and 
Prove that
of positive integers such that
over 
is linear
be positive integers satisfying ![\[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\]](http://latex.artofproblemsolving.com/b/8/c/b8c3fe14640a2dcb30b755b2347721da5d080a78.png)
Determine all possible values of 
.
Y by Adventure10, Mango247
Let us define cutting a convex polygon with 
sides by choosing a pair of consecutive sides 
and 
and substituting them by three segments 
, and 
, where 
is the midpoint of and 
is the midpoint of . In other words, the triangle 
is removed and a convex polygon with 
sides is obtained.
Let
be a regular hexagon with area 
. is cut and the polygon 
is obtained. Then is cut in one of seven ways and polygon 
is obtained, and so on. Prove that, regardless of how the cuts are made, the area of 
is always greater than 
.
sides by choosing a pair of consecutive sides 
and 
and substituting them by three segments 
, and 
, where 
is the midpoint of and 
is the midpoint of . In other words, the triangle 
is removed and a convex polygon with 
sides is obtained.Let
be a regular hexagon with area 
. is cut and the polygon 
is obtained. Then is cut in one of seven ways and polygon 
is obtained, and so on. Prove that, regardless of how the cuts are made, the area of 
is always greater than 
.
