Question bank AIME 2005

A game uses a deck of $n$ different cards, where $n$ is an integer and $n \ge 6.$ The number of possible sets of $6$ cards that can be drawn from the deck is $6$ times the number of possible sets of $3$ cards that can be drawn. Find $n.$

A hotel packed a breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls, and, once they were wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability that each guest got one roll of each type is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$

An infinite geometric series has sum $2005.$ A new series, obtained by squaring each term of the original series, has sum $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Find the number of positive integers that are divisors of at least one of $10^{10},$ $15^7,$ $18^{11}.$

Determine the number of ordered pairs $(a, b)$ of integers such that $\log_a b + 6\log_b a = 5,$ $2 \le a \le 2005,$ and $2 \le b \le 2005.$

The cards in a stack of $2n$ cards are numbered consecutively from $1$ through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaining cards form pile $B.$ The cards are now restacked into a single stack by taking cards alternately from the tops of pile $B$ and pile $A,$ respectively. In this process, card number $(n + 1)$ is the bottom card of the new stack, card number $1$ is on top of this card, and so on, until piles $A$ and $B$ are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is called *magical*. For example, eight cards form a magical stack because cards number $3$ and number $6$ retain their original positions. Find the number of cards in the magical stack in which card number $131$ retains its original position.

Let $$
\begin{aligned} x &= \scriptsize \frac{4}{(\sqrt{5} + 1)(\sqrt[4]{5} + 1)(\sqrt[8]{5} + 1)(\sqrt[16]{5} + 1)}. \end{aligned}
$$ Find $(x + 1)^{48}.$

Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are externally tangent, and they are both internally tangent to circle $\mathcal{C}_3.$ The radii of $\mathcal{C}_1$ and $\mathcal{C}_2$ are $4$ and $10,$ respectively, and the centers of the three circles are all collinear. A chord of $\mathcal{C}_3$ is also a common external tangent of $\mathcal{C}_1$ and $\mathcal{C}_2.$ Given that the length of the chord is $\frac{m\sqrt{n}}{p},$ 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 $m + n + p.$

For how many positive integers $n$ less than or equal to $1000$ is $$
\begin{aligned} &(\sin t + i \cos t)^n \\ &= \sin nt + i \cos nt \end{aligned}
$$ true for all real $t?$

Given that $\mathcal{O}$ is a regular octahedron, that $\mathcal{C}$ is the cube whose vertices are the centers of the faces of $\mathcal{O},$ and that the ratio of the volume of $\mathcal{O}$ to that of $\mathcal{C}$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$

Let $m$ be a positive integer, and let $a_0,$ $a_1,$ $\ldots,$ $a_m$ be a sequence of real numbers such that $a_0 = 37,$ $a_1 = 72,$ $a_m = 0,$ and $$
a_{k+1} = a_{k-1} - \frac{3}{a_k}
$$ for $k = 1,$ $2,$ $\ldots,$ $m - 1.$ Find $m.$

Square $ABCD$ has center $O,$ $AB = 900,$ $E$ and $F$ are on $\overline{AB}$ with $AE \lt BF$ and $E$ between $A$ and $F,$ $m\angle EOF = 45^\circ,$ and $EF = 400.$ Given that $BF = p + q\sqrt{r},$ where $p,$ $q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p + q + r.$

Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17) = 10$ and $P(24) = 17.$ Given that the equation $P(n) = n + 3$ has two distinct integer solutions $n_1$ and $n_2,$ find the product $n_1 \cdot n_2.$

In triangle $ABC,$ $AB = 13,$ $BC = 15,$ and $CA = 14.$ Point $D$ is on $\overline{BC}$ with $CD = 6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE \cong \angle CAD.$ Given that $BE = \frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers, find $q.$

Let $\omega_1$ and $\omega_2$ denote the circles $x^2 + y^2 + 10x - 24y - 87 = 0$ and $x^2 + y^2 - 10x - 24y + 153 = 0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y = ax$ contains the center of a circle that is internally tangent to $\omega_1$ and externally tangent to $\omega_2.$ Given that $m^2 = \frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers, find $p + q.$

Six congruent circles form a ring with each circle externally tangent to the two circles adjacent to it. All six circles are internally tangent to a circle $\mathcal{C}$ with radius $30.$ Let $K$ be the area of the region inside $\mathcal{C}$ and outside all of the six circles in the ring. Find $\lfloor K\rfloor.$ (The notation $\lfloor K\rfloor$ denotes the greatest integer that is less than or equal to $K.$)

For each positive integer $k,$ let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k.$ For example, $S_3$ is the sequence $1,$ $4,$ $7,$ $\ldots.$ For how many values of $k$ does $S_k$ contain the term $2005?$

How many positive integers have exactly three proper divisors, each of which is less than $50?$ (A *proper divisor* of a positive integer $n$ is a positive integer divisor of $n$ other than $n$ itself.)

The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are $5$ members left over. The director finds that if they are arranged in a rectangular formation with $7$ more rows than columns, the desired result can be obtained. Find the maximum number of members this band can have.

Robert has $4$ indistinguishable gold coins and $4$ indistinguishable silver coins. Each coin has an engraving of a face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the $8$ coins.