# Difference between revisions of "2014 AMC 10B Problems"

(→Problem 20) |
(→Problem 21) |
||

Line 79: | Line 79: | ||

==Problem 21== | ==Problem 21== | ||

Trapezoid <math>ABCD</math> has parallel sides <math>\overline{AB}</math> of length <math>33</math> and <math>\overline{CD}</math> of length <math>21</math>. The other two sides are of lengths <math>10</math> and <math>14</math>. The angles at <math>A</math> and <math>B</math> are acute. What is the length of the shorter diagonal of <math>ABCD</math>? | Trapezoid <math>ABCD</math> has parallel sides <math>\overline{AB}</math> of length <math>33</math> and <math>\overline{CD}</math> of length <math>21</math>. The other two sides are of lengths <math>10</math> and <math>14</math>. The angles at <math>A</math> and <math>B</math> are acute. What is the length of the shorter diagonal of <math>ABCD</math>? | ||

+ | |||

+ | <math> \textbf{(A) } 10\sqrt{6} \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 8\sqrt{10} \qquad\textbf{(D) } 18\sqrt{2} \qquad\textbf{(E) } 26</math> | ||

==Problem 22== | ==Problem 22== |

## Revision as of 13:25, 20 February 2014

## 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

## Problem 1

Leah has coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?

## Problem 2

What is ?

## Problem 3

Randy drove the first third of his trip on a gravel road, the next miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Randy's trip?

## Problem 4

Susie pays for muffins and bananas. Calvin spends twice as much paying for muffins and bananas. A muffin is how many times as expensive as a banana?

## Problem 5

## Problem 6

Orvin went to the store with just enough money to buy balloons. When he arrived, he discovered that the store had a special sale on balloons: buy balloon at the regular price and get a second at off the regular price. What is the greatest number of balloons Orvin could buy?

## Problem 7

Suppose and A is $x%$ (Error compiling LaTeX. ! Missing $ inserted.) greater than . What is ?

## Problem 8

A truck travels feet ever seconds. There are feet in a yard. How many yards does the truck travel in minutes?

## Problem 9

## Problem 10

## Problem 11

## Problem 12

The largest divisor of 2,014,000,000 is itself. What is the fifth-largest divisor?

## Problem 13

## Problem 14

## Problem 15

## Problem 16

Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?

## Problem 17

## Problem 18

A list of positive integers has a mean of , a median of , and a unique mode of . What is the largest possible value of an integer in the list?

## Problem 19

Two concentric circles have radii and . Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

## Problem 20

For how many integers is the number negative?

## Problem 21

Trapezoid has parallel sides of length and of length . The other two sides are of lengths and . The angles at and are acute. What is the length of the shorter diagonal of ?

## Problem 22

## Problem 23

## Problem 24

The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is *bad* if it is not true that for every from to one can find a subset of the numbers that appear consecutively on the circle that sum to . Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

## Problem 25

In a small pond there are eleven lily pads in a row labeled through . A frog is sitting on pad . When the frog is on pad , , it will jump to pad with probability and to pad with probability . Each jump is independent of the previous jumps. If the frog reaches pad it will be eaten by a patiently waiting snake. If the frog reaches pad it will exit the pond, never to return. what is the probability that the frog will escape being eaten by the snake?