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Difference between revisions of "2014 AMC 10B Problems"

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[[2014 AMC 10B Problems/Problem 1|Solution]]
 
[[2014 AMC 10B Problems/Problem 1|Solution]]
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==Problem 2==
 
==Problem 2==
 
What is <math>\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}</math>?
 
What is <math>\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}</math>?
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[[2014 AMC 10B Problems/Problem 2|Solution]]
 
[[2014 AMC 10B Problems/Problem 2|Solution]]
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==Problem 3==
 
==Problem 3==
  
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[[2014 AMC 10B Problems/Problem 3|Solution]]
 
[[2014 AMC 10B Problems/Problem 3|Solution]]
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==Problem 4==
 
==Problem 4==
  
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[[2014 AMC 10B Problems/Problem 4|Solution]]
 
[[2014 AMC 10B Problems/Problem 4|Solution]]
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==Problem 5==
 
==Problem 5==
  
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[[2014 AMC 10B Problems/Problem 5|Solution]]
 
[[2014 AMC 10B Problems/Problem 5|Solution]]
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==Problem 6==
 
==Problem 6==
  
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[[2014 AMC 10B Problems/Problem 6|Solution]]
 
[[2014 AMC 10B Problems/Problem 6|Solution]]
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==Problem 7==
 
==Problem 7==
  
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[[2014 AMC 10B Problems/Problem 7|Solution]]
 
[[2014 AMC 10B Problems/Problem 7|Solution]]
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==Problem 8==
 
==Problem 8==
 
A truck travels <math>\frac{b}{6}</math> feet every <math>t</math> seconds. There are <math>3</math> feet in a yard. How many yards does the truck travel in <math>3</math> minutes?
 
A truck travels <math>\frac{b}{6}</math> feet every <math>t</math> seconds. There are <math>3</math> feet in a yard. How many yards does the truck travel in <math>3</math> minutes?
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[[2014 AMC 10B Problems/Problem 8|Solution]]
 
[[2014 AMC 10B Problems/Problem 8|Solution]]
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==Problem 9==
 
==Problem 9==
 
For real numbers <math>w</math> and <math>z</math>,  
 
For real numbers <math>w</math> and <math>z</math>,  
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[[2014 AMC 10B Problems/Problem 9|Solution]]
 
[[2014 AMC 10B Problems/Problem 9|Solution]]
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==Problem 10==
 
==Problem 10==
  
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[[2014 AMC 10B Problems/Problem 11|Solution]]
 
[[2014 AMC 10B Problems/Problem 11|Solution]]
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==Problem 12==
 
==Problem 12==
 
The largest divisor of <math>2,014,000,000</math> is itself. What is its fifth-largest divisor?
 
The largest divisor of <math>2,014,000,000</math> is itself. What is its fifth-largest divisor?
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[[2014 AMC 10B Problems/Problem 12|Solution]]
 
[[2014 AMC 10B Problems/Problem 12|Solution]]
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==Problem 13==
 
==Problem 13==
  
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[[2014 AMC 10B Problems/Problem 14|Solution]]
 
[[2014 AMC 10B Problems/Problem 14|Solution]]
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==Problem 15==
 
==Problem 15==
  
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[[2014 AMC 10B Problems/Problem 16|Solution]]
 
[[2014 AMC 10B Problems/Problem 16|Solution]]
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==Problem 17==
 
==Problem 17==
 
What is the greatest power of <math>2</math> that is a factor of <math>10^{1002} - 4^{501}</math>?
 
What is the greatest power of <math>2</math> that is a factor of <math>10^{1002} - 4^{501}</math>?
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[[2014 AMC 10B Problems/Problem 17|Solution]]
 
[[2014 AMC 10B Problems/Problem 17|Solution]]
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==Problem 18==
 
==Problem 18==
 
A list of <math>11</math> positive integers has a mean of <math>10</math>, a median of <math>9</math>, and a unique mode of <math>8</math>. What is the largest possible value of an integer in the list?
 
A list of <math>11</math> positive integers has a mean of <math>10</math>, a median of <math>9</math>, and a unique mode of <math>8</math>. What is the largest possible value of an integer in the list?
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[[2014 AMC 10B Problems/Problem 18|Solution]]
 
[[2014 AMC 10B Problems/Problem 18|Solution]]
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==Problem 19==
 
==Problem 19==
 
Two concentric circles have radii <math>1</math> and <math>2</math>. 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?
 
Two concentric circles have radii <math>1</math> and <math>2</math>. 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?
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[[2014 AMC 10B Problems/Problem 19|Solution]]
 
[[2014 AMC 10B Problems/Problem 19|Solution]]
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==Problem 20==
 
==Problem 20==
 
For how many integers <math>x</math> is the number <math>x^4 - 51x^2 + 50</math> negative?
 
For how many integers <math>x</math> is the number <math>x^4 - 51x^2 + 50</math> negative?
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[[2014 AMC 10B Problems/Problem 20|Solution]]
 
[[2014 AMC 10B Problems/Problem 20|Solution]]
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==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>?
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[[2014 AMC 10B Problems/Problem 21|Solution]]
 
[[2014 AMC 10B Problems/Problem 21|Solution]]
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==Problem 22==
 
==Problem 22==
  

Revision as of 15:08, 20 February 2014

Problem 1

Leah has $13$ 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?

$\textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 41$

Solution

Problem 2

What is $\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$?

$\textbf {(A) } 16 \qquad \textbf {(B) } 24 \qquad \textbf {(C) } 32 \qquad \textbf {(D) } 48 \qquad \textbf {(E) } 64$

Solution

Problem 3

Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Randy's trip?

$\textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7}$

Solution

Problem 4

Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?

$\textbf {(A) } \frac{3}{2} \qquad \textbf {(B) } \frac{5}{3} \qquad \textbf {(C) } \frac{7}{4} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \frac{13}{4}$

Solution

Problem 5

Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width of each pane is 5 : 2, and the borders around and between the panes are 2 inches wide. In inches, what is the side length of the square window?

$\textbf {(A) } 26 \qquad \textbf {(B) } 28 \qquad \textbf {(C) } 30 \qquad \textbf {(D) } 32 \qquad \textbf {(E) } 34$

Solution

Problem 6

Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?

$\textbf {(A) } 33 \qquad \textbf {(B) } 34 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 38 \qquad \textbf {(E) } 39$

Solution

Problem 7

Suppose $A>B>0$ and $A$ is $x\%$ greater than $B$. What is $x$?

$\textbf {(A) } 100(\frac{A-B}{B}) \qquad \textbf {(B) } 100(\frac{A+B}{B}) \qquad \textbf {(C) } 100(\frac{A+B}{A})\qquad \textbf {(D) } 100(\frac{A-B}{A}) \qquad \textbf {(E) } 100(\frac{A}{B})$

Solution

Problem 8

A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes?

$\textbf {(A) } \frac{b}{1080t} \qquad \textbf {(B) } \frac{30t}{b} \qquad \textbf {(C) } \frac{30b}{t}\qquad \textbf {(D) } \frac{10t}{b} \qquad \textbf {(E) } \frac{10b}{t}$

Solution

Problem 9

For real numbers $w$ and $z$, \[\frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014.\] What is $\frac{w+z}{w-z}$?

$\texbf{(A) } -2014 \qquad\texbf{(B) } \frac{-1}{2014} \qquad\texbf{(C) } \frac{1}{2014} \qquad\texbf{(D) } 1 \qquad\texbf{(E) } 2014$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 10

Problem 11

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

(1) two successive $15\%$ discounts (2) three successive $10\%$ discounts (3) a $25\%$ discount followed by a $5\%$ discount

What is the possible positive integer value of $n$?

$\textbf{(A)}\ \ 27\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}}\ 31\qquad\textbf{(E)}\ 33$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 12

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

$\textbf {(A) } 125, 875, 000 \qquad \textbf {(B) } 201, 400, 000 \qquad \textbf {(C) } 251, 750, 000 \qquad \textbf {(D) } 402, 800, 000 \qquad \textbf {(E) } 503, 500, 000$

Solution

Problem 13

Problem 14

Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \ge 1$ and $a + b + c \ge 7$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2 + b^2 + c^2$ ?

$\textbf {(A) } 26 \qquad \textbf {(B) } 27 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 37 \qquad \textbf {(E) }41$

Solution

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?

$\textbf {(A) } \frac{1}{36} \qquad \textbf {(B) } \frac{7}{72} \qquad \textbf {(C) } \frac{1}{9}\qquad \textbf {(D) } \frac{5}{36} \qquad \textbf {(E) } \frac{1}{6}$

Solution

Problem 17

What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?

$\textbf{(A) } 2^{1002} \qquad\textbf{(B) } 2^{1003} \qquad\textbf{(C) } 2^{1004} \qquad\textbf{(D) } 2^{1005} \qquad\textbf{(E) }2^{1006}$

Solution

Problem 18

A list of $11$ positive integers has a mean of $10$, a median of $9$, and a unique mode of $8$. What is the largest possible value of an integer in the list?

$\textbf {(A) } 24 \qquad \textbf {(B) } 30 \qquad \textbf {(C) } 31\qquad \textbf {(D) } 33 \qquad \textbf {(E) } 35$

Solution

Problem 19

Two concentric circles have radii $1$ and $2$. 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?

$\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{2-\sqrt{2}}{2}\qquad\textbf{(D) }\frac{1}{3}\qquad\textbf{(E) }\frac{1}{2}\qquad$

Solution

Problem 20

For how many integers $x$ is the number $x^4 - 51x^2 + 50$ negative?

$\textbf {(A) } 8 \qquad \textbf {(B) } 10 \qquad \textbf {(C) } 12\qquad \textbf {(D) } 14 \qquad \textbf {(E) }16$

Solution

Problem 21

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

$\textbf{(A) } 10\sqrt{6} \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 8\sqrt{10} \qquad\textbf{(D) } 18\sqrt{2} \qquad\textbf{(E) } 26$

Solution

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 $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

$\textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5$

Solution

Problem 25

In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. what is the probability that the frog will escape being eaten by the snake?

$\textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2}$

Solution

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