number
is 
, and the circumcircle of 
meets 
and 
at 
and 
(
). Show that the extension of the line segment 
is perpendicular to 
.
gives 
so 
and similarly 
so M is the orthocenter of 
and 
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)
![[asy]
size(8cm);
defaultpen(fontsize(10pt));
pair A = dir(130), B = dir(210), C = dir(330), M = origin, P = extension((A+C)/2,M,B,C), Q = extension((B+C)/2,M,A,C);
dot("$A$", A, dir(130));
dot("$B$", B, dir(210));
dot("$C$", C, dir(330));
dot("$Q$", Q, dir(45));
dot("$P$", P, dir(270));
dot("$M$", M, dir(315));
draw(A--B--C--cycle);
draw(circumcircle(A,B,M), dotted);
draw(A--M--B, dashed);
draw(B--Q, dotted);
draw(A--P, dotted);
[/asy]](http://latex.artofproblemsolving.com/e/b/9/eb923682bc5fc5981a2e2974ecb9ed388521542b.png)
is cyclic, we must have 
so triangles 
and 
are isosceles, which implies 
and 
.