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

For any positive integer $\displaystyle x_{}$ , let $\displaystyle S(x)$ be the sum of the digits of $\displaystyle x_{}$ , and let $\displaystyle T(x)$ be $\displaystyle |S(x+2)-S(x)|.$ For example, $\displaystyle T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values $\displaystyle T(x)$ do not exceed 1999?

Solution

Problem 6

A transformation of the first quadrant of the coordinate plane maps each point $\displaystyle (x,y)$ to the point $\displaystyle (\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $\displaystyle ABCD$ are $\displaystyle A=(900,300), B=(1800,600), C=(600,1800),$ and $\displaystyle D=(300,900).$ Let $\displaystyle k_{}$ be the area of the region enclosed by the image of quadrilateral $\displaystyle ABCD.$ Find the greatest integer that does not exceed $\displaystyle k_{}.$

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 $A$ ?

Solution

Problem 8

Let $\displaystyle \mathcal{T}$ be the set of ordered triples $\displaystyle (x,y,z)$ of nonnegative real numbers that lie in the plane $\displaystyle x+y+z=1.$ Let us say that $\displaystyle (x,y,z)$ supports $\displaystyle (a,b,c)$ when exactly two of the following are true: $\displaystyle x\ge a, y\ge b, z\ge c.$ Let $\displaystyle \mathcal{S}$ consist of those triples in $\displaystyle \mathcal{T}$ that support $\displaystyle \left(\frac 12,\frac 13,\frac 16\right).$ The area of $\displaystyle \mathcal{S}$ divided by the area of $\displaystyle \mathcal{T}$ is $\displaystyle m/n,$ where $\displaystyle m_{}$ and $\displaystyle n_{}$ are relatively prime positive integers, find $\displaystyle m+n.$

Solution

Problem 9

A function $\displaystyle f$ is defined on the complex numbers by $\displaystyle f(z)=(a+bi)z,$ where $\displaystyle a_{}$ and $\displaystyle b_{}$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $\displaystyle |a+bi|=8$ and that $\displaystyle b^2=m/n,$ where $\displaystyle m_{}$ and $\displaystyle n_{}$ are relatively prime positive integers. Find $\displaystyle m+n.$