1990 AIME Problem 3

Question

Let

P_1

be a regular

r

-gon and

P_2

be a regular

s

-gon

(r\geq s\geq3)

such that each interior angle of

P_1

is

\frac{59}{58}

as large as each interior angle of

P_2.

What is the largest possible value of

s?

Answer

Use

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

for the interior angle of a regular

n

-gon

After simplifying, set

t=118-s

and use the positivity of

t

The angle condition gives

\frac{\frac{r-2}{r}}{\frac{s-2}{s}}=\frac{59}{58},

which simplifies to

r(118-s)=116s.

Put

t=118-s.

Then

r=\frac{13688}{t}-116.

In particular,

t

is a positive divisor of

13688,

and maximizing

s=118-t

means taking the least possible

t.

The choice

t=1

gives

s=117

and the positive integer

r=13572,

which also satisfies

r\geq s.

Hence the largest possible

s

is

117.

Problem 3 in other years