# On convex $n$-gons with special properties

This problem is pretty hard :

Does there exist a convex $n$-gon with all sides equal, whose vertices lie on the graph $y=x^2$ when $n$ is an even number ?

The case of $n$ being odd can be delt with after some manipulations (the answer is yes). The even case seems more tricky.

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The highest vertex $L$ on the left, and the two highest vertices on the right $R_1$ (highest) and $R_2$ form a triangle with $\measuredangle LR_2R_1$ being larger then 90 degrees. But you also know that the distances $R_1R_2$ and $R_1L$ are the same which gives a contradiction.
$L$ and $R_2$ are at the same height, $R_1$ is higher and to the right of $R_2$, so the angle is larger than 90 degrees. – Phira Apr 20 '12 at 11:31