2012 UNCO Math Contest II Problems/Problem 10
Problem
An integer equiangular hexagon is a six-sided polygon whose side
lengths are all integers and whose internal angles all measure
.
(a) How many distinct (i.e., non-congruent) integer equiangular
hexagons have no side length greater than 
? Two such hexagons
are shown.
![[asy] draw((0,0)--(1,0)--(4,3*sqrt(3))--(3,4*sqrt(3))--(-1,4*sqrt(3))--(-1-3*sqrt(3)/2,4*sqrt(3)-1.5)--cycle,black); MP("1",(.5,0),S);MP("6",(2.5,1.5*sqrt(3)),SE);MP("2",(3.5,3.5*sqrt(3)),NE);MP("4",(1,4*sqrt(3)),N);MP("3",(-1-.75*sqrt(3),4*sqrt(3)-.75),NW);MP("5",(-.5-.75*sqrt(3),2*sqrt(3)-.75),W); draw((8,0)--(11,0)--(13,2*sqrt(3))--(11.5,3.5*sqrt(3))--(7.5,3.5*sqrt(3))--(6,2*sqrt(3))--cycle,black); MP("3",(9.5,0),S);MP("4",(12,sqrt(3)),SE);MP("3",(12.25,2.75*sqrt(3)),NE);MP("4",(9.5,3.5*sqrt(3)),N);MP("3",(6.75,2.75*sqrt(3)),NW);MP("4",(7,sqrt(3)),W); [/asy]](http://latex.artofproblemsolving.com/d/f/b/dfb405f827518b1a98db6990b7d66679bac2f293.png)
(b) How many distinct integer equiangular hexagons have no side
greater than 
? Give a closed formula in terms of .
(A figure and its mirror image are congruent and are not considered distinct. Translations and rotations of one another are also congruent and not distinct.)
