## 2014AMC10B真题

**答案解析请参考文末**

## 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

Peter 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 Peter'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

Camden constructs a square window using equal-size panes of glass, as shown. The ratio of the height to width for each pane is , and the borders around and between the panes are inches wide. In inches, what is the side length of the square window?

## 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 is greater than . What is ?

## Problem 8

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

## Problem 9

For real numbers and ,What is ?

## Problem 10

In the addition shown below and are distinct digits. How many different values are possible for ?

## Problem 11

For the consumer, a single discount of is more advantageous than any of the following discounts:

(1) two successive discounts

(2) three successive discounts

(3) a discount followed by a discount

What is the smallest possible positive integer value of ?

## Problem 12

The largest divisor of is itself. What is its fifth largest divisor?

## Problem 13

Six regular hexagons surround a regular hexagon of side length as shown. What is the area of ?

## Problem 14

Danica drove her new car on a trip for a whole number of hours, averaging miles per hour. At the beginning of the trip, miles was displayed on the odometer, where is a 3-digit number with and . At the end of the trip, the odometer showed miles. What is ?

## Problem 15

In rectangle , and points and lie on so that and trisect as shown. What is the ratio of the area of to the area of rectangle ?

Solution

## 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?

Solution

## Problem 17

What is the greatest power of that is a factor of ?

## 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

Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?

## Problem 23

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?

## 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 without being eaten by the snake?

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