Difference between revisions of "2018 AMC 12A Problems"
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==Problem 3== | ==Problem 3== | ||
− | How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.) | + | How many ways can a student schedule <math>3</math> mathematics courses -- algebra, geometry, and number theory -- in a <math>6</math>-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other <math>3</math> periods is of no concern here.) |
<math>\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24</math> | <math>\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24</math> | ||
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==Problem 4== | ==Problem 4== | ||
− | Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let <math>d</math> be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of <math>d</math>? | + | Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least <math>6</math> miles away," Bob replied, "We are at most <math>5</math> miles away." Charlie then remarked, "Actually the nearest town is at most <math>4</math> miles away." It turned out that none of the three statements were true. Let <math>d</math> be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of <math>d</math>? |
<math>\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty) </math> | <math>\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty) </math> | ||
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[[2018 AMC 12A Problems/Problem 5|Solution]] | [[2018 AMC 12A Problems/Problem 5|Solution]] | ||
+ | |||
==Problem 6== | ==Problem 6== | ||
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==Problem 8== | ==Problem 8== | ||
− | All of the triangles in the diagram below are similar to isosceles triangle <math>ABC</math>, in which <math>AB=AC</math>. Each of the 7 smallest triangles has area 1, and <math>\triangle ABC</math> has area 40. What is the area of trapezoid <math>DBCE</math>? | + | All of the triangles in the diagram below are similar to isosceles triangle <math>ABC</math>, in which <math>AB=AC</math>. Each of the <math>7</math> smallest triangles has area <math>1,</math> and <math>\triangle ABC</math> has area <math>40</math>. What is the area of trapezoid <math>DBCE</math>? |
<asy> | <asy> | ||
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Which of the following describes the largest subset of values of <math>y</math> within the closed interval <math>[0,\pi]</math> for which | Which of the following describes the largest subset of values of <math>y</math> within the closed interval <math>[0,\pi]</math> for which | ||
<cmath>\sin(x+y)\leq \sin(x)+\sin(y)</cmath>for every <math>x</math> between <math>0</math> and <math>\pi</math>, inclusive? | <cmath>\sin(x+y)\leq \sin(x)+\sin(y)</cmath>for every <math>x</math> between <math>0</math> and <math>\pi</math>, inclusive? | ||
− | < | + | |
+ | <math>\textbf{(A) } y=0 \qquad \textbf{(B) } 0\leq y\leq \frac{\pi}{4} \qquad \textbf{(C) } 0\leq y\leq \frac{\pi}{2} \qquad \textbf{(D) } 0\leq y\leq \frac{3\pi}{4} \qquad \textbf{(E) } 0\leq y\leq \pi </math> | ||
[[2018 AMC 12A Problems/Problem 9|Solution]] | [[2018 AMC 12A Problems/Problem 9|Solution]] | ||
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How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of equations? | How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of equations? | ||
− | <cmath>x+3y=3 | + | <cmath>\begin{align*} |
− | + | x+3y&=3 \\ | |
+ | \big||x|-|y|\big|&=1 | ||
+ | \end{align*}</cmath> | ||
<math>\textbf{(A) } 1 \qquad | <math>\textbf{(A) } 1 \qquad | ||
\textbf{(B) } 2 \qquad | \textbf{(B) } 2 \qquad | ||
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==Problem 11== | ==Problem 11== | ||
− | + | A paper triangle with sides of lengths <math>3,4,</math> and <math>5</math> inches, as shown, is folded so that point <math>A</math> falls on point <math>B</math>. What is the length in inches of the crease? | |
− | A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point <math>A</math> falls on point <math>B</math>. What is the length in inches of the crease? | ||
<asy> | <asy> | ||
draw((0,0)--(4,0)--(4,3)--(0,0)); | draw((0,0)--(4,0)--(4,3)--(0,0)); | ||
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==Problem 12== | ==Problem 12== | ||
− | Let <math>S</math> be a set of 6 integers taken from <math>\{1,2,\dots,12\}</math> with the property that if <math>a</math> and <math>b</math> are elements of <math>S</math> with <math>a<b</math>, then <math>b</math> is not a multiple of <math>a</math>. What is the least possible value of an element in <math>S | + | Let <math>S</math> be a set of <math>6</math> integers taken from <math>\{1,2,\dots,12\}</math> with the property that if <math>a</math> and <math>b</math> are elements of <math>S</math> with <math>a<b</math>, then <math>b</math> is not a multiple of <math>a</math>. What is the least possible value of an element in <math>S</math>? |
<math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7</math> | <math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7</math> | ||
[[2018 AMC 12A Problems/Problem 12|Solution]] | [[2018 AMC 12A Problems/Problem 12|Solution]] | ||
+ | |||
==Problem 13== | ==Problem 13== | ||
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==Problem 14== | ==Problem 14== | ||
− | The | + | The solution to the equation <math>\log_{3x} 4 = \log_{2x} 8</math>, where <math>x</math> is a positive real number other than <math>\frac{1}{3}</math> or <math>\frac{1}{2}</math>, can be written as <math>\frac {p}{q}</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. What is <math>p + q</math>? |
<math>\textbf{(A) } 5 \qquad | <math>\textbf{(A) } 5 \qquad | ||
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[[2018 AMC 12A Problems/Problem 14|Solution]] | [[2018 AMC 12A Problems/Problem 14|Solution]] | ||
+ | |||
==Problem 15== | ==Problem 15== | ||
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==Problem 17== | ==Problem 17== | ||
− | Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square <math>S</math> so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from <math>S</math> to the hypotenuse is 2 units. What fraction of the field is planted? | + | Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths <math>3</math> and <math>4</math> units. In the corner where those sides meet at a right angle, he leaves a small unplanted square <math>S</math> so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from <math>S</math> to the hypotenuse is <math>2</math> units. What fraction of the field is planted? |
<asy> | <asy> | ||
− | + | /* Edited by MRENTHUSIASM */ | |
− | + | size(160); | |
− | + | pair A, B, C, D, F; | |
− | label("$4$", ( | + | A = origin; |
− | label("$3$", ( | + | B = (4,0); |
− | label("$2$", ( | + | C = (0,3); |
− | label("$S$", ( | + | D = (2/7,2/7); |
− | draw( | + | F = foot(D,B,C); |
+ | fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray); | ||
+ | draw(A--B--C--cycle); | ||
+ | draw((2/7,0)--D--(0,2/7)); | ||
+ | label("$4$", midpoint(A--B), N); | ||
+ | label("$3$", midpoint(A--C), E); | ||
+ | label("$2$", midpoint(D--F), SE); | ||
+ | label("$S$", midpoint(A--D)); | ||
+ | draw(D--F, dashed); | ||
</asy> | </asy> | ||
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Let <math>A</math> be the set of positive integers that have no prime factors other than <math>2</math>, <math>3</math>, or <math>5</math>. The infinite sum <cmath>\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots</cmath>of the reciprocals of the elements of <math>A</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>? | Let <math>A</math> be the set of positive integers that have no prime factors other than <math>2</math>, <math>3</math>, or <math>5</math>. The infinite sum <cmath>\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots</cmath>of the reciprocals of the elements of <math>A</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>? | ||
− | <math>\textbf{(A)} | + | <math>\textbf{(A) } 16 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 23 \qquad \textbf{(E) } 36</math> |
[[2018 AMC 12A Problems/Problem 19|Solution]] | [[2018 AMC 12A Problems/Problem 19|Solution]] | ||
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Which of the following polynomials has the greatest real root? | Which of the following polynomials has the greatest real root? | ||
+ | |||
<math>\textbf{(A) } x^{19}+2018x^{11}+1 \qquad \textbf{(B) } x^{17}+2018x^{11}+1 \qquad \textbf{(C) } x^{19}+2018x^{13}+1 \qquad \textbf{(D) } x^{17}+2018x^{13}+1 \qquad \textbf{(E) } 2019x+2018 </math> | <math>\textbf{(A) } x^{19}+2018x^{11}+1 \qquad \textbf{(B) } x^{17}+2018x^{11}+1 \qquad \textbf{(C) } x^{19}+2018x^{13}+1 \qquad \textbf{(D) } x^{17}+2018x^{13}+1 \qquad \textbf{(E) } 2019x+2018 </math> | ||
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==Problem 24== | ==Problem 24== | ||
− | Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between <math>\tfrac{1}{2}</math> and <math>\tfrac{2}{3}.</math> Armed with this information, what number should Carol choose to maximize her chance of winning? | + | Alice, Bob, and Carol play a game in which each of them chooses a real number between <math>0</math> and <math>1.</math> The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between <math>0</math> and <math>1,</math> and Bob announces that he will choose his number uniformly at random from all the numbers between <math>\tfrac{1}{2}</math> and <math>\tfrac{2}{3}.</math> Armed with this information, what number should Carol choose to maximize her chance of winning? |
− | |||
<math> | <math> | ||
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For a positive integer <math>n</math> and nonzero digits <math>a</math>, <math>b</math>, and <math>c</math>, let <math>A_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>a</math>; let <math>B_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>b</math>, and let <math>C_n</math> be the <math>2n</math>-digit (not <math>n</math>-digit) integer each of whose digits is equal to <math>c</math>. What is the greatest possible value of <math>a + b + c</math> for which there are at least two values of <math>n</math> such that <math>C_n - B_n = A_n^2</math>? | For a positive integer <math>n</math> and nonzero digits <math>a</math>, <math>b</math>, and <math>c</math>, let <math>A_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>a</math>; let <math>B_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>b</math>, and let <math>C_n</math> be the <math>2n</math>-digit (not <math>n</math>-digit) integer each of whose digits is equal to <math>c</math>. What is the greatest possible value of <math>a + b + c</math> for which there are at least two values of <math>n</math> such that <math>C_n - B_n = A_n^2</math>? | ||
− | <math>\textbf{(A)} | + | <math>\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20</math> |
[[2018 AMC 12A Problems/Problem 25|Solution]] | [[2018 AMC 12A Problems/Problem 25|Solution]] | ||
− | |||
==See also== | ==See also== | ||
{{AMC12 box|year=2018|ab=A|before=[[2017 AMC 12B Problems]]|after=[[2018 AMC 12B Problems]]}} | {{AMC12 box|year=2018|ab=A|before=[[2017 AMC 12B Problems]]|after=[[2018 AMC 12B Problems]]}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:39, 3 July 2024
2018 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
A large urn contains balls, of which
are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be
? (No red balls are to be removed.)
Problem 2
While exploring a cave, Carl comes across a collection of -pound rocks worth
each,
-pound rocks worth
each, and
-pound rocks worth
each. There are at least
of each size. He can carry at most
pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
Problem 3
How many ways can a student schedule mathematics courses -- algebra, geometry, and number theory -- in a
-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other
periods is of no concern here.)
Problem 4
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least miles away," Bob replied, "We are at most
miles away." Charlie then remarked, "Actually the nearest town is at most
miles away." It turned out that none of the three statements were true. Let
be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of
?
Problem 5
What is the sum of all possible values of for which the polynomials
and
have a root in common?
Problem 6
For positive integers and
such that
, both the mean and the median of the set
are equal to
. What is
?
Problem 7
For how many (not necessarily positive) integer values of is the value of
an integer?
Problem 8
All of the triangles in the diagram below are similar to isosceles triangle , in which
. Each of the
smallest triangles has area
and
has area
. What is the area of trapezoid
?
Problem 9
Which of the following describes the largest subset of values of within the closed interval
for which
for every
between
and
, inclusive?
Problem 10
How many ordered pairs of real numbers satisfy the following system of equations?
Problem 11
A paper triangle with sides of lengths and
inches, as shown, is folded so that point
falls on point
. What is the length in inches of the crease?
Problem 12
Let be a set of
integers taken from
with the property that if
and
are elements of
with
, then
is not a multiple of
. What is the least possible value of an element in
?
Problem 13
How many nonnegative integers can be written in the form
where
for
?
Problem 14
The solution to the equation , where
is a positive real number other than
or
, can be written as
where
and
are relatively prime positive integers. What is
?
Problem 15
A scanning code consists of a grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of
squares. A scanning code is called
if its look does not change when the entire square is rotated by a multiple of
counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
Problem 16
Which of the following describes the set of values of for which the curves
and
in the real
-plane intersect at exactly
points?
Problem 17
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths and
units. In the corner where those sides meet at a right angle, he leaves a small unplanted square
so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from
to the hypotenuse is
units. What fraction of the field is planted?
Problem 18
Triangle with
and
has area
. Let
be the midpoint of
, and let
be the midpoint of
. The angle bisector of
intersects
and
at
and
, respectively. What is the area of quadrilateral
?
Problem 19
Let be the set of positive integers that have no prime factors other than
,
, or
. The infinite sum
of the reciprocals of the elements of
can be expressed as
, where
and
are relatively prime positive integers. What is
?
Problem 20
Triangle is an isosceles right triangle with
. Let
be the midpoint of hypotenuse
. Points
and
lie on sides
and
, respectively, so that
and
is a cyclic quadrilateral. Given that triangle
has area
, the length
can be written as
, where
,
, and
are positive integers and
is not divisible by the square of any prime. What is the value of
?
Problem 21
Which of the following polynomials has the greatest real root?
Problem 22
The solutions to the equations and
where
form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form
where
and
are positive integers and neither
nor
is divisible by the square of any prime number. What is
Problem 23
In
and
Points
and
lie on sides
and
respectively, so that
Let
and
be the midpoints of segments
and
respectively. What is the degree measure of the acute angle formed by lines
and
Problem 24
Alice, Bob, and Carol play a game in which each of them chooses a real number between and
The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between
and
and Bob announces that he will choose his number uniformly at random from all the numbers between
and
Armed with this information, what number should Carol choose to maximize her chance of winning?
Problem 25
For a positive integer and nonzero digits
,
, and
, let
be the
-digit integer each of whose digits is equal to
; let
be the
-digit integer each of whose digits is equal to
, and let
be the
-digit (not
-digit) integer each of whose digits is equal to
. What is the greatest possible value of
for which there are at least two values of
such that
?
See also
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2017 AMC 12B Problems |
Followed by 2018 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.