# Difference between revisions of "Mock AIME 6 2006-2007 Problems"

## Problem 1

Let $T$ be the sum of all positive integers of the form $2^r\cdot3^s$, where $r$ and $s$ are nonnegative integers that do not exceed $4$. Find the remainder when $T$ is divided by $1000$.

## Problem 2

Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degrees, formed by the intersections of the diagonals in the interior of the octagon?

## Problem 3

Alvin, Simon, and Theodore are running around a $1000$-meter circular track starting at different positions. Alvin is running in the opposite direction of Simon and Theodore. He is also the fastest, running twice as fast as Simon and three times as fast as Theodore. If Alvin meets Simon for the first time after running $312$ meters, and Simon meets Theodore for the first time after running $2526$ meters, how far apart along the track (shorter distance) did Alvin and Theodore meet?

## Problem 4

Let $R$ be a set of $13$ points in the plane, no three of which lie on the same line. At most how many ordered triples of points $(A,B,C)$ in $R$ exist such that $\angle ABC$ is obtuse?

## Problem 5

Let $S(n)$ be the sum of the squares of the digits of $n$. How many positive integers $n>2007$ satisfy the inequality $n-S(n)\le 2007$?

## Problem 6

$C_1$ is a circle with radius $164$ and $C_2$ is a circle internally tangent to $C_1$ that passes through the center of $C_1$. $\overline{AB}$ is a chord in $C_1$ of length $320$ tangent to $C_2$ at $D$ where $AD>BD$. Given that $BD=a-b\sqrt{c}$ where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime, what is $a+b+c$?

## Problem 7

Let $P_n(x)=1+x+x^2+\cdots+x^n$ and $Q_n(x)=P_1\cdot P_2\cdots P_n$ for all integers $n\ge 1$. How many more distinct complex roots does $Q_{1004}$ have than $Q_{1003}$?

## Problem 8

A sequence of positive reals defined by $a_0=x$, $a_1=y$, and $a_n\cdot a_{n+2}=a_{n+1}$ for all integers $n\ge 0$. Given that $a_{2007}+a_{2008}=3$ and $a_{2007}\cdot a_{2008}=\frac 13$, find $x^3+y^3$.

## Problem 9

$ABC$ is a triangle with integer side lengths. Extend $\overline{AC}$ beyond $C$ to point $D$ such that $CD=120$. Similarly, extend $\overline{CB}$ beyond $B$ to point $E$ such that $BE=112$ and $\overline{BA}$ beyond $A$ to point $F$ such that $AF=104$. If triangles $CBD$, $BAE$, and $ACF$ all have the same area, what is the minimum possible area of triangle $ABC$?

## Problem 10

Given a point $P$ in the coordinate plane, let $T_k(P)$ be the $90\degree$ (Error compiling LaTeX. ! Undefined control sequence.) rotation of $P$ around the point $(2000-k,k)$. Let $P_0$ be the point $(2007,0)$ and $P_{n+1}=T_n(P_n)$ for all integers $n\ge 0$. If $P_m$ has a $y$-coordinate of $433$, what is $m$?

## Problem 11

Each face of an octahedron is randomly colored blue or red. A caterpillar is on a vertex of the octahedron and wants to get to the opposite vertex by traversing the edges. The probability that it can do so without traveling along an edge that is shared by two faces of the same color is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

## Problem 12

Let $x_k$ be the largest positive rational solution $x$ to the equation $(2007-x)(x+2007^{-k})^k=1$ for all integers $k\ge 2$. For each $k$, let $x_k=\frac{a_k}{b_k}$, where $a_k$ and $b_k$ are relatively prime positive integers. If $$S=\sum_{k=2}^{2007} (2007b_k-a_k),$$ what is the remainder when $S$ is divided by $1000$?

## Problem 13

Consider two circles of different sizes that do not intersect. The smaller circle has center $O$. Label the intersection of their common external tangents $P$. A common internal tangent interesects the common external tangents at points $A$ and $B$. Given that the radius of the larger circle is $11$, $PO=3$, and $AB=20\sqrt{2}$, what is the square of the area of triangle $PBA$?

## Problem 14

A rational $\frac{1}{k}$, where $k$ is a positive integer, is said to be $\textit{n-unsound}$ if its base $N$ representation terminates. Let $S_n$ be the set of all $\textit{n-unsound}$ rationals. The sum of all the elements in the union set $S_2\cupS_3\cup\cdots\cup S_{14}$ (Error compiling LaTeX. ! Undefined control sequence.) is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

## Problem 15

For any finite sequence of positive integers $A=(a_1,a_2,\cdots,a_n)$, let $f(A)$ be the sequence of the differences between consecutive terms of $A$. i.e. $f(A)=(a_2-a_1,a_3-a_2,\cdots,a_n-a_{n-1})$. Let $F^k(A)$ denote $F$ applied $k$ times to $A$. If all of the sequences $A, f(A), f^2(A),\cdots, f^{n-2}(A)$ are strictly increasing and the only term of $f^{n01}(A)$ is $1$, we call the sequence $A$ $\textit{superpositive}$. How many sequences $A$ with at least two terms and no terms exceeding $18$ are $\textit{superpositive}$?