2009 AIME I

Call a $3$-digit number *geometric* if it has $3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

There is a complex number $z$ with imaginary part $164$ and a positive integer $n$ such that $$
\frac{z}{z + n} = 4i.
$$ Find $n.$

A coin that comes up heads with probability $p \gt 0$ and tails with probability $1 - p \gt 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $\frac{1}{25}$ of the probability of five heads and three tails. Let $p = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

In parallelogram $ABCD,$ point $M$ is on $\overline{AB}$ so that $\frac{AM}{AB} = \frac{17}{1000},$ and point $N$ is on $\overline{AD}$ so that $\frac{AN}{AD} = \frac{17}{2009}.$ Let $P$ be the point of intersection of $\overline{AC}$ and $\overline{MN}.$ Find $\frac{AC}{AP}.$

Triangle $ABC$ has $AC = 450$ and $BC = 300.$ Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK,$ and $\overline{CL}$ is the angle bisector of angle $C.$ Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL},$ and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\overline{PM}.$ If $AM = 180,$ find $LP.$

How many positive integers $N$ less than $1000$ are there such that the equation $x^{\lfloor x \rfloor} = N$ has a solution for $x?$ (The notation $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x.$)

The sequence $(a_n)$ satisfies $a_1 = 1$ and $5^{(a_{n+1} - a_n)} - 1 = \frac{1}{n + \frac{2}{3}}$ for $n \ge 1.$ Let $k$ be the least integer greater than $1$ for which $a_k$ is an integer. Find $k.$

Let $S = \{2^0, 2^1, 2^2, \ldots, 2^{10}\}.$ Consider all possible positive differences of pairs of elements of $S.$ Let $N$ be the sum of all of these differences. Find the remainder when $N$ is divided by $1000.$

A game show offers a contestant three prizes $A,$ $B$ and $C,$ each of which is worth a whole number of dollars from $\$1$ to $\$9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order $A,$ $B,$ $C.$ As a hint, the digits of the three prices are given. On a particular day, the digits given were $1,$ $1,$ $1,$ $1,$ $3,$ $3,$ $3.$ Find the total number of possible guesses for all three prizes consistent with the hint.

The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a Martian must occupy chair $1$ and an Earthling must occupy chair $15.$ Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $N \cdot (5!)^3.$ Find $N.$

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x, y)$ such that $41x + y = 2009.$ Find the number of such distinct triangles whose area is a positive integer.

In right $\triangle ABC$ with hypotenuse $\overline{AB},$ $AC = 12,$ $BC = 35,$ and $\overline{CD}$ is the altitude to $\overline{AB}.$ Let $\omega$ be the circle having $\overline{CD}$ as a diameter. Let $I$ be a point outside $\triangle ABC$ such that $\overline{AI}$ and $\overline{BI}$ are both tangent to circle $\omega.$ The ratio of the perimeter of $\triangle ABI$ to the length $AB$ can be expressed in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

The terms of the sequence $(a_i)$ defined by $a_{n+2} = \frac{a_n + 2009}{1 + a_{n+1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $a_1 + a_2.$

For $t = 1,$ $2,$ $3,$ $4,$ define $S_t = \sum_{i=1}^{350} a_i^t,$ where $a_i \in \{1, 2, 3, 4\}.$ If $S_1 = 513$ and $S_4 = 4745,$ find the minimum possible value for $S_2.$

In triangle $ABC,$ $AB = 10,$ $BC = 14,$ and $CA = 16.$ Let $D$ be a point in the interior of $\overline{BC}.$ Let $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD,$ respectively. The circumcircles of triangles $BI_BD$ and $CI_CD$ meet at distinct points $P$ and $D.$ The maximum possible area of $\triangle BPC$ can be expressed in the form $a - b\sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c.$