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

Answer: 117

Concepts:regular polygon divisibility Diophantine Equation

Difficulty rating: 2040

Small Hint:

Use $\frac{180(n-2)}{n}$ for the interior angle of a regular $n$ -gon

Big Hint:

After simplifying, set $t=118-s$ and use the positivity of $t$

Solution:

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 $117.$

1990 AIME Problem 3

Answer: 117

Difficulty rating: 2040

Small Hint:

Big Hint:

Solution:

Problem 3 in Other Years