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?

Solution

Problem 3

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

Solution

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 $\frac1{1985}$ .

Solution

Problem 5

A sequence of integers $a_1, a_2, a_3, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$ . What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?

Solution

Problem 6

As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$ .

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Solution

Problem 7

Assume that $a$ , $b$ , $c$ , and $d$ are positive integers such that $a^5 = b^4$ , $c^3 = d^2$ , and $c - a = 19$ . Determine $d - b$ .

Solution

Problem 8

The sum of the following seven numbers is exactly 19: $a_1 = 2.56$ , $a_2 = 2.61$ , $a_3 = 2.65$ , $a_4 = 2.71$ , $a_5 = 2.79$ , $a_6 = 2.81$ , $a_7 = 2.86$ . It is desired to replace each $a_i$ by an integer approximation $A_i$ , $1\le i \le 7$ , so that the sum of the $A_i$ 's is also 19 and so that $M$ , the maximum of the "errors" $\| A_i-a_i\|$ , the maximum absolute value of the difference, is as small as possible. For this minimum $M$ , what is $100M$ ?

Solution

Problem 9

In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$ , $\beta$ , and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$ . If $\cos \alpha$ , which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?