Mock AIME 2 2006-2007 Problems

Problem 1

A positive integer is called a dragon if it can be written as the sum of four positive integers $a,b,c,$ and $d$ such that $a+4=b-4=4c=d/4.$ Find the smallest dragon.


Problem 2

The set $S$ consists of all integers from $1$ to $2007,$ inclusive. For how many elements $n$ in $S$ is $f(n) = \frac{2n^3+n^2-n-2}{n^2-1}$ an integer?


Problem 3

Let $S$ be the sum of all positive integers $n$ such that $n^2+12n-2007$ is a perfect square. Find the remainder when $S$ is divided by $1000.$


Problem 4

Let $n$ be the smallest positive integer for which there exist positive real numbers $a$ and $b$ such that $(a+bi)^n=(a-bi)^n$. Compute $\frac{b^2}{a^2}$.


Problem 5

Given that $iz^2=1+\frac 2z + \frac{3}{z^2}+\frac{4}{z ^3}+\frac{5}{z^4}+\cdots$ and $z=n\pm \sqrt{-i},$ find $\lfloor 100n \rfloor.$


Problem 6

If $\tan 15^\circ \tan 25^\circ \tan 35^\circ =\tan \theta$ and $0^\circ \le \theta \le 180^\circ,$ find $\theta.$


Problem 7

A right circular cone of base radius $17$cm and slant height $51$cm is given. $P$ is a point on the circumference of the base and the shortest path from $P$ around the cone and back is drawn (see diagram). If the length of this path is $m\sqrt{n},$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Mock AIME 2 2007 Problem8.jpg


Problem 8

The positive integers $x_1, x_2, ... , x_7$ satisfy $x_6 = 144$ and $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n = 1, 2, 3, 4$. Find the last three digits of $x_7$.


Problem 9

In right triangle $ABC,$ $\angle C=90^\circ.$ Cevians $AX$ and $BY$ intersect at $P$ and are drawn to $BC$ and $AC$ respectively such that $\frac{BX}{CX}=\frac23$ and $\frac{AY}{CY}=\sqrt 3.$ If $\tan \angle APB= \frac{a+b\sqrt{c}}{d},$ where $a,b,$ and $d$ are relatively prime and $c$ has no perfect square divisors excluding $1,$ find $a+b+c+d.$


Problem 10

Find the number of solutions, in degrees, to the equation $\sin^{10}x + \cos^{10}x = \frac{29}{16}\cos^4 2x,$ where $0^\circ \le x^\circ \le 2007^\circ.$


Problem 11

Find the sum of the squares of the roots, real or complex, of the system of simultaneous equations

$x+y+z=3,~x^2+y^2+z^2=3,~x^3+y^3+z^3 =3.$


Problem 12

In quadrilateral $ABCD,$ $m \angle DAC= m\angle DBC$ and $\frac{[ADB]}{[ABC]}=\frac12.$ If $AD=4,$ $BC=6$, $BO=1,$ and the area of $ABCD$ is $\frac{a\sqrt{b}}{c},$ where $a,b,c$ are relatively prime positive integers, find $a+b+c.$

Note*: $[ABC]$ and $[ADB]$ refer to the areas of triangles $ABC$ and $ADB.$


Problem 13

In his spare time, Richard Rusczyk shuffles a standard deck of 52 playing cards. He then turns the cards up one by one from the top of the deck until the third ace appears. If the expected (average) number of cards Richard will turn up is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$


Problem 14

In triangle ABC, $AB = 308$ and $AC=35.$ Given that $AD$, $BE,$ and $CF,$ intersect at $P$ and are an angle bisector, median, and altitude of the triangle, respectively, compute the length of $BC.$

Mock AIME 2 2007 Problem14.jpg


Problem 15

A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are colored red. An arrangement of the cubes is $\mathfrak{Intriguing}$ if there is exactly $1$ red unit cube in every $1\times1\times4$ rectangular box composed of $4$ unit cubes. Determine the number of $\mathfrak{Intriguing}$ colorings.