# 1985 AIME Problems

## Problem 1

Let $x_1=97$, and for $n>1$ let$x_n=\frac{n}{x_{n-1}}$. Calculate the product $x_1x_2x_3x_4x_5x_6x_7x_8$.

## Problem 2

When a right triangle is rotated about one leg, the volume of the cone produced is $800\pi \;\textrm{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\pi \;\textrm{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?

## Problem 3

Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.

## Problem 4

A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\frac1{1985}$.