Track / Stage 5 / 27 of 400 #627 of 2000

Problem 627

AIME late

Geometry Difficulty 5.1 Find the answer

If we double the number of sides of a regular $n$ -sided polygon inscribed in a circle, then each of its angles will increase by $15^{\circ}$ . How many sides does the polygon have?

Your answer

Official solution

One of the angles of an $n$ -sided polygon is $\frac{2 n R-4 R}{n};$ since $15^{\circ}=\frac{R}{6}$ , according to the problem,

$\frac{2 n R-4 R}{n}=\frac{4 n R-4 R}{2 n}-\frac{R}{6}$

from which $n-12$ .

(Sarolta Petrik, Budapest.)

The problem was also solved by: L. Bánó, E. Freund, P. Füstös, Z. Harsányi, P. Heimlich, Gy. Jánosy, J. Kiss, E. Makó, A. Paunz, S. Pichler, Z. Reich, E. Sárközy, Gy. Schilhanek, Gy. Schuster, J. Steiger, J. Szabó, 1. Szécsi, B. Tóth.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.