# 2006 iTest Problems

## Contents

## Multiple Choice Section

### Problem 1

Find the number of positive integral divisors of 2006.

### Problem 2

Find the harmonic mean of 10 and 20.

### Problem 3

Let be distinct positive integers such that the product . What is the largest possible value of the sum ?

### Problem 4

Four couples go ballroom dancing one evening. Their first names are Henry, Peter, Louis, Roger, Elizabeth, Jeanne, Mary, and Anne. If Henry's wife is not dancing with her husband (but with Elizabeth's husband), Roger and Anne are not dancing, Peter is playing the trumpet, and Mary is playing the piano, and Anne's husband is not Peter, who is Roger's wife?

### Problem 5

A line has y-intercept and forms a right angle to the line . Find the x-intercept of the line.

### Problem 6

What is the remainder when is divided by 7?

### Problem 7

The sum of consecutive integers is . Find the second largest integer.

### Problem 8

The point is a point on a circle with center . Perpendicular lines are drawn from to perpendicular diameters, and , meeting them at points and , respectively. If the diameter of the circle is , what is the length of ?

### Problem 9

If and is in the third quadrant, what is the absolute value of ?

### Problem 10

Find the number of elements in the first rows of Pascal's Triangle that are divisible by .

### Problem 11

Find the radius of the inscribed circle of a triangle with sides of length , , and .

### Problem 12

What is the highest possible probability of getting of these multiple choice questions correct, given that you don't know how to work any of them and are forced to blindly guess on each one?

### Problem 13

Suppose that are three distinct prime numbers such that . Find the maximum possible value for the product .

### Problem 14

Find , where is the smallest positive integer such that leaves a remainder of when divided by , , and .

### Problem 15

How many integers between and , inclusive, are perfect squares?