# Difference between revisions of "2006 AMC 10B Problems"

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== Problem 18 == | == Problem 18 == | ||

+ | Let <math> a_1 , a_2 , ... </math> be a sequence for which | ||

+ | |||

+ | <math> a_1=2 </math> , <math> a_2=3 </math>, and <math>a_n=\frac{a_{n-1}}{a_{n-2}} </math> for each positive integer <math> n \ge 3 </math>. | ||

+ | |||

+ | What is <math> a_{2006} </math>? | ||

+ | |||

+ | <math> \mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \frac{3}{2}\qquad \mathrm{(D) \ } 2\qquad \mathrm{(E) \ } 3 </math> | ||

[[2006 AMC 12B Problems/Problem 18|Solution]] | [[2006 AMC 12B Problems/Problem 18|Solution]] |

## Revision as of 15:45, 13 July 2006

## 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

What is ?

## Problem 2

For real numbers and , define . What is ?

## Problem 3

A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?

## Problem 4

Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?

## Problem 5

A rectangle and a rectangle are contained within a square without overlapping at any point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?

## Problem 6

A region is bounded by semicircular arcs constructed on the side of a square whose sides measure , as shown. What is the perimeter of this region?

## Problem 7

Which of the folowing is equivalent to when

## Problem 8

A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?

## Problem 9

Francesca uses 100 grams of lemon juce, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?

## Problem 10

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?

## Problem 11

What is the tens digit in the sum

## Problem 12

The lines and intersect at the point . What is ?

## Problem 13

Joe and JoAnn each bought 12 ounces of coffee in a 16 ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the ammount of cream in Joe's coffee to that in JoAnn's coffee?

## Problem 14

Let and be the roots of the equation . Suppose that and are the roots of the equation . What is ?

## Problem 15

Rhombus is similar to rhombus . The area of rhombus is and . What is the area of rhombus ?

## Problem 16

## Problem 17

Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same?

## Problem 18

Let be a sequence for which

, , and for each positive integer .

What is ?