# Difference between revisions of "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.

## 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 $\displaystyle m+n.$

## Problem 3

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

## 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 $\displaystyle m+n.$

## 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?

## 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 $\displaystyle k_{}.$

## 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 $A$?

## 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 $\displaystyle m+n.$

## 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 $\displaystyle m+n.$

## Problem 10

Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $\displaystyle m/n,$ where $\displaystyle m_{}$ and $\displaystyle n_{}$ are relatively prime positive integers. Find $\displaystyle m+n.$

## Problem 11

Given that $\displaystyle \sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $\displaystyle m_{}$ and $\displaystyle n_{}$ are relatively prime positive integers that satisfy $\displaystyle \frac mn<90,$ find $\displaystyle m+n.$

## Problem 12

The inscribed circle of triangle $\displaystyle ABC$ is tangent to $\displaystyle \overline{AB}$ at $\displaystyle P_{},$ and its radius is 21. Given that $\displaystyle AP=23$ and $\displaystyle PB=27,$ find the perimeter of the triangle.

## Problem 13

Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $\displaystyle 50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\displaystyle m/n,$ where $\displaystyle m_{}$ and $\displaystyle n_{}$ are relatively prime positive integers. Find $\displaystyle \log_2 n.$