Difference between revisions of "2014 AMC 10B Problems"
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+ | {{AMC10 Problems|year=2014|ab=B}} | ||
==Problem 1== | ==Problem 1== | ||
− | Leah has <math>13</math> 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 | + | Leah has <math>13</math> 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 as nickels. In cents, how much are Leah's coins worth? |
<math> \textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 41 </math> | <math> \textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 41 </math> | ||
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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>? | ||
− | <math> \textbf {(A) } 16 \qquad \textbf {(B) } 24 \qquad \textbf {(C) } 32 \qquad \textbf {(D) } 48 \qquad \textbf {(E) } 64 </math> | + | <math>\textbf {(A) } 16 \qquad \textbf {(B) } 24 \qquad \textbf {(C) } 32 \qquad \textbf {(D) } 48 \qquad \textbf {(E) } 64</math> |
[[2014 AMC 10B Problems/Problem 2|Solution]] | [[2014 AMC 10B Problems/Problem 2|Solution]] | ||
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==Problem 3== | ==Problem 3== | ||
− | + | Peter drove the first third of his trip on a gravel road, the next <math>20</math> miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Peter's trip? | |
<math> \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}</math> | <math> \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}</math> | ||
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[[2014 AMC 10B Problems/Problem 4|Solution]] | [[2014 AMC 10B Problems/Problem 4|Solution]] | ||
− | |||
==Problem 5== | ==Problem 5== | ||
− | + | Camden constructs a square window using <math> 8 </math> equal-size panes of glass, as shown. The ratio of the height to width for each pane is <math> 5 : 2 </math>, and the borders around and between the panes are <math> 2 </math> inches wide. In inches, what is the side length of the square window? | |
− | + | <asy> | |
− | <math> \textbf {(A) } 26 \qquad \textbf {(B) } 28 \qquad \textbf {(C) } 30 \qquad \textbf {(D) } 32 \qquad \textbf {(E) } 34</math> | + | fill((0,0)--(25,0)--(25,25)--(0,25)--cycle,grey); |
+ | for(int i = 0; i < 4; ++i){ | ||
+ | for(int j = 0; j < 2; ++j){ | ||
+ | fill((6*i+2,11*j+3)--(6*i+5,11*j+3)--(6*i+5,11*j+11)--(6*i+2,11*j+11)--cycle,white); | ||
+ | } | ||
+ | }</asy> | ||
+ | <math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 </math> | ||
[[2014 AMC 10B Problems/Problem 5|Solution]] | [[2014 AMC 10B Problems/Problem 5|Solution]] | ||
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[[2014 AMC 10B Problems/Problem 6|Solution]] | [[2014 AMC 10B Problems/Problem 6|Solution]] | ||
− | |||
==Problem 7== | ==Problem 7== | ||
Suppose <math>A>B>0</math> and <math>A</math> is <math>x\%</math> greater than <math>B</math>. What is <math>x</math>? | Suppose <math>A>B>0</math> and <math>A</math> is <math>x\%</math> greater than <math>B</math>. What is <math>x</math>? | ||
− | <math> \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})</math> | + | <math> \textbf {(A) } 100\left(\frac{A-B}{B}\right) \qquad \textbf {(B) } 100\left(\frac{A+B}{B}\right) \qquad \textbf {(C) } 100\left(\frac{A+B}{A}\right)\qquad \textbf {(D) } 100\left(\frac{A-B}{A}\right) \qquad \textbf {(E) } 100\left(\frac{A}{B}\right)</math> |
[[2014 AMC 10B Problems/Problem 7|Solution]] | [[2014 AMC 10B Problems/Problem 7|Solution]] | ||
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[[2014 AMC 10B Problems/Problem 8|Solution]] | [[2014 AMC 10B Problems/Problem 8|Solution]] | ||
− | |||
==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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What is <math>\frac{w+z}{w-z}</math>? | What is <math>\frac{w+z}{w-z}</math>? | ||
− | <math>\ | + | <math>\textbf{(A) } -2014 \qquad\textbf{(B) } \frac{-1}{2014} \qquad\textbf{(C) } \frac{1}{2014} \qquad\textbf{(D) } 1 \qquad\textbf{(E) } 2014</math> |
[[2014 AMC 10B Problems/Problem 9|Solution]] | [[2014 AMC 10B Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
+ | |||
+ | In the addition shown below <math>A, B, C,</math> and <math>D</math> are distinct digits. How many different values are possible for <math>D</math>? | ||
+ | |||
+ | <cmath> \begin{array}{lr}&ABBCB\\ +& BCADA\\ \hline & DBDDD\end{array} </cmath> | ||
+ | |||
+ | <math> \textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9 </math> | ||
+ | |||
+ | [[2014 AMC 10B Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
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(1) two successive <math>15\%</math> discounts | (1) two successive <math>15\%</math> discounts | ||
+ | |||
(2) three successive <math>10\%</math> discounts | (2) three successive <math>10\%</math> discounts | ||
+ | |||
(3) a <math>25\%</math> discount followed by a <math>5\%</math> discount | (3) a <math>25\%</math> discount followed by a <math>5\%</math> discount | ||
− | What is the possible positive integer value of <math>n</math>? | + | What is the smallest possible positive integer value of <math>n</math>? |
− | <math> \textbf{(A)}\ \ 27\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 29\qquad\textbf{(D) | + | <math> \textbf{(A)}\ \ 27\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}\ 31\qquad\textbf{(E)}\ 33 </math> |
[[2014 AMC 10B Problems/Problem 11|Solution]] | [[2014 AMC 10B Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | The largest divisor of <math>2,014,000,000</math> is itself. What is its fifth | + | The largest divisor of <math>2,014,000,000</math> is itself. What is its fifth largest divisor? |
− | <math> \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 </math> | + | <math> \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 </math> |
[[2014 AMC 10B Problems/Problem 12|Solution]] | [[2014 AMC 10B Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
+ | |||
+ | Six regular hexagons surround a regular hexagon of side length <math>1</math> as shown. What is the area of <math>\triangle ABC</math>? | ||
+ | |||
+ | <asy> | ||
+ | for(int i = 0; i < 6; ++i){ | ||
+ | for(int j = 0; j < 6; ++j){ | ||
+ | draw(sqrt(3)*dir(60*i+30)+dir(60*j)--sqrt(3)*dir(60*i+30)+dir(60*j+60)); | ||
+ | } | ||
+ | } | ||
+ | |||
+ | draw(2*dir(60)--2*dir(180)--2*dir(300)--cycle); | ||
+ | label("A",2*dir(180),dir(180)); | ||
+ | label("B",2*dir(60),dir(60)); | ||
+ | label("C",2*dir(300),dir(300)); | ||
+ | </asy> | ||
+ | |||
+ | |||
+ | <math> \textbf {(A) } 2\sqrt{3} \qquad \textbf {(B) } 3\sqrt{3} \qquad \textbf {(C) } 1+3\sqrt{2} \qquad \textbf {(D) } 2+2\sqrt{3} \qquad \textbf {(E) } 3+2\sqrt{3} </math> | ||
+ | |||
+ | [[2014 AMC 10B Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
− | Danica drove her new car on a trip for a whole number of hours, averaging <math>55</math> miles per hour. At the beginning of the trip, <math>abc</math> miles was displayed on the odometer, where <math>abc</math> is a 3-digit number with <math>a \ge 1</math> and <math>a + b + c \ | + | Danica drove her new car on a trip for a whole number of hours, averaging <math>55</math> miles per hour. At the beginning of the trip, <math>abc</math> miles was displayed on the odometer, where <math>abc</math> is a 3-digit number with <math>a \ge 1</math> and <math>a + b + c \le 7</math>. At the end of the trip, the odometer showed <math>cba</math> miles. What is <math>a^2 + b^2 + c^2</math> ? |
<math> \textbf {(A) } 26 \qquad \textbf {(B) } 27 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 37 \qquad \textbf {(E) }41</math> | <math> \textbf {(A) } 26 \qquad \textbf {(B) } 27 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 37 \qquad \textbf {(E) }41</math> | ||
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==Problem 15== | ==Problem 15== | ||
+ | In rectangle <math>ABCD</math>, <math>DC=2 \cdot CB</math> and points <math>E</math> and <math>F</math> lie on <math>\overline{AB}</math> so that <math>\overline{ED}</math> and <math>\overline{FD}</math> trisect <math>\angle ADC</math> as shown. What is the ratio of the area of <math>\triangle DEF</math> to the area of rectangle <math>ABCD</math>? | ||
+ | |||
+ | <asy> | ||
+ | pair A = (0,1), B = (2,1), C = (2,0), D = (0,0); | ||
+ | pair E = intersectionpoint(A--B,D--2*dir(60)), F = intersectionpoint(A--B,D--3*dir(30)); | ||
+ | |||
+ | draw(A--D--C--B--cycle); | ||
+ | draw(E--D--F); | ||
+ | label("A",A,N); | ||
+ | label("B",B,N); | ||
+ | label("C",C,S); | ||
+ | label("D",D,S); | ||
+ | label("E",E,N); | ||
+ | label("F",F,N); | ||
+ | </asy> | ||
+ | |||
+ | <math> \textbf{(A)}\ \ \frac{\sqrt{3}}{6}\qquad\textbf{(B)}\ \frac{\sqrt{6}}{8}\qquad\textbf{(C)}\ \frac{3\sqrt{3}}{16}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{4} </math> | ||
+ | |||
+ | [[2014 AMC 10B Problems/Problem 15|Solution]] | ||
==Problem 16== | ==Problem 16== | ||
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[[2014 AMC 10B Problems/Problem 16|Solution]] | [[2014 AMC 10B Problems/Problem 16|Solution]] | ||
− | |||
==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]] | ||
− | |||
==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]] | ||
− | |||
==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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<math>\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</math> | <math>\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</math> | ||
+ | [[2014 AMC 10B Problems/Problem 19|Solution]] | ||
==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? | ||
<math> \textbf {(A) } 8 \qquad \textbf {(B) } 10 \qquad \textbf {(C) } 12\qquad \textbf {(D) } 14 \qquad \textbf {(E) }16</math> | <math> \textbf {(A) } 8 \qquad \textbf {(B) } 10 \qquad \textbf {(C) } 12\qquad \textbf {(D) } 14 \qquad \textbf {(E) }16</math> | ||
+ | |||
+ | [[2014 AMC 10B Problems/Problem 20|Solution]] | ||
==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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<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> | <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> | ||
+ | [[2014 AMC 10B Problems/Problem 21|Solution]] | ||
==Problem 22== | ==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? | ||
+ | |||
+ | <math>\text{(A) } \dfrac{1+\sqrt2}4 \quad \text{(B) } \dfrac{\sqrt5-1}2 \quad \text{(C) } \dfrac{\sqrt3+1}4 \quad \text{(D) } \dfrac{2\sqrt3}5 \quad \text{(E) } \dfrac{\sqrt5}3</math> | ||
+ | |||
+ | <asy> | ||
+ | scale(200); | ||
+ | draw(scale(.5)*((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle)); | ||
+ | path p = arc((.25,-.5),.25,0,180)--arc((-.25,-.5),.25,0,180); | ||
+ | draw(p); | ||
+ | p=rotate(90)*p; draw(p); | ||
+ | p=rotate(90)*p; draw(p); | ||
+ | p=rotate(90)*p; draw(p); | ||
+ | draw(scale((sqrt(5)-1)/4)*unitcircle); | ||
+ | </asy> | ||
+ | |||
+ | [[2014 AMC 10B Problems/Problem 22|Solution]] | ||
==Problem 23== | ==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? | ||
+ | |||
+ | <asy> | ||
+ | real r=(3+sqrt(5))/2; | ||
+ | real s=sqrt(r); | ||
+ | real Brad=r; | ||
+ | real brad=1; | ||
+ | real Fht = 2*s; | ||
+ | import graph3; | ||
+ | import solids; | ||
+ | currentprojection=orthographic(1,0,.2); | ||
+ | currentlight=(10,10,5); | ||
+ | revolution sph=sphere((0,0,Fht/2),Fht/2); | ||
+ | //draw(surface(sph),green+white+opacity(0.5)); | ||
+ | //triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));} | ||
+ | triple f(pair t) { | ||
+ | triple v0 = Brad*(cos(t.x),sin(t.x),0); | ||
+ | triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht); | ||
+ | return (v0 + t.y*(v1-v0)); | ||
+ | } | ||
+ | triple g(pair t) { | ||
+ | return (t.y*cos(t.x),t.y*sin(t.x),0); | ||
+ | } | ||
+ | surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2); | ||
+ | surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2); | ||
+ | surface base = surface(g,(0,0),(2pi,Brad),80,2); | ||
+ | draw(sback,gray(0.3)); | ||
+ | draw(sfront,gray(0.5)); | ||
+ | draw(base,gray(0.9)); | ||
+ | draw(surface(sph),gray(0.4));</asy> | ||
+ | |||
+ | <math>\text{(A) } \dfrac32 \quad \text{(B) } \dfrac{1+\sqrt5}2 \quad \text{(C) } \sqrt3 \quad \text{(D) } 2 \quad \text{(E) } \dfrac{3+\sqrt5}2</math> | ||
+ | |||
+ | [[2014 AMC 10B Problems/Problem 23|Solution]] | ||
==Problem 24== | ==Problem 24== | ||
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<math> \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 </math> | <math> \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 </math> | ||
+ | [[2014 AMC 10B Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== | ||
− | In a small pond there are eleven lily pads in a row labeled <math>0</math> through <math>10</math>. A frog is sitting on pad <math>1</math>. When the frog is on pad <math>N</math>, <math>0<N<10</math>, it will jump to pad <math>N-1</math> with probability <math>\frac{N}{10}</math> and to pad <math>N+1</math> with probability <math>1-\frac{N}{10}</math>. Each jump is independent of the previous jumps. If the frog reaches pad <math>0</math> it will be eaten by a patiently waiting snake. If the frog reaches pad <math>10</math> it will exit the pond, never to return. | + | In a small pond there are eleven lily pads in a row labeled <math>0</math> through <math>10</math>. A frog is sitting on pad <math>1</math>. When the frog is on pad <math>N</math>, <math>0<N<10</math>, it will jump to pad <math>N-1</math> with probability <math>\frac{N}{10}</math> and to pad <math>N+1</math> with probability <math>1-\frac{N}{10}</math>. Each jump is independent of the previous jumps. If the frog reaches pad <math>0</math> it will be eaten by a patiently waiting snake. If the frog reaches pad <math>10</math> it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake? |
<math> \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} </math> | <math> \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} </math> | ||
+ | |||
+ | [[2014 AMC 10B Problems/Problem 25|Solution]] | ||
+ | |||
+ | == See also == | ||
+ | {{AMC10 box|year=2014|ab=B|before=[[2014 AMC 10A Problems]]|after=[[2015 AMC 10A Problems]]}} | ||
+ | * [[AMC 10]] | ||
+ | * [[AMC 10 Problems and Solutions]] | ||
+ | * [[2014 AMC 10B]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 21:54, 8 November 2022
2014 AMC 10B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
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
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 as 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
?
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
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?
See also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2014 AMC 10A Problems |
Followed by 2015 AMC 10A Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.