Question bank AIME 1998

For how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6,$ $8^8,$ and $k?$

Find the number of ordered pairs $(x, y)$ of positive integers that satisfy $x \le 2y \le 60$ and $y \le 2x \le 60.$

The graph of $y^2 + 2xy + 40|x| = 400$ partitions the plane into several regions. What is the area of the bounded region?

Nine tiles are numbered $1,$ $2,$ $3,$ $\ldots,$ $9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Given that $A_k = \frac{k(k - 1)}{2}\cos\frac{k(k - 1)\pi}{2},$ find $|A_{19} + A_{20} + \cdots + A_{98}|.$

Let $ABCD$ be a parallelogram. Extend $\overline{DA}$ through $A$ to a point $P,$ and let $\overline{PC}$ meet $\overline{AB}$ at $Q$ and $\overline{DB}$ at $R.$ Given that $PQ = 735$ and $QR = 112,$ find $RC.$

Let $n$ be the number of ordered quadruples $(x_1, x_2, x_3, x_4)$ of positive odd integers that satisfy $x_1 + x_2 + x_3 + x_4 = 98.$ Find $\frac{n}{100}.$

Except for the first two terms, each term of the sequence $1000,$ $x,$ $1000 - x,$ $\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $x$ produces a sequence of maximum length?

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between $9$ a.m. and $10$ a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40\%,$ and $m = a - b\sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c.$

Eight spheres of radius $100$ are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is $a + b\sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c.$

Three of the edges of a cube are $\overline{AB},$ $\overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P,$ $Q,$ and $R$ are on $\overline{AB},$ $\overline{BC},$ and $\overline{CD},$ respectively, so that $AP = 5,$ $PB = 15,$ $BQ = 15,$ and $CR = 10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?

Let $ABC$ be equilateral, and $D,$ $E,$ and $F$ be the midpoints of $\overline{BC},$ $\overline{CA},$ and $\overline{AB},$ respectively. There exist points $P,$ $Q,$ and $R$ on $\overline{DE},$ $\overline{EF},$ and $\overline{FD},$ respectively, with the property that $P$ is on $\overline{CQ},$ $Q$ is on $\overline{AR},$ and $R$ is on $\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a + b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers, and $c$ is not divisible by the square of any prime. What is $a^2 + b^2 + c^2?$

If $\{a_1, a_2, a_3, \ldots, a_n\}$ is a set of real numbers, indexed so that $a_1 \lt a_2 \lt a_3 \lt \cdots \lt a_n,$ its complex power sum is defined to be $a_1 i + a_2 i^2 + a_3 i^3 + \cdots + a_n i^n,$ where $i^2 = -1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1, 2, \ldots, n\}.$ Given that $S_8 = -176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$

An $m \times n \times p$ rectangular box has half the volume of an $(m + 2) \times (n + 2) \times (p + 2)$ rectangular box, where $m,$ $n,$ and $p$ are integers, and $m \le n \le p.$ What is the largest possible value of $p?$

Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i, j)$ and $(j, i)$ do not both appear for any $i$ and $j.$ Let $D_{40}$ be the set of all dominos whose coordinates are no larger than $40.$ Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$