# 2008 AIME II Problems

## Problem 1

Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$.

## Problem 2

Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the $50$-mile mark at exactly the same time. How many minutes has it taken them?

## Problem 3

A block of cheese in the shape of a rectangular solid measures $10$ cm by $13$ cm by $14$ cm. Ten slices are cut from the cheese. Each slice has a width of $1$ cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?

## Problem 4

There exist $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ unique integers $a_k$ ($1\le k\le r$) with each $a_k$ either $1$ or $- 1$ such that $$a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2008.$$ Find $n_1 + n_2 + \cdots + n_r$.

## Problem 5

In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\angle A = 37^\circ$, $\angle D = 53^\circ$, and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $MN$.

## Problem 6

The sequence $\{a_n\}$ is defined by $$a_0 = 1,a_1 = 1, \text{ and } a_n = a_{n - 1} + \frac {a_{n - 1}^2}{a_{n - 2}}\text{ for }n\ge2.$$ The sequence $\{b_n\}$ is defined by $$b_0 = 1,b_1 = 3, \text{ and } b_n = b_{n - 1} + \frac {b_{n - 1}^2}{b_{n - 2}}\text{ for }n\ge2.$$ Find $\frac {b_{32}}{a_{32}}$.

## Problem 7

Let $r$, $s$, and $t$ be the three roots of the equation $$8x^3 + 1001x + 2008 = 0.$$ Find $(r + s)^3 + (s + t)^3 + (t + r)^3$.

## Problem 8

Let $a = \pi/2008$. Find the smallest positive integer $n$ such that $$2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]$$ is an integer.

## Problem 9

A particle is located on the coordinate plane at $(5,0)$. Define a move for the particle as a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Given that the particle's position after $150$ moves is $(p,q)$, find the greatest integer less than of equal to $|p| + |q|$.

## Problem 11

In triangle $ABC$, $AB = AC = 100$, and $BC = 56$. Circle $P$ has radius $16$ and is tangent to $\overbar{AC}$ (Error compiling LaTeX. ! Undefined control sequence.) and $\overbar{BC}$ (Error compiling LaTeX. ! Undefined control sequence.). Circle $Q$ is externally tangent to circle $P$ and is tangent to $\overbar{AB}$ (Error compiling LaTeX. ! Undefined control sequence.) and $\overbar{BC}$ (Error compiling LaTeX. ! Undefined control sequence.). No point of circle $Q$ lies outside of $\bigtriangleup\overbar{ABC}$ (Error compiling LaTeX. ! Undefined control sequence.). The radius of circle $Q$ can be expressed in the form $m - n\sqrt{k}$,where $m$, $n$, and $k$ are positive integers and $k$ is the product of distinct primes. Find $m +nk$.