Difference between revisions of "2014 AMC 10A Problems"
(Undo revision 115472 by Happycupcake (talk)) (Tag: Undo) |
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− | + | ==Problem 1== | |
What is <math>10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?</math> | What is <math>10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?</math> | ||
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[[2014 AMC 10A Problems/Problem 1|Solution]] | [[2014 AMC 10A Problems/Problem 1|Solution]] | ||
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− | |||
==Problem 2== | ==Problem 2== | ||
− | Roy's | + | Roy's cat eats <math>\frac{1}{3}</math> of a can of cat food every morning and <math>\frac{1}{4}</math> of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing <math>6</math> cans of cat food. On what day of the week did Roy's cat finish eating all the cat food in the box? |
<math> \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}\qquad\textbf{(F)}\ \text{Option 6}</math> | <math> \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}\qquad\textbf{(F)}\ \text{Option 6}</math> | ||
− | [[2014 AMC 10A Problems/Problem 2| | + | [[2014 AMC 10A Problems/Problem 2|solution]] |
− | == | + | ==Problem 3== |
− | + | Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for <math>\textdollar 2.50</math> each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs <math>\textdollar 0.75</math> for her to make. In dollars, what is her profit for the day? | |
<math>\textbf{(A)}\ 24\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52</math> | <math>\textbf{(A)}\ 24\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52</math> | ||
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==Problem 4== | ==Problem 4== | ||
− | + | Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? | |
<math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math> | <math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math> | ||
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[[2014 AMC 10A Problems/Problem 4|Solution]] | [[2014 AMC 10A Problems/Problem 4|Solution]] | ||
− | == | + | ==Problem 5== |
On an algebra quiz, <math>10\%</math> of the students scored <math>0</math> points, <math>35\%</math> scored <math>10</math> points, <math>30\%</math> scored <math>15</math> points, and the rest scored <math>30</math> points. What is the difference between the mean and median score of the students' scores on this quiz? | On an algebra quiz, <math>10\%</math> of the students scored <math>0</math> points, <math>35\%</math> scored <math>10</math> points, <math>30\%</math> scored <math>15</math> points, and the rest scored <math>30</math> points. What is the difference between the mean and median score of the students' scores on this quiz? | ||
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==Problem 6== | ==Problem 6== | ||
− | Suppose that | + | Suppose that <math>a</math> cows give <math>b</math> gallons of milk in <math>c</math> days. At this rate, how many gallons of milk will <math>d</math> cows give in <math>e</math> days? |
− | <math>\ \frac{bcde}{a} | + | <math>\textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}</math> |
[[2014 AMC 10A Problems/Problem 6|Solution]] | [[2014 AMC 10A Problems/Problem 6|Solution]] | ||
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\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}</math> | \textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}</math> | ||
− | <math> \textbf{( | + | <math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4</math> |
[[2014 AMC 10A Problems/Problem 7|Solution]] | [[2014 AMC 10A Problems/Problem 7|Solution]] | ||
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==Problem 8== | ==Problem 8== | ||
− | Which of the following numbers is | + | Which of the following numbers is a perfect square? |
<math> \textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2 </math> | <math> \textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2 </math> | ||
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[[2014 AMC 10A Problems/Problem 8|Solution]] | [[2014 AMC 10A Problems/Problem 8|Solution]] | ||
− | == | + | ==Problem 9== |
The two legs of a right triangle, which are altitudes, have lengths <math>2\sqrt3</math> and <math>6</math>. How long is the third altitude of the triangle? | The two legs of a right triangle, which are altitudes, have lengths <math>2\sqrt3</math> and <math>6</math>. How long is the third altitude of the triangle? | ||
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defaultpen(linewidth(0.8)); | defaultpen(linewidth(0.8)); | ||
path hexagon=(2*dir(0))--(2*dir(60))--(2*dir(120))--(2*dir(180))--(2*dir(240))--(2*dir(300))--cycle; | path hexagon=(2*dir(0))--(2*dir(60))--(2*dir(120))--(2*dir(180))--(2*dir(240))--(2*dir(300))--cycle; | ||
− | fill(hexagon, | + | fill(hexagon,grey); |
− | for(int i=0;i<= | + | for(int i=0;i<=5;i=i+1) |
{ | { | ||
− | path arc=2*dir( | + | path arc=2*dir(60*i)--arc(2*dir(60*i),1,120+60*i,240+60*i)--cycle; |
unfill(arc); | unfill(arc); | ||
draw(arc); | draw(arc); | ||
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import graph; | import graph; | ||
size(6cm); | size(6cm); | ||
− | pen dps = linewidth(0.7) + fontsize( | + | pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps); |
pair B = (0,0); | pair B = (0,0); | ||
pair C = (1,0); | pair C = (1,0); | ||
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pair E = rotate(270,A)*B; | pair E = rotate(270,A)*B; | ||
− | pair D = rotate( | + | pair D = rotate(270,E)*A; |
− | pair F = rotate( | + | pair F = rotate(90,A)*C; |
pair G = rotate(90,F)*A; | pair G = rotate(90,F)*A; | ||
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draw(A--B--C--cycle); | draw(A--B--C--cycle); | ||
− | draw(A--E-- | + | draw(A--E--D--B); |
draw(A--F--G--C); | draw(A--F--G--C); | ||
draw(B--I--H--C); | draw(B--I--H--C); | ||
draw(E--F); | draw(E--F); | ||
− | draw( | + | draw(D--I); |
draw(I--H); | draw(I--H); | ||
draw(H--G); | draw(H--G); | ||
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label("$I$",I,SW); | label("$I$",I,SW); | ||
</asy> | </asy> | ||
− | |||
<math> \textbf{(A)}\ \dfrac{12+3\sqrt3}4\qquad\textbf{(B)}\ \dfrac92\qquad\textbf{(C)}\ 3+\sqrt3\qquad\textbf{(D)}\ \dfrac{6+3\sqrt3}2\qquad\textbf{(E)}\ 6 </math> | <math> \textbf{(A)}\ \dfrac{12+3\sqrt3}4\qquad\textbf{(B)}\ \dfrac92\qquad\textbf{(C)}\ 3+\sqrt3\qquad\textbf{(D)}\ \dfrac{6+3\sqrt3}2\qquad\textbf{(E)}\ 6 </math> | ||
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The <math>y</math>-intercepts, <math>A</math> and <math>B</math>, of two perpendicular lines intersecting at the point <math>D(6,8)</math> have a sum of zero. What is the area of <math>\triangle DAB</math>? | The <math>y</math>-intercepts, <math>A</math> and <math>B</math>, of two perpendicular lines intersecting at the point <math>D(6,8)</math> have a sum of zero. What is the area of <math>\triangle DAB</math>? | ||
− | <math> \textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72 | + | <math> \textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72 </math> |
− | |||
[[2014 AMC 10A Problems/Problem 14|Solution]] | [[2014 AMC 10A Problems/Problem 14|Solution]] | ||
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==Problem 15== | ==Problem 15== | ||
− | + | David drives from his home to the airport to catch a flight. He drives <math>35</math> miles in the first hour, but realizes that he will be <math>1</math> hour late if she continues at this speed. He increases his speed by <math>15</math> miles per hour for the rest of the way to the airport and arrives <math>30</math> minutes early. How many miles is the airport from his home? | |
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− | |||
− | |||
− | How many miles is the airport from his home? | ||
<math>\textbf{(A) }140\qquad | <math>\textbf{(A) }140\qquad | ||
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size(9cm); | size(9cm); | ||
pen dps = fontsize(10); defaultpen(dps); | pen dps = fontsize(10); defaultpen(dps); | ||
− | pair D = (0, | + | pair D = (0,0); |
− | pair F = (1/2, | + | pair F = (1/2,0); |
− | pair C = (1, | + | pair C = (1,0); |
− | pair G = ( | + | pair G = (0,1); |
− | pair E = (1, | + | pair E = (1,1); |
pair A = (0,2); | pair A = (0,2); | ||
pair B = (1,2); | pair B = (1,2); | ||
− | pair H = (1/ | + | pair H = (1/2,1); |
// do not look | // do not look | ||
− | pair X = (1/ | + | pair X = (1/3,2/3); |
− | pair Y = (2/3,2/ | + | pair Y = (2/3,2/3); |
− | draw(A-- | + | draw(A--B--C--D--cycle); |
draw(G--E); | draw(G--E); | ||
− | draw(A-- | + | draw(A--F--B); |
− | draw(D-- | + | draw(D--H--C); |
− | filldraw(H-- | + | filldraw(H--X--F--Y--cycle,grey); |
− | label("$A$", | + | label("$A$",A,NW); |
label("$B$",B,NE); | label("$B$",B,NE); | ||
− | label("$C$", | + | label("$C$",C,SE); |
− | label("$D$", | + | label("$D$",D,SW); |
− | label("$E$", | + | label("$E$",E,E); |
− | label("$F$", | + | label("$F$",F,S); |
− | label("$G$", | + | label("$G$",G,W); |
− | label("$H$", | + | label("$H$",H,N); |
label("$\frac12$",(0.25,0),S); | label("$\frac12$",(0.25,0),S); | ||
label("$\frac12$",(0.75,0),S); | label("$\frac12$",(0.75,0),S); | ||
− | label("$1$",( | + | label("$1$",(1,0.5),E); |
label("$1$",(1,1.5),E); | label("$1$",(1,1.5),E); | ||
</asy> | </asy> | ||
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==Problem 17== | ==Problem 17== | ||
− | |||
− | |||
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die? | Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die? | ||
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A square in the coordinate plane has vertices whose <math>y</math>-coordinates are <math>0</math>, <math>1</math>, <math>4</math>, and <math>5</math>. What is the area of the square? | A square in the coordinate plane has vertices whose <math>y</math>-coordinates are <math>0</math>, <math>1</math>, <math>4</math>, and <math>5</math>. What is the area of the square? | ||
− | <math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 | + | <math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 </math> |
− | + | ||
[[2014 AMC 10A Problems/Problem 18|Solution]] | [[2014 AMC 10A Problems/Problem 18|Solution]] | ||
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label("$2$", (2,8), W,fontsize(8pt)); | label("$2$", (2,8), W,fontsize(8pt)); | ||
label("$3$", (3,5.5), W,fontsize(8pt)); | label("$3$", (3,5.5), W,fontsize(8pt)); | ||
− | |||
</asy> | </asy> | ||
[[2014 AMC 10A Problems/Problem 19|Solution]] | [[2014 AMC 10A Problems/Problem 19|Solution]] | ||
Line 323: | Line 312: | ||
<asy> | <asy> | ||
import graph; | import graph; | ||
− | size( | + | size(6cm); |
real L = 0.05; | real L = 0.05; |
Revision as of 17:52, 24 January 2020
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
Roy's cat eats of a can of cat food every morning and of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing cans of cat food. On what day of the week did Roy's cat finish eating all the cat food in the box?
Problem 3
Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs for her to make. In dollars, what is her profit for the day?
Problem 4
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
Problem 5
On an algebra quiz, of the students scored points, scored points, scored points, and the rest scored points. What is the difference between the mean and median score of the students' scores on this quiz?
Problem 6
Suppose that cows give gallons of milk in days. At this rate, how many gallons of milk will cows give in days?
Problem 7
Nonzero real numbers , , , and satisfy and . How many of the following inequalities must be true?
Problem 8
Which of the following numbers is a perfect square?
Problem 9
The two legs of a right triangle, which are altitudes, have lengths and . How long is the third altitude of the triangle?
Problem 10
Five positive consecutive integers starting with have average . What is the average of consecutive integers that start with ?
Problem 11
A customer who intends to purchase an appliance has three coupons, only one of which may be used:
Coupon 1: off the listed price if the listed price is at least
Coupon 2: off the listed price if the listed price is at least
Coupon 3: off the amount by which the listed price exceeds
For which of the following listed prices will coupon offer a greater price reduction than either coupon or coupon ?
Problem 12
A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. What is the area of the shaded region?
Problem 13
Equilateral has side length , and squares , , lie outside the triangle. What is the area of hexagon ?
Problem 14
The -intercepts, and , of two perpendicular lines intersecting at the point have a sum of zero. What is the area of ?
Problem 15
David drives from his home to the airport to catch a flight. He drives miles in the first hour, but realizes that he will be hour late if she continues at this speed. He increases his speed by miles per hour for the rest of the way to the airport and arrives minutes early. How many miles is the airport from his home?
Problem 16
In rectangle , , , and points , , and are midpoints of , , and , respectively. Point is the midpoint of . What is the area of the shaded region?
Problem 17
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?
Problem 18
A square in the coordinate plane has vertices whose -coordinates are , , , and . What is the area of the square?
Problem 19
Four cubes with edge lengths , , , and are stacked as shown. What is the length of the portion of contained in the cube with edge length ?
Problem 20
The product , where the second factor has digits, is an integer whose digits have a sum of . What is ?
Problem 21
Positive integers and are such that the graphs of and intersect the -axis at the same point. What is the sum of all possible -coordinates of these points of intersection?
Problem 22
In rectangle , and . Let be a point on such that . What is ?
Problem 23
A rectangular piece of paper whose length is times the width has area . The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area . What is the ratio ?
Problem 24
A sequence of natural numbers is constructed by listing the first , then skipping one, listing the next , skipping , listing , skipping , and, on the th iteration, listing and skipping . The sequence begins . What is the th number in the sequence?
Problem 25
The number is between and . How many pairs of integers are there such that and
See also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2013 AMC 10B Problems |
Followed by 2014 AMC 10B 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.