Difference between revisions of "2014 AMC 10A Problems"
(→Half-Life 3) |
(→Problem 20) |
||
(89 intermediate revisions by 14 users not shown) | |||
Line 1: | Line 1: | ||
− | == | + | {{AMC10 Problems|year=2014|ab=A}} |
+ | ==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> | ||
− | <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ | + | <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}\ \frac{170}{3}\qquad\textbf{(E)}\ 170</math> |
[[2014 AMC 10A Problems/Problem 1|Solution]] | [[2014 AMC 10A Problems/Problem 1|Solution]] | ||
Line 9: | Line 10: | ||
==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 the 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 | + | <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}</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> | ||
[[2014 AMC 10A Problems/Problem 3|Solution]] | [[2014 AMC 10A Problems/Problem 3|Solution]] | ||
− | |||
==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? | 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{( | + | <math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math> |
[[2014 AMC 10A Problems/Problem 4|Solution]] | [[2014 AMC 10A Problems/Problem 4|Solution]] | ||
− | |||
==Problem 5== | ==Problem 5== | ||
Line 41: | Line 39: | ||
==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]] | ||
Line 49: | Line 47: | ||
==Problem 7== | ==Problem 7== | ||
− | Nonzero real numbers <math>x</math>, <math>y</math>, <math>a</math>, and <math>b</math> satisfy <math>x < a</math> and <math>y < b</math>. How many of the following inequalities must be true? | + | Nonzero real numbers <math>x</math>, <math>y</math>, <math>a</math>, and <math>b</math> satisfy <math>x < a</math> and <math>y < b</math>. How many of the following inequalities must be true? |
<math>\textbf{(I)}\ x+y < a+b\qquad</math> | <math>\textbf{(I)}\ x+y < a+b\qquad</math> | ||
Line 81: | Line 79: | ||
[[2014 AMC 10A Problems/Problem 9|Solution]] | [[2014 AMC 10A Problems/Problem 9|Solution]] | ||
− | |||
==Problem 10== | ==Problem 10== | ||
Line 101: | Line 98: | ||
For which of the following listed prices will coupon <math>1</math> offer a greater price reduction than either coupon <math>2</math> or coupon <math>3</math>? | For which of the following listed prices will coupon <math>1</math> offer a greater price reduction than either coupon <math>2</math> or coupon <math>3</math>? | ||
+ | |||
<math>\textbf{(A) }\textdollar179.95\qquad | <math>\textbf{(A) }\textdollar179.95\qquad | ||
\textbf{(B) }\textdollar199.95\qquad | \textbf{(B) }\textdollar199.95\qquad | ||
Line 108: | Line 106: | ||
[[2014 AMC 10A Problems/Problem 11|Solution]] | [[2014 AMC 10A Problems/Problem 11|Solution]] | ||
− | |||
==Problem 12== | ==Problem 12== | ||
Line 117: | Line 114: | ||
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,lightgrey); |
− | for(int i=0;i<= | + | for(int i=0;i<=5;i=i+1) |
{ | { | ||
path arc=2*dir(60*i)--arc(2*dir(60*i),1,120+60*i,240+60*i)--cycle; | path arc=2*dir(60*i)--arc(2*dir(60*i),1,120+60*i,240+60*i)--cycle; | ||
Line 133: | Line 130: | ||
Equilateral <math>\triangle ABC</math> has side length <math>1</math>, and squares <math>ABDE</math>, <math>BCHI</math>, <math>CAFG</math> lie outside the triangle. What is the area of hexagon <math>DEFGHI</math>? | Equilateral <math>\triangle ABC</math> has side length <math>1</math>, and squares <math>ABDE</math>, <math>BCHI</math>, <math>CAFG</math> lie outside the triangle. What is the area of hexagon <math>DEFGHI</math>? | ||
+ | |||
<asy> | <asy> | ||
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); | ||
Line 142: | Line 140: | ||
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; | ||
Line 151: | Line 149: | ||
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); | ||
Line 174: | Line 172: | ||
[[2014 AMC 10A Problems/Problem 13|Solution]] | [[2014 AMC 10A Problems/Problem 13|Solution]] | ||
− | |||
==Problem 14== | ==Problem 14== | ||
Line 184: | Line 181: | ||
==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 he 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? | |
− | |||
− | |||
− | |||
− | How many miles is the airport from his home? | ||
<math>\textbf{(A) }140\qquad | <math>\textbf{(A) }140\qquad | ||
Line 197: | Line 190: | ||
[[2014 AMC 10A Problems/Problem 15|Solution]] | [[2014 AMC 10A Problems/Problem 15|Solution]] | ||
− | |||
==Problem 16== | ==Problem 16== | ||
Line 246: | Line 238: | ||
==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? |
<math> \textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29 </math> | <math> \textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29 </math> | ||
[[2014 AMC 10A Problems/Problem 17|Solution]] | [[2014 AMC 10A Problems/Problem 17|Solution]] | ||
− | |||
==Problem 18== | ==Problem 18== | ||
Line 285: | Line 276: | ||
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)); | ||
− | label("$ | + | label("$4$", (4,2), W,fontsize(8pt)); |
</asy> | </asy> | ||
[[2014 AMC 10A Problems/Problem 19|Solution]] | [[2014 AMC 10A Problems/Problem 19|Solution]] | ||
− | |||
==Problem 20== | ==Problem 20== | ||
− | The product <math>(8)(888\dots8)</math>, where the second factor has <math> | + | The product <math>(8)(888\dots8)</math>, where the second factor has <math>k</math> digits, is an integer whose digits have a sum of <math>1000</math>. What is <math>k</math>? |
− | <math>\textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}\ 991\qquad\textbf{(E)}\ 999</math> | + | <math> \textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}\ 991\qquad\textbf{(E)}\ 999 </math> |
[[2014 AMC 10A Problems/Problem 20|Solution]] | [[2014 AMC 10A Problems/Problem 20|Solution]] | ||
Line 347: | Line 337: | ||
==Problem 24== | ==Problem 24== | ||
− | A sequence of natural numbers is constructed by listing the first <math>4</math>, then skipping one, listing the next <math>5</math>, skipping <math>2</math>, listing <math>6</math>, skipping <math>3</math>, and, on the <math>n</math>th iteration, listing <math>n+3</math> and skipping <math>n</math>. The sequence begins <math>1,2,3,4,6,7,8,9,10,13</math>. What is the <math>500,000</math> | + | A sequence of natural numbers is constructed by listing the first <math>4</math>, then skipping one, listing the next <math>5</math>, skipping <math>2</math>, listing <math>6</math>, skipping <math>3</math>, and, on the <math>n</math>th iteration, listing <math>n+3</math> and skipping <math>n</math>. The sequence begins <math>1,2,3,4,6,7,8,9,10,13</math>. What is the <math>500,000^{\text{th}}</math> number in the sequence? |
<math> \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996,507\qquad\textbf{(C)}\ 996,508\qquad\textbf{(D)}\ 996,509\qquad\textbf{(E)}\ 996,510 </math> | <math> \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996,507\qquad\textbf{(C)}\ 996,508\qquad\textbf{(D)}\ 996,509\qquad\textbf{(E)}\ 996,510 </math> | ||
Line 353: | Line 343: | ||
[[2014 AMC 10A Problems/Problem 24|Solution]] | [[2014 AMC 10A Problems/Problem 24|Solution]] | ||
− | == | + | ==Problem 25== |
The number <math>5^{867}</math> is between <math>2^{2013}</math> and <math>2^{2014}</math>. How many pairs of integers <math>(m,n)</math> are there such that <math>1\leq m\leq 2012</math> and <cmath>5^n<2^m<2^{m+2}<5^{n+1}?</cmath> | The number <math>5^{867}</math> is between <math>2^{2013}</math> and <math>2^{2014}</math>. How many pairs of integers <math>(m,n)</math> are there such that <math>1\leq m\leq 2012</math> and <cmath>5^n<2^m<2^{m+2}<5^{n+1}?</cmath> | ||
Line 360: | Line 350: | ||
[[2014 AMC 10A Problems/Problem 25|Solution]] | [[2014 AMC 10A Problems/Problem 25|Solution]] | ||
− | |||
==See also== | ==See also== | ||
{{AMC10 box|year=2014|ab=A|before=[[2013 AMC 10B Problems]]|after=[[2014 AMC 10B Problems]]}} | {{AMC10 box|year=2014|ab=A|before=[[2013 AMC 10B Problems]]|after=[[2014 AMC 10B Problems]]}} |
Latest revision as of 15:54, 28 July 2024
2014 AMC 10A (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
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 the 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 he 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 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.