1962 AMC 12 Problem 31

Question

The ratio of the interior angles of two regular polygons with sides of unit length is

3:2.

How many such pairs are there?

1

2

3

4

infinitely many

For an

n

-gon, an interior angle is

\frac{180^\circ(n-2)}{n}

If the smaller polygon has

n

sides, solve for the larger side count and test the possible integers

3\le n\lt6

Let the smaller and larger polygons have

n

and

N

sides. Then

\frac{\frac{N-2}{N}}{\frac{n-2}{n}}=\frac32,

which gives

N=\frac{4n}{6-n}.

Positivity and

N\gt n

require

n=3,

n=4,

or

n=5.

These give

N=4,

N=8,

and

N=20,

respectively. Thus there are

3

pairs.

Therefore, the correct answer is C.

Problem 31 in other years