1999 AIME Problems

Problem 1

Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.

Solution

Problem 2

Consider the parallelogram with vertices $\displaystyle (10,45),$ $\displaystyle (10,114),$ $\displaystyle (28,153),$ and $\displaystyle (28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $\displaystyle m/n,$ where $\displaystyle m_{}$ and $\displaystyle n_{}$ are relatively prime positive integers. Find $\displaystyle m+n.$

Solution

Problem 3

Find the sum of all positive integers $\displaystyle n$ for which $\displaystyle n^2-19n+99$ is a perfect square.

Solution

Problem 4

The two squares shown share the same center $\displaystyle O_{}$ and have sides of length 1. The length of $\displaystyle \overline{AB}$ is $\displaystyle 43/99$ and the area of octagon $\displaystyle ABCDEFGH$ is $\displaystyle m/n,$ where $\displaystyle m_{}$ and $\displaystyle n_{}$ are relatively prime positive integers. Find $\displaystyle m+n.$

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

There is a set of 1000 switches, each of which has four positions, called $A, B, C$ , and $D$ . When the position of any switch changes, it is only from $A$ to $B$ , from $B$ to $C$ , from $C$ to $D$ , or from $D$ to $A$ . Initially each switch is in position $A$ . The switches are labeled with the 1000 different integers $(2^{x})(3^{y})(5^{z})$ , where $x, y$ , and $z$ take on the values $0, 1, \ldots, 9$ . At step i of a 1000-step process, the $i$ -th switch is advanced one step, and so are all the other switches whose labels divide the label on the $i$ -th switch. After step 1000 has been completed, how many switches will be in position $A$ ?