Problem 997

The side lengths of the polygon can only be from the set $\{1, \sqrt{2}, \sqrt{3}, 2\}$, which take up $60^{\circ}, 90^{\circ}, 120^{\circ}, 180^{\circ}$ of the circle respectively. By working modulo 60 degrees we see that $\sqrt{2}$ must be used an even number of times. We now proceed to casework on the longest side of the polygon. Case 1: If the longest side has length 2, then the remaining sides must contribute the remaining 180 degrees. There are 3 possibilities: $(1,1,1,2),(1, \sqrt{3}, 2),(\sqrt{2}, \sqrt{2}, 2)$. Case 2: If the longest side has length $\sqrt{3}$, then it takes up either $120^{\circ}$ or $240^{\circ}$ of the circle. In the former case we have 6 possibilities: $(1,1,1,1, \sqrt{3}),(1, \sqrt{2}, \sqrt{2}, \sqrt{3}),(\sqrt{2}, 1, \sqrt{2}, \sqrt{3}),(1,1, \sqrt{3}, \sqrt{3})$, $(1, \sqrt{3}, 1, \sqrt{3}),(\sqrt{3}, \sqrt{3}, \sqrt{3})$. In the latter case there is only 1 possibility: $(1,1, \sqrt{3})$. Case 3: If the longest side has length $\sqrt{2}$, then it shows up either twice or four times. In the former case we have 2 possibilities: $(1,1,1, \sqrt{2}, \sqrt{2}),(1,1, \sqrt{2}, 1, \sqrt{2})$. In the latter case there is only 1 possibility: $(\sqrt{2}, \sqrt{2}, \sqrt{2}, \sqrt{2})$. Case 4: If all sides have length 1, then there is 1 possibility: $(1,1,1,1,1,1)$. Adding up all cases, we have $3+6+1+2+1+1=14$ polygons.