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Mock AIME II 2012 Problems

Problem 1

Given that \[\left(\dfrac{6^2-1}{6^2+11}\right)\left(\dfrac{7^2-2}{7^2+12}\right)\left(\dfrac{8^2-3}{8^2+13}\right)\cdots\left(\dfrac{2012^2-2007}{2012^2+2017}\right)=\dfrac{m}{n},\] where $m$ and $n$ are positive relatively prime integers, find the remainder when $m+n$ is divided by $1000$.

Solution

Problem 2

Let $\{a_n\}$ be a recursion defined such that $a_1=1, a_2=20$, and $a_n=\sqrt{\left| a_{n-1}^2-a_{n-2}^2 \right|}$ where $n\ge 3$, and $n$ is an integer. If $a_m=k$ for $k$ being a positive integer greater than $1$ and $m$ being a positive integer greater than 2, find the smallest possible value of $m+k$.

Solution

Problem 3

The $\textit{digital root}$ of a number is defined as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the $\textit{digital root}$ of $237$ is $3$ ($2+3+7=12, 1+2=3$). Find the $\textit{digital root}$ of $2012^{2012^{2012}}$.

Solution

Problem 4

Let $\triangle ABC$ be a triangle, and let $I_A$, $I_B$, and $I_C$ be the points where the angle bisectors of $A$, $B$, and $C$, respectfully, intersect the sides opposite them. Given that $AI_B=5$, $CI_B=4$, and $CI_A=3$, then the ratio $AI_C:BI_C$ can be written in the form $m/n$ where $m$ and $n$ are positive relatively prime integers. Find $m+n$.

Solution

Problem 5

A fair die with $12$ sides numbered $1$ through $12$ inclusive is rolled $n$ times. The probability that the sum of the rolls is $2012$ is nonzero and is equivalent to the probability that a sum of $k$ is rolled. Find the minimum value of k.

Solution


Problem 6

A circle with radius $5$ and center in the first quadrant is placed so that it is tangent to the $y$-axis. If the line passing through the origin that is tangent to the circle has slope $\dfrac{1}{2}$, then the $y$-coordinate of the center of the circle can be written in the form $\dfrac{m+\sqrt{n}}{p}$ where $m$, $n$, and $p$ are positive integers, and $\text{gcd}(m,p)=1$. Find $m+n+p$.

Solution

Problem 7

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