# Mock AIME 4 2006-2007 Problems

## Contents

## Problem 1

Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits he wrote (the 998th, 999th, and 1000th digits, in that order).

## Problem 2

Two points and are chosen on the graph of , with . The points and trisect , with . Through a horizontal line is drawn to cut the curve at . Find if and .

## Problem 3

Find the largest prime factor of the smallest positive integer such that are distinct integers such that the polynomial has exactly nonzero coefficients.

## Problem 4

Points , , and are on the circumference of a unit circle so that the measure of is , the measure of is , and the measure of is . The area of the triangular shape bounded by and line segments and can be written in the form , where and are relatively prime positive integers. Find .

## Problem 5

How many 10-digit positive integers have all digits either 1 or 2, and have two consecutive 1's?

## Problem 6

For how many positive integers does there exist a regular -sided polygon such that the number of diagonals is a nonzero perfect square?

## Problem 7

Find the remainder when is divided by 1000.

## Problem 8

The number of increasing sequences of positive integers such that is even for can be expressed as for some positive integers . Compute the remainder when is divided by 1000.

## Problem 9

Compute the smallest positive integer such that the fraction

is reducible.

## Problem 10

Compute the remainder when

is divided by 1000.

## Problem 11

Let be an equilateral triangle. Two points and are chosen on and , respectively, such that . Let be the intersection of and . The area of is 13 and the area of is 3. If , where , , and are relatively prime positive integers, compute .

## Problem 12

The number of partitions of 2007 that have an even number of even parts can be expressed as , where and are positive integers and is prime. Find the sum of the digits of .

## Problem 13

The sum

can be written in the form , where . Compute the remainder when is divided by 100.

## Problem 14

Let be the arithmetic mean of all positive integers such that

.

Find the greatest integer less than or equal to .

## Problem 15

Triangle has sides , , and of length 43, 13, and 48, respectively. Let be the circle circumscribed around and let be the intersection of and the perpendicular bisector of that is not on the same side of as . The length of can be expressed as , where and are positive integers and is not divisible by the square of any prime. Find the greatest integer less than or equal to .