Contest Problem

Deadline: September 30, 4PM.

A regular polygon is a convex polygon with all sides equal and all angles equal. A regular $n$-gon is a regular polygon with $n$ sides. We normalize our regular polygons by assuming that the circumscribed circle has radius $1$.

a) Fix a vertex of the regular $8$-gon (or octagon). Compute the sum of the squares of the lengths of the $2$ edges and the $5$ diagonals originating at this vertex.
b) Fix a vertex of the regular $n$-gon. Find a formula for the sum of the squares of the lengths of the $2$ edges and the $n-3$ diagonals originating at this vertex.

General Information:

This is a monthly contest for Pitt undergraduate students. The monthly problem will be posted here and on the Undergraduate Contest section of the bulletin board in 705 Thackeray Hall.

Written solutions to the monthly problem should be submitted by 4PM of the last business day of the corresponding month. Please submit your solutions to the mailbox labelled "Undergraduate Contest" in 301 Thackeray Hall. Late submissions are not accepted. Please remember to include your name and e-mail address.

The winner will be selected in a random drawing from all eligible entries that contain a full and correct solutions to the month's contest problem. The winner will be notified by e-mail. The winner will receive a prize and the names of all the people that submitted a correct solution will be posted on the Undergraduate Contest section of the bulletin board in 705 Thackeray Hall.

Also keep an eye on the math home page and your Pitt emails to see updates regarding this event.