Question bank AIME 1997

How many of the integers between $1$ and $1000,$ inclusive, can be expressed as the difference of the squares of two nonnegative integers?

The nine horizontal and nine vertical lines on an $8 \times 8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $\frac{s}{r}$ can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?

Circles of radii $5,$ $5,$ $8,$ and $\frac{m}{n}$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a,$ $b,$ $c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is $1$ or $2$ and whose denominator is an integer. The closest such fraction to $r$ is $\frac{2}{7}.$ What is the number of possible values for $r?$

Point $B$ is in the exterior of the regular $n$-sided polygon $A_1A_2\cdots A_n,$ and $A_1A_2B$ is an equilateral triangle. What is the largest value of $n$ for which $A_n,$ $A_1,$ and $B$ are consecutive vertices of a regular polygon?

A car travels due east at $\frac{2}{3}$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\frac{1}{2}\sqrt{2}$ mile per minute. At time $t = 0,$ the center of the storm is $110$ miles due north of the car. At time $t = t_1$ minutes, the car enters the storm circle, and at time $t = t_2$ minutes, the car leaves the storm circle. Find $\frac{1}{2}(t_1 + t_2).$

How many different $4 \times 4$ arrays whose entries are all $1$’s and $-1$’s have the property that the sum of the entries in each row is $0$ and the sum of the entries in each column is $0?$

Given a nonnegative real number $x,$ let $\langle x\rangle$ denote the fractional part of $x;$ that is, $\langle x\rangle = x - \lfloor x\rfloor,$ where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x.$ Suppose that $a$ is positive, $\langle a^{-1}\rangle = \langle a^2\rangle,$ and $2 \lt a^2 \lt 3.$ Find the value of $a^{12} - 144a^{-1}.$

Every card in a deck has a picture of one shape — circle, square, or triangle, which is painted in one of the three colors — red, blue, or green. Furthermore, each color is applied in one of three shades — light, medium, or dark. The deck has $27$ cards, with every shape-color-shade combination represented. A set of three cards from the deck is called *complementary* if all of the following statements are true:

• Either each of the three cards has a different shape or all three of the cards have the same shape.

• Either each of the three cards has a different color or all three of the cards have the same color.

• Either each of the three cards has a different shade or all three of the cards have the same shade.

How many different complementary three-card sets are there?

Let $$
x = \frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle\sum_{n=1}^{44} \sin n^\circ}.
$$ What is the greatest integer that does not exceed $100x?$

The function $f$ defined by $f(x) = \frac{ax + b}{cx + d},$ where $a,$ $b,$ $c,$ and $d$ are nonzero real numbers, has the properties $f(19) = 19,$ $f(97) = 97,$ and $f(f(x)) = x$ for all values except $\frac{-d}{c}.$ Find the unique number that is not in the range of $f.$

Let $S$ be the set of points in the Cartesian plane that satisfy $$
\begin{aligned} &\Bigl|\bigl||x| - 2\bigr| - 1\Bigr| \\ &\quad {}+ \Bigl|\bigl||y| - 2\bigr| - 1\Bigr| = 1. \end{aligned}
$$ If a model of $S$ were built from wire of negligible thickness, then the total length of wire required would be $a\sqrt{b},$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime number. Find $a + b.$

Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997} - 1 = 0.$ Let $\frac{m}{n}$ be the probability that $\sqrt{2 + \sqrt{3}} \le |v + w|,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

The sides of rectangle $ABCD$ have lengths $10$ and $11.$ An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$ The maximum possible area of such a triangle can be written in the form $p\sqrt{q} - r,$ where $p,$ $q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p + q + r.$