Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Interior Angle.

## Interior Angle Problem – AIME I, 1990

Let \(P_1\) be a regular r gon and \(P_2\) be a regular s gon \((r \geq s \geq 3)\) such that each interior angle of \(P_1\) is \(\frac{59}{58}\) as large as each interior angle of \(P_2\), find the largest possible value of s.

- is 107
- is 117
- is 840
- cannot be determined from the given information

**Key Concepts**

Integers

Polygons

Algebra

## Check the Answer

But try the problem first…

Answer: is 117.

AIME I, 1990, Question 3

Elementary Algebra by Hall and Knight

## Try with Hints

First hint

Interior angle of a regular sided polygon=\(\frac{(n-2)180}{n}\)

or, \(\frac{\frac{(r-2)180}{r}}{\frac{(s-2)180}{s}}=\frac{59}{58}\)

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

Second Hint

or, 58rs-58(2s)=59rs-59(2r)

or, 118r-116s=rs

or, r=\(\frac{116s}{118-s}\)

Final Step

for 118-s>0, s<118

or, s=117

or, r=(116)(117)

or, s=117.

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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