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 $1000$ .

Solution

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?

Solution

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?

Solution

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?

Solution

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 $n-S(n)\le 2007$ ?

Solution

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 $a+b+c$ ?

Solution

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 $Q_{1003}$ ?

Solution

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 $x^3+y^3$ .

Solution

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 $ABC$ ?