# 2008 Mock ARML 1 Problems

## Contents

# Set 1

## Problem 1

Compute all real values of such that .

## Problem 2

A positive integer is a yo-yo if the absolute value of the difference between any two consecutive digits of is at least . Compute the number of -digit yo-yos.

# Set 2

## Problem 3

In regular hexagon with side length , intersects at , and intersects at . Compute the length of .

## Problem 4

There are black balls and white ball in a hat. A turn consists of picking a ball from the hat and replacing it with one of the opposite color. Compute the probability that, after a sequence of turns, there are black balls in the hat before there are white balls.

# Set 3

## Problem 5

The positive real numbers are in arithmetic progression in that order. They also satisfy

Compute the common difference of this arithmetic progression.

## Problem 6

Square has side length . is the midpoint of , and is the midpoint of . is on such that is between and , and . Compute the length of .

# Set 4

## Problem 7

Compute the number of -digit base- positive integer multiples of that are also divisible by when read in base instead of base .

## Problem 8

For positive real numbers ,

Compute .