# Difference between revisions of "2002 AMC 12B Problems"

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== Problem 4 == | == Problem 4 == | ||

− | Let <math>n</math> be a positive integer such that <math>\frac 12 + \frac 13 + \frac 17 + \frac 1n</math> is an integer. Which of the following statements is '''not ''' true: | + | Let <math>n</math> be a positive integer such that <math>\frac 12 + \frac 13 + \frac 17 + \frac 1n</math> is an integer. Which of the following statements is '''not''' true: |

<math>\mathrm{(A)}\ 2\ \text{divides\ }n | <math>\mathrm{(A)}\ 2\ \text{divides\ }n |

## Revision as of 22:03, 19 August 2012

## 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 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 2

What is the value of

when ?

## Problem 3

For how many positive integers is a prime number?

## Problem 4

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

## Problem 5

Let and be the degree measures of the five angles of a pentagon. Suppose that and and form an arithmetic sequence. Find the value of .

## Problem 6

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

## Problem 7

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

## Problem 8

Suppose July of year has five Mondays. Which of the following must occur five times in August of year ? (Note: Both months have 31 days.)

## Problem 9

If are positive real numbers such that form an increasing arithmetic sequence and form a geometric sequence, then is

## Problem 10

How many different integers can be expressed as the sum of three distinct members of the set ?

## Problem 11

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

## Problem 12

For how many integers is the square of an integer?

## Problem 13

The sum of consecutive positive integers is a perfect square. The smallest possible value of this sum is

## Problem 14

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

## Problem 15

How many four-digit numbers have the property that the three-digit number obtained by removing the leftmost digit is one ninth of ?

## Problem 16

Juan rolls a fair regular octahedral die marked with the numbers through . Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?

## Problem 17

Andy’s lawn has twice as much area as Beth’s lawn and three times as much area 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 18

A point is randomly selected from the rectangular region with vertices . What is the probability that is closer to the origin than it is to the point ?

## Problem 19

If and are positive real numbers such that and , then is

## Problem 20

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

## Problem 21

For all positive integers less than , let

Calculate .

## Problem 22

For all integers greater than , define . Let and . Then equals

## Problem 23

In , we have and . Side and the median from to have the same length. What is ?

## Problem 24

A convex quadrilateral with area contains a point in its interior such that . Find the perimeter of .

## Problem 25

Let , and let denote the set of points in the coordinate plane such that The area of is closest to