# 2002 AMC 10B Problems

## Contents

- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also

## Problem 1

The ratio is:

## Problem 2

For the nonzero numbers and define Find .

## Problem 3

The arithmetic mean of the nine numbers in the set is a -digit number , all of whose digits are distinct. The number does not contain the digit

## Problem 4

What is the value of

when ?

## Problem 5

Circles of radius and are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.

## Problem 6

For how many positive integers is a prime number?

## Problem 7

Let be a positive integer such that is an integer. Which of the following statements is **not** true?

## Problem 8

Suppose July of year has five Mondays. Which of the following must occurs five times in the August of year ? (Note: Both months have days.)

## Problem 9

Using the letters , , , , and , we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" occupies position

## Problem 10

Suppose that and are nonzero real numbers, and that the equation has positive solutions and . Then the pair is

## Problem 11

The product of three consecutive positive integers is times their sum. What is the sum of the squares?

## Problem 12

For which of the following values of does the equation have no solution for ?

## Problem 13

Find the value(s) of such that is true for all values of .

## Problem 14

The number is the square of a positive integer . In decimal representation, the sum of the digits of is

## Problem 15

The positive integers , , , and are all prime numbers. The sum of these four primes is

## Problem 16

For how many integers is the square of an integer?

## Problem 17

A regular octagon has sides of length two. Find the area of .

## Problem 18

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

## Problem 19

Suppose that is an arithmetic sequence with What is the value of

## Problem 20

Let and be real numbers such that and Then is

## Problem 21

Andy's lawn has twice as much area as Beth's lawn and three times as much as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first?

## Problem 22

Let be a right-angled triangle with . Let and be the midpoints of the legs and , respectively. Given and , find .

## Problem 23

Let be a sequence of integers such that and for all positive integers and Then is

## Problem 24

Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point vertical feet above the bottom?

## Problem 25

When is appended to a list of integers, the mean is increased by . When is appended to the enlarged list, the mean of the enlarged list is decreased by . How many integers were in the original list?