Difference between revisions of "2014 AMC 12A Problems"
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+ | {{AMC12 Problems|year=2014|ab=A}} | ||
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==Problem 1== | ==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)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D) | + | <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]] | ||
==Problem 2== | ==Problem 2== | ||
− | At the theater children get in for half price. The price for <math>5</math> adult tickets and <math>4</math> child tickets is <math>24.50</math>. How much would <math>8</math> adult tickets and <math>6</math> child tickets cost? | + | At the theater children get in for half price. The price for <math>5</math> adult tickets and <math>4</math> child tickets is <math>\$24.50</math>. How much would <math>8</math> adult tickets and <math>6</math> child tickets cost? |
− | <math>\textbf{(A) }35\qquad | + | <math>\textbf{(A) }\$35\qquad |
− | \textbf{(B) }38.50\qquad | + | \textbf{(B) }\$38.50\qquad |
− | \textbf{(C) }40\qquad | + | \textbf{(C) }\$40\qquad |
− | \textbf{(D) }42\qquad | + | \textbf{(D) }\$42\qquad |
− | \textbf{(E) }42.50</math> | + | \textbf{(E) }\$42.50</math> |
[[2014 AMC 12A Problems/Problem 2|Solution]] | [[2014 AMC 12A Problems/Problem 2|Solution]] | ||
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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{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D) | + | <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]] | ||
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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? | 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> \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D) | + | <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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On an algebra quiz, <math>10\%</math> of the students scored <math>70</math> points, <math>35\%</math> scored <math>80</math> points, <math>30\%</math> scored <math>90</math> points, and the rest scored <math>100</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>70</math> points, <math>35\%</math> scored <math>80</math> points, <math>30\%</math> scored <math>90</math> points, and the rest scored <math>100</math> points. What is the difference between the mean and median score of the students' scores on this quiz? | ||
− | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math> |
[[2014 AMC 10A Problems/Problem 5|Solution]] | [[2014 AMC 10A Problems/Problem 5|Solution]] | ||
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A customer who intends to purchase an appliance has three coupons, only one of which may be used: | A customer who intends to purchase an appliance has three coupons, only one of which may be used: | ||
− | Coupon 1: <math>10\%</math> off the listed price if the listed price is at least <math>50</math> | + | Coupon 1: <math>10\%</math> off the listed price if the listed price is at least <math>\$50</math> |
− | Coupon 2: <math>20</math> | + | Coupon 2: <math>\$20</math> off the listed price if the listed price is at least <math>\$100</math> |
− | Coupon 3: <math>18\%</math> off the amount by which the listed price exceeds <math>100</math> | + | Coupon 3: <math>18\%</math> off the amount by which the listed price exceeds <math>\$100</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>? | 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) }179.95\qquad | + | <math>\textbf{(A) }\$179.95\qquad |
− | \textbf{(B) }199.95\qquad | + | \textbf{(B) }\$199.95\qquad |
− | \textbf{(C) }219.95\qquad | + | \textbf{(C) }\$219.95\qquad |
− | \textbf{(D) }239.95\qquad | + | \textbf{(D) }\$239.95\qquad |
− | \textbf{(E) }259.95\qquad</math> | + | \textbf{(E) }\$259.95\qquad</math> |
[[2014 AMC 10A Problems/Problem 11|Solution]] | [[2014 AMC 10A Problems/Problem 11|Solution]] | ||
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Five positive consecutive integers starting with <math>a</math> have average <math>b</math>. What is the average of <math>5</math> consecutive integers that start with <math>b</math>? | Five positive consecutive integers starting with <math>a</math> have average <math>b</math>. What is the average of <math>5</math> consecutive integers that start with <math>b</math>? | ||
− | <math> \textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D) | + | <math> \textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}\ a+6\qquad\textbf{(E)}\ a+7</math> |
[[2014 AMC 10A Problems/Problem 10|Solution]] | [[2014 AMC 10A Problems/Problem 10|Solution]] | ||
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==Problem 15== | ==Problem 15== | ||
− | A five-digit palindrome is a positive integer with respective digits <math>abcba</math>, where <math>a</math> is non-zero. Let <math>S</math> be the sum of all five-digit palindromes. What is the sum of the digits of <math>S</math> | + | A five-digit palindrome is a positive integer with respective digits <math>abcba</math>, where <math>a</math> is non-zero. Let <math>S</math> be the sum of all five-digit palindromes. What is the sum of the digits of <math>S</math>? |
<math>\textbf{(A) }9\qquad | <math>\textbf{(A) }9\qquad | ||
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<center><asy> | <center><asy> | ||
− | import | + | import graph3; |
import solids; | import solids; | ||
real h=2+2*sqrt(7); | real h=2+2*sqrt(7); | ||
currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); | currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); | ||
− | currentlight=light( | + | currentlight=light(4,-4,4); |
draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); | draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); | ||
− | draw(shift((1,3,1))*unitsphere, | + | draw(shift((1,3,1))*unitsphere,gray(0.85)); |
− | draw(shift((3,3,1))*unitsphere, | + | draw(shift((3,3,1))*unitsphere,gray(0.85)); |
− | draw(shift((3,1,1))*unitsphere, | + | draw(shift((3,1,1))*unitsphere,gray(0.85)); |
− | draw(shift((1,1,1))*unitsphere, | + | draw(shift((1,1,1))*unitsphere,gray(0.85)); |
− | draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere, | + | draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); |
− | draw(shift((1,3,h-1))*unitsphere, | + | draw(shift((1,3,h-1))*unitsphere,gray(0.85)); |
− | draw(shift((3,3,h-1))*unitsphere, | + | draw(shift((3,3,h-1))*unitsphere,gray(0.85)); |
− | draw(shift((3,1,h-1))*unitsphere, | + | draw(shift((3,1,h-1))*unitsphere,gray(0.85)); |
− | draw(shift((1,1,h-1))*unitsphere, | + | draw(shift((1,1,h-1))*unitsphere,gray(0.85)); |
draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); | draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); | ||
</asy></center> | </asy></center> | ||
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==Problem 25== | ==Problem 25== | ||
− | The parabola <math>P</math> has focus <math>(0,0)</math> and goes through the points <math>(4,3)</math> and <math>(-4,-3)</math>. For how many points <math>(x,y)\in P</math> with integer | + | The parabola <math>P</math> has focus <math>(0,0)</math> and goes through the points <math>(4,3)</math> and <math>(-4,-3)</math>. For how many points <math>(x,y)\in P</math> with integer coordinates is it true that <math>|4x+3y|\leq 1000</math>? |
<math>\textbf{(A) }38\qquad | <math>\textbf{(A) }38\qquad | ||
Line 280: | Line 283: | ||
[[2014 AMC 12A Problems/Problem 25|Solution]] | [[2014 AMC 12A Problems/Problem 25|Solution]] | ||
+ | |||
+ | ==See also== | ||
+ | |||
+ | {{AMC12 box|year=2014|ab=A|before=[[2013 AMC 12B Problems]]|after=[[2014 AMC 12B Problems]]}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 19:04, 11 April 2024
2014 AMC 12A (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
At the theater children get in for half price. The price for adult tickets and
child tickets is
. How much would
adult tickets and
child tickets cost?
Problem 3
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 4
Suppose that cows give
gallons of milk in
days. At this rate, how many gallons of milk will
cows give in
days?
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
The difference between a two-digit number and the number obtained by reversing its digits is times the sum of the digits of either number. What is the sum of the two digit number and its reverse?
Problem 7
The first three terms of a geometric progression are ,
, and
. What is the fourth term?
Problem 8
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 9
Five positive consecutive integers starting with have average
. What is the average of
consecutive integers that start with
?
Problem 10
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length . The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?
Problem 11
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 12
Two circles intersect at points and
. The minor arcs
measure
on one circle and
on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?
Problem 13
A fancy bed and breakfast inn has rooms, each with a distinctive color-coded decor. One day
friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than
friends per room. In how many ways can the innkeeper assign the guests to the rooms?
Problem 14
Let be three integers such that
is an arithmetic progression and
is a geometric progression. What is the smallest possible value of
?
Problem 15
A five-digit palindrome is a positive integer with respective digits , where
is non-zero. Let
be the sum of all five-digit palindromes. What is the sum of the digits of
?
Problem 16
The product , where the second factor has
digits, is an integer whose digits have a sum of
. What is
?
Problem 17
A rectangular box contains a sphere of radius
and eight smaller spheres of radius
. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is
?
![[asy] import graph3; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(4,-4,4); draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); draw(shift((1,3,1))*unitsphere,gray(0.85)); draw(shift((3,3,1))*unitsphere,gray(0.85)); draw(shift((3,1,1))*unitsphere,gray(0.85)); draw(shift((1,1,1))*unitsphere,gray(0.85)); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); draw(shift((1,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,1,h-1))*unitsphere,gray(0.85)); draw(shift((1,1,h-1))*unitsphere,gray(0.85)); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy]](http://latex.artofproblemsolving.com/9/d/8/9d894dbf42883508d51bf4973c78d1368570c6fa.png)
Problem 18
The domain of the function is an interval of length
, where
and
are relatively prime positive integers. What is
?
Problem 19
There are exactly distinct rational numbers
such that
and
has at least one integer solution for
. What is
?
Problem 20
In ,
,
, and
. Points
and
lie on
and
respectively. What is the minimum possible value of
?
Problem 21
For every real number , let
denote the greatest integer not exceeding
, and let
The set of all numbers
such that
and
is a union of disjoint intervals. What is the sum of the lengths of those intervals?
Problem 22
The number is between
and
. How many pairs of integers
are there such that
and
Problem 23
The fraction where
is the length of the period of the repeating decimal expansion. What is the sum
?
Problem 24
Let , and for
, let
. For how many values of
is
?
Problem 25
The parabola has focus
and goes through the points
and
. For how many points
with integer coordinates is it true that
?
See also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2013 AMC 12B Problems |
Followed by 2014 AMC 12B 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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.