1984 AIME Problems

Problem 1

Find the value of $\displaystyle a_2+a_4+a_6+a_8+\ldots+a_{98}$ if $\displaystyle a_1$ , $\displaystyle a_2$ , $\displaystyle a_3\ldots$ is an arithmetic progression with common difference 1, and $\displaystyle a_1+a_2+a_3+\ldots+a_{98}=137$ .

Solution

Problem 2

The integer $n$ is the smallest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$ . Compute $\frac{n}{15}$ .

Solution

Problem 3

A point $P$ is chosen in the interior of $\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$ , the resulting smaller triangles $t_{1}$ , $t_{2}$ , and $t_{3}$ in the figure, have areas $4$ , $9$ , and $49$ , respectively. Find the area of $\triangle ABC$ .

Solution

Problem 4

Let $S$ be a list of positive integers - not necessarily distinct - in which the number $68$ appears. The arithmetic mean of the numbers in $S$ is $56$ . However, if $68$ is removed, the arithmetic mean of the numbers is $55$ . What's the largest number that can appear in $S$ ?

Solution

Problem 5

Determine the value of $ab$ if $\log_8a+\log_4b^2=5$ and $\log_8b+\log_4a^2=7$ .

Solution

Problem 6

Three circles, each of radius 3, are drawn with centers at $(14, 92)$ , $(17, 76)$ , and $(19, 84)$ . A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?

Solution

Problem 7

The function f is defined on the set of integers and satisfies $f(n)= \begin{cases} n-3 & \mbox{if }n\ge 1000 \\ f(f(n+5)) & \mbox{if }n<1000 \end{cases}$

Find $\displaystyle f(84)$ .

Solution

Problem 8

The equation $\displaystyle z^6+z^3+1$ has complex roots with argument $\theta$ between $90^\circ$ and $180^\circ$ in thet complex plane. Determine the degree measure of $\theta$ .