Question bank AIME 2003

The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N.$

Let $N$ be the greatest integer multiple of $8,$ no two of whose digits are the same. What is the remainder when $N$ is divided by $1000?$

Define a *good word* as a sequence of letters that consists only of the letters $A,$ $B,$ and $C$ — some of these letters may not appear in the sequence — and in which $A$ is never immediately followed by $B,$ $B$ is never immediately followed by $C,$ and $C$ is never immediately followed by $A.$ How many seven-letter good words are there?

In a regular tetrahedron, the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as $n\pi,$ where $n$ is a positive integer. Find $n.$

In $\triangle ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C'$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area of the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$

Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25,$ respectively.

Find the eighth term of the sequence $1440,$ $1716,$ $1848,$ $\ldots,$ whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.

Consider the polynomials $P(x) = x^6 - x^5 - x^3 - x^2 - x$ and $Q(x) = x^4 - x^3 - x^2 - 1.$ Given that $z_1,$ $z_2,$ $z_3,$ and $z_4$ are the roots of $Q(x) = 0,$ find $P(z_1) + P(z_2) + P(z_3) + P(z_4).$

Two positive integers differ by $60.$ The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?

Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $\overline{AB},$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of $\triangle CDM$ can be expressed as $\frac{m\sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m + n + p.$

The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of the committee?

A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$

Let $A = (0, 0)$ and $B = (b, 2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB} \parallel \overline{DE},$ $\overline{BC} \parallel \overline{EF},$ $\overline{CD} \parallel \overline{FA},$ and the $y$-coordinates of its vertices are distinct elements of the set $\{0, 2, 4, 6, 8, 10\}.$ The area of the hexagon can be written in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n.$

Let $$
\begin{aligned} &P(x) = 24x^{24} \\ &\quad {}+ \sum_{j=1}^{23} (24 - j)\left(x^{24-j} + x^{24+j}\right). \end{aligned}
$$ Let $z_1,$ $z_2,$ $\ldots,$ $z_r$ be the distinct zeros of $P(x),$ and let $z_k^2 = a_k + b_k i$ for $k = 1,$ $2,$ $\ldots,$ $r,$ where $i = \sqrt{-1},$ and $a_k$ and $b_k$ are real numbers. Let $$
\sum_{k=1}^{r} |b_k| = m + n\sqrt{p},
$$ where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m + n + p.$

Given that $\frac{((3!)!)!}{3!} = k \cdot n!,$ where $k$ and $n$ are positive integers and $n$ is as large as possible, find $k + n.$

One hundred concentric circles with radii $1,$ $2,$ $3,$ $\ldots,$ $100$ are drawn in a plane. The interior of the circle of radius $1$ is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius $100$ can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Let the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}.$ Susan makes a list as follows: for each two-element subset of $\mathcal{S},$ she writes on her list the greater of the set’s two elements. Find the sum of the numbers on the list.

Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10}(\sin x + \cos x)$ $= \frac{1}{2}(\log_{10} n - 1),$ find $n.$

Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\frac{m + n\pi}{p},$ where $m,$ $n,$ and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p.$