# Difference between revisions of "2006 AMC 12B Problems"

(→Problem 10) |
(→Problem 23) |
||

Line 301: | Line 301: | ||

== Problem 23 == | == Problem 23 == | ||

− | Isosceles <math>\triangle ABC</math> has a right angle at <math>C</math>. Point <math>P</math> is inside <math>\triangle ABC</math>, such that <math>PA=11</math>, <math>PB=7</math>, and <math>PC=6</math>. Legs <math>\overline{AC}</math> and <math>\overline{BC}</math> have length <math>s=\sqrt{a+b\sqrt{2} | + | Isosceles <math>\triangle ABC</math> has a right angle at <math>C</math>. Point <math>P</math> is inside <math>\triangle ABC</math>, such that <math>PA=11</math>, <math>PB=7</math>, and <math>PC=6</math>. Legs <math>\overline{AC}</math> and <math>\overline{BC}</math> have length <math>s=\sqrt{a+b\sqrt{2}}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? |

<asy> | <asy> |

## Revision as of 00:31, 20 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

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

Mary is about to pay for five items at the grocery store. The prices of the items are , , , , and . Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the that she will receive in change?

## Problem 5

John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?

## Problem 6

Francesca uses 100 grams of lemon juice, 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 7

Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?

## Problem 8

The lines and intersect at the point . What is ?

## Problem 9

How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?

## 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

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 amount of cream in Joe's coffee to that in JoAnn's coffee?

## Problem 12

The parabola has vertex and -intercept , where . What is ?

## Problem 13

Rhombus is similar to rhombus . The area of rhombus is 24, and $\angle BAD \equal{} 60^\circ$ (Error compiling LaTeX. ! Undefined control sequence.). What is the area of rhombus ?

## Problem 14

Elmo makes sandwiches for a fundraiser. For each sandwich he uses globs of peanut butter at cents per glob and blobs of jam at cents per glob. The cost of the peanut butter and jam to make all the sandwiches is . Assume that , and are all positive integers with . What is the cost of the jam Elmo uses to make the sandwiches?

## Problem 15

Circles with centers and have radii 2 and 4, respectively, and are externally tangent. Points and are on the circle centered at , and points and are on the circle centered at , such that and are common external tangents to the circles. What is the area of hexagon ?

## Problem 16

Regular hexagon has vertices and at and , respectively. What is its area?

## Problem 17

For a particular peculiar pair of dice, the probabilities of rolling , , , , and on each die are in the ratio . What is the probability of rolling a total of on the two dice?

## Problem 18

An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?

## Problem 19

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?

## Problem 20

Let be chosen at random from the interval . What is the probability that ? Here denotes the greatest integer that is less than or equal to .

## Problem 21

Rectangle has area . An ellipse with area passes through and and has foci at and . What is the perimeter of the rectangle? (The area of an ellipse is where and are the lengths of the axes.)

## Problem 22

Suppose , and are positive integers with , and , where and are integers and is not divisible by . What is the smallest possible value of ?

## Problem 23

Isosceles has a right angle at . Point is inside , such that , , and . Legs and have length , where and are positive integers. What is ?

## Problem 24

Let be the set of all points in the coordinate plane such that and . What is the area of the subset of for which ?

## Problem 25

A sequence of non-negative integers is defined by the rule for . If , and , how many different values of are possible?