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Geometry Difficulty 4.8 AIME Find the answer

Regular polygons ICAO,VENTII C A O, V E N T I, and ALBEDOA L B E D O lie on a plane. Given that IN=1I N=1, compute the number of possible values of ONO N.

A number or a short expression. Spacing and $ signs are ignored.

Solution

First, place ALBEDOA L B E D O. We then note that ICAOI C A O has two orientations, both of which have II on EOE O. Next, we note that for any given orientation of ICAOI C A O, the two orientations of VENTIV E N T I have NN symmetric to line EIE I. Thus, for any given orientation of ICAOI C A O, we have that ONO N is the same in both orientations of VENTIV E N T I, which gives a total of 2 possible values for ONO N.

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