Difference between revisions of "2018 AMC 12A Problems"

Latest revision as of 12:39, 3 July 2024

2018 AMC 12A (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.
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Problem 1

A large urn contains $100$ balls, of which $36 \%$ 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 $72 \%$ ? (No red balls are to be removed.)

$\textbf{(A)}\ 28 \qquad\textbf{(B)}\ 32 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 64$

Solution

Problem 2

While exploring a cave, Carl comes across a collection of $5$ -pound rocks worth $$14$ each, $4$ -pound rocks worth $$11$ each, and $1$ -pound rocks worth $$2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?

$\textbf{(A) } 48 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 50 \qquad \textbf{(D) } 51 \qquad \textbf{(E) } 52$

Solution

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.)

$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$

Solution

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 $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$ ?

$\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty)$

Solution

Problem 5

What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common?

$\textbf{(A) }3 \qquad\textbf{(B) }4 \qquad\textbf{(C) }5 \qquad\textbf{(D) }6 \qquad\textbf{(E) }10 \qquad$

Solution

Problem 6

For positive integers $m$ and $n$ such that $m+10<n+1$ , both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$ . What is $m+n$ ?

$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$

Solution

Problem 7

For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer?

$\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }6 \qquad \textbf{(D) }8 \qquad \textbf{(E) }9 \qquad$

Solution

Problem 8

All of the triangles in the diagram below are similar to isosceles triangle $ABC$ , in which $AB=AC$ . Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$ . What is the area of trapezoid $DBCE$ ?

$[asy] unitsize(5); dot((0,0)); dot((60,0)); dot((50,10)); dot((10,10)); dot((30,30)); draw((0,0)--(60,0)--(50,10)--(30,30)--(10,10)--(0,0)); draw((10,10)--(50,10)); label("$B$",(0,0),SW); label("$C$",(60,0),SE); label("$E$",(50,10),E); label("$D$",(10,10),W); label("$A$",(30,30),N); draw((10,10)--(15,15)--(20,10)--(25,15)--(30,10)--(35,15)--(40,10)--(45,15)--(50,10)); draw((15,15)--(45,15)); [/asy]$

$\textbf{(A) } 16 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 22 \qquad \textbf{(E) } 24$

Solution

Problem 9

Which of the following describes the largest subset of values of $y$ within the closed interval $[0,\pi]$ for which $\sin(x+y)\leq \sin(x)+\sin(y)$ for every $x$ between $0$ and $\pi$ , inclusive?

$\textbf{(A) } y=0 \qquad \textbf{(B) } 0\leq y\leq \frac{\pi}{4} \qquad \textbf{(C) } 0\leq y\leq \frac{\pi}{2} \qquad \textbf{(D) } 0\leq y\leq \frac{3\pi}{4} \qquad \textbf{(E) } 0\leq y\leq \pi$

Solution

Problem 10

How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? $\begin{align*} x+3y&=3 \\ \big||x|-|y|\big|&=1 \end{align*}$ $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 8$

Solution

Problem 11

A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$ . What is the length in inches of the crease? $[asy] draw((0,0)--(4,0)--(4,3)--(0,0)); label("$A$", (0,0), SW); label("$B$", (4,3), NE); label("$C$", (4,0), SE); label("$4$", (2,0), S); label("$3$", (4,1.5), E); label("$5$", (2,1.5), NW); fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray); [/asy]$ $\textbf{(A) } 1+\frac12 \sqrt2 \qquad \textbf{(B) } \sqrt3 \qquad \textbf{(C) } \frac74 \qquad \textbf{(D) } \frac{15}{8} \qquad \textbf{(E) } 2$

Solution

Problem 12

Let $S$ be a set of $6$ integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$ , then $b$ is not a multiple of $a$ . What is the least possible value of an element in $S$ ?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$

Solution

Problem 13

How many nonnegative integers can be written in the form $a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$ where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$ ?

$\textbf{(A) } 512 \qquad \textbf{(B) } 729 \qquad \textbf{(C) } 1094 \qquad \textbf{(D) } 3281 \qquad \textbf{(E) } 59,048$

Solution

Problem 14

The solution to the equation $\log_{3x} 4 = \log_{2x} 8$ , where $x$ is a positive real number other than $\frac{1}{3}$ or $\frac{1}{2}$ , can be written as $\frac {p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$ ?

$\textbf{(A) } 5 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 31 \qquad \textbf{(E) } 35$

Solution

Problem 15

A scanning code consists of a $7 \times 7$ 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 $49$ squares. A scanning code is called $\textit{symmetric}$ if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ 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?

$\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$

Solution

Problem 16

Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$ -plane intersect at exactly $3$ points?

$\textbf{(A) }a=\frac14 \qquad \textbf{(B) }\frac14 < a < \frac12 \qquad \textbf{(C) }a>\frac14 \qquad \textbf{(D) }a=\frac12 \qquad \textbf{(E) }a>\frac12 \qquad$

Solution

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 $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is $2$ units. What fraction of the field is planted?

$[asy] /* Edited by MRENTHUSIASM */ size(160); pair A, B, C, D, F; A = origin; B = (4,0); C = (0,3); D = (2/7,2/7); 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]$

$\textbf{(A) } \frac{25}{27} \qquad \textbf{(B) } \frac{26}{27} \qquad \textbf{(C) } \frac{73}{75} \qquad \textbf{(D) } \frac{145}{147} \qquad \textbf{(E) } \frac{74}{75}$

Solution

Problem 18

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$ . Let $D$ be the midpoint of $\overline{AB}$ , and let $E$ be the midpoint of $\overline{AC}$ . The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$ , respectively. What is the area of quadrilateral $FDBG$ ?

$\textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad$

Solution

Problem 19

Let $A$ be the set of positive integers that have no prime factors other than $2$ , $3$ , or $5$ . The infinite sum $\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots$ of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

$\textbf{(A) } 16 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 23 \qquad \textbf{(E) } 36$

Solution

Problem 20

Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$ . Let $M$ be the midpoint of hypotenuse $\overline{BC}$ . Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$ , respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$ , the length $CI$ can be written as $\frac{a-\sqrt{b}}{c}$ , where $a$ , $b$ , and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$ ?

$\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }11 \qquad \textbf{(D) }12 \qquad \textbf{(E) }13 \qquad$

Solution

Problem 21

Which of the following polynomials has the greatest real root?

$\textbf{(A) } x^{19}+2018x^{11}+1 \qquad \textbf{(B) } x^{17}+2018x^{11}+1 \qquad \textbf{(C) } x^{19}+2018x^{13}+1 \qquad \textbf{(D) } x^{17}+2018x^{13}+1 \qquad \textbf{(E) } 2019x+2018$

Solution

Problem 22

The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt 3i,$ where $i=\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q-r\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p+q+r+s?$

$\textbf{(A) } 20 \qquad \textbf{(B) } 21 \qquad \textbf{(C) } 22 \qquad \textbf{(D) } 23 \qquad \textbf{(E) } 24$

Solution

Problem 23

In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA?$

$\textbf{(A) } 76 \qquad \textbf{(B) } 77 \qquad \textbf{(C) } 78 \qquad \textbf{(D) } 79 \qquad \textbf{(E) } 80$

Solution

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 $\tfrac{1}{2}$ and $\tfrac{2}{3}.$ Armed with this information, what number should Carol choose to maximize her chance of winning?

$\textbf{(A) }\frac{1}{2}\qquad \textbf{(B) }\frac{13}{24} \qquad \textbf{(C) }\frac{7}{12} \qquad \textbf{(D) }\frac{5}{8} \qquad \textbf{(E) }\frac{2}{3}\qquad$

Solution

Problem 25

For a positive integer $n$ and nonzero digits $a$ , $b$ , and $c$ , let $A_n$ be the $n$ -digit integer each of whose digits is equal to $a$ ; let $B_n$ be the $n$ -digit integer each of whose digits is equal to $b$ , and let $C_n$ be the $2n$ -digit (not $n$ -digit) integer each of whose digits is equal to $c$ . What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$ ?

$\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20$

Solution

2018 AMC 12A (Problems • Answer Key • Resources)
Preceded by 2017 AMC 12B Problems	Followed by 2018 AMC 12B Problems
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All AMC 12 Problems and Solutions

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Difference between revisions of "2018 AMC 12A Problems"

Latest revision as of 12:39, 3 July 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also

@@ Line 1: / Line 1: @@
+{{AMC12 Problems|year=2018|ab=A}}
 ==Problem 1==
-Pablo buys popsicles for his friends. The store sells single popsicles for \$1 each, 3-popsicle boxes for \$2, and 5-popsicle boxes for \$3. What is the greatest number of popsicles that Pablo can buy with \$8?
+A large urn contains <math>100</math> balls, of which <math>36 \%</math> 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 <math>72 \%</math>? (No red balls are to be removed.)
-<math>\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15</math>
+<math> \textbf{(A)}\ 28 \qquad\textbf{(B)}\  32 \qquad\textbf{(C)}\  36 \qquad\textbf{(D)}\
+\qquad\textbf{(E)}\ 64 </math>
 [[2018 AMC 12A  Problems/Problem 1|Solution]]
+==Problem 2==
+While exploring a cave, Carl comes across a collection of <math>5</math>-pound rocks worth <math>\$14</math> each, <math>4</math>-pound rocks worth <math>\$11</math> each, and <math>1</math>-pound rocks worth <math>\$2</math> each. There are at least <math>20</math> of each size. He can carry at most <math>18</math> pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
+<math>\textbf{(A) } 48 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 50 \qquad \textbf{(D) } 51 \qquad \textbf{(E) } 52 </math>
+[[2018 AMC 12A  Problems/Problem 2|Solution]]
+==Problem 3==
+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>
+[[2018 AMC 12A  Problems/Problem 3|Solution]]
+==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 <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>
+[[2018 AMC 12A  Problems/Problem 4|Solution]]
+==Problem 5==
+What is the sum of all possible values of <math>k</math> for which the polynomials <math>x^2 - 3x + 2</math> and <math>x^2 - 5x + k</math> have a root in common?
+<math>\textbf{(A) }3 \qquad\textbf{(B) }4 \qquad\textbf{(C) }5 \qquad\textbf{(D) }6 \qquad\textbf{(E) }10 \qquad</math>
+[[2018 AMC 12A  Problems/Problem 5|Solution]]
+==Problem 6==
+For positive integers <math>m</math> and <math>n</math> such that <math>m+10<n+1</math>, both the mean and the median of the set <math>\{m, m+4, m+10, n+1, n+2, 2n\}</math> are equal to <math>n</math>. What is <math>m+n</math>?
+<math>\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24</math>
+[[2018 AMC 12A  Problems/Problem 6|Solution]]
+==Problem 7==
+For how many (not necessarily positive) integer values of <math>n</math> is the value of <math>4000\cdot \left(\tfrac{2}{5}\right)^n</math> an integer?
+<math>
+\textbf{(A) }3 \qquad
+\textbf{(B) }4 \qquad
+\textbf{(C) }6 \qquad
+\textbf{(D) }8 \qquad
+\textbf{(E) }9 \qquad
+</math>
+[[2018 AMC 12A  Problems/Problem 7|Solution]]
+==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 <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>
+unitsize(5);
+dot((0,0));
+dot((60,0));
+dot((50,10));
+dot((10,10));
+dot((30,30));
+draw((0,0)--(60,0)--(50,10)--(30,30)--(10,10)--(0,0));
+draw((10,10)--(50,10));
+label("$B$",(0,0),SW);
+label("$C$",(60,0),SE);
+label("$E$",(50,10),E);
+label("$D$",(10,10),W);
+label("$A$",(30,30),N);
+draw((10,10)--(15,15)--(20,10)--(25,15)--(30,10)--(35,15)--(40,10)--(45,15)--(50,10));
+draw((15,15)--(45,15));
+</asy>
+<math>\textbf{(A) }   16   \qquad        \textbf{(B) }   18   \qquad    \textbf{(C) }   20   \qquad   \textbf{(D) }  22 \qquad  \textbf{(E) }   24 </math>
+[[2018 AMC 12A  Problems/Problem 8|Solution]]
+==Problem 9==
+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?
+<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]]
+==Problem 10==
+How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of equations?
+<cmath>\begin{align*}
+x+3y&=3 \\
+\big||x|-|y|\big|&=1
+\end{align*}</cmath>
+<math>\textbf{(A) } 1 \qquad
+\textbf{(B) } 2 \qquad
+\textbf{(C) } 3 \qquad
+\textbf{(D) } 4 \qquad
+\textbf{(E) } 8 </math>
+[[2018 AMC 12A  Problems/Problem 10|Solution]]
+==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?
+<asy>
+draw((0,0)--(4,0)--(4,3)--(0,0));
+label("$A$", (0,0), SW);
+label("$B$", (4,3), NE);
+label("$C$", (4,0), SE);
+label("$4$", (2,0), S);
+label("$3$", (4,1.5), E);
+label("$5$", (2,1.5), NW);
+fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray);
+</asy>
+<math>\textbf{(A) }   1+\frac12 \sqrt2   \qquad        \textbf{(B) }   \sqrt3   \qquad    \textbf{(C) }   \frac74   \qquad   \textbf{(D) }  \frac{15}{8} \qquad  \textbf{(E) }   2 </math>
+[[2018 AMC 12A  Problems/Problem 11|Solution]]
+==Problem 12==
+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>
+[[2018 AMC 12A  Problems/Problem 12|Solution]]
+==Problem 13==
+How many nonnegative integers can be written in the form <cmath>a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,</cmath>
+where <math>a_i\in \{-1,0,1\}</math> for <math>0\le i \le 7</math>?
+<math>\textbf{(A) } 512 \qquad
+\textbf{(B) } 729 \qquad
+\textbf{(C) } 1094 \qquad
+\textbf{(D) } 3281 \qquad
+\textbf{(E) } 59,048 </math>
+[[2018 AMC 12A  Problems/Problem 13|Solution]]
+==Problem 14==
+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
+\textbf{(B) } 13   \qquad
+\textbf{(C) } 17   \qquad
+\textbf{(D) } 31 \qquad
+\textbf{(E) } 35 </math>
+[[2018 AMC 12A  Problems/Problem 14|Solution]]
+==Problem 15==
+A scanning code consists of a <math>7 \times 7</math> 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 <math>49</math> squares. A scanning code is called <math>\textit{symmetric}</math> if its look does not change when the entire square is rotated by a multiple of <math>90 ^{\circ}</math> 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?
+<math>\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}</math>
+[[2018 AMC 12A  Problems/Problem 15|Solution]]
+==Problem 16==
+Which of the following describes the set of values of <math>a</math> for which the curves <math>x^2+y^2=a^2</math> and <math>y=x^2-a</math> in the real <math>xy</math>-plane intersect at exactly <math>3</math> points?
+<math>
+\textbf{(A) }a=\frac14 \qquad
+\textbf{(B) }\frac14 < a < \frac12 \qquad
+\textbf{(C) }a>\frac14 \qquad
+\textbf{(D) }a=\frac12 \qquad
+\textbf{(E) }a>\frac12 \qquad
+</math>
+[[2018 AMC 12A  Problems/Problem 16|Solution]]
+==Problem 17==
+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>
+/* Edited by MRENTHUSIASM */
+size(160);
+pair A, B, C, D, F;
+A = origin;
+B = (4,0);
+C = (0,3);
+D = (2/7,2/7);
+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>
+<math>\textbf{(A) }   \frac{25}{27}   \qquad        \textbf{(B) }   \frac{26}{27}   \qquad    \textbf{(C) }   \frac{73}{75}   \qquad   \textbf{(D) } \frac{145}{147} \qquad  \textbf{(E) }   \frac{74}{75} </math>
+[[2018 AMC 12A  Problems/Problem 17|Solution]]
+==Problem 18==
+Triangle <math>ABC</math> with <math>AB=50</math> and <math>AC=10</math> has area <math>120</math>. Let <math>D</math> be the midpoint of <math>\overline{AB}</math>, and let <math>E</math> be the midpoint of <math>\overline{AC}</math>. The angle bisector of <math>\angle BAC</math> intersects <math>\overline{DE}</math> and <math>\overline{BC}</math> at <math>F</math> and <math>G</math>, respectively. What is the area of quadrilateral <math>FDBG</math>?
+<math>
+\textbf{(A) }60 \qquad
+\textbf{(B) }65 \qquad
+\textbf{(C) }70 \qquad
+\textbf{(D) }75 \qquad
+\textbf{(E) }80 \qquad
+</math>
+[[2018 AMC 12A  Problems/Problem 18|Solution]]
+==Problem 19==
+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) } 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]]
+==Problem 20==
+Triangle <math>ABC</math> is an isosceles right triangle with <math>AB=AC=3</math>. Let <math>M</math> be the midpoint of hypotenuse <math>\overline{BC}</math>. Points <math>I</math> and <math>E</math> lie on sides <math>\overline{AC}</math> and <math>\overline{AB}</math>, respectively, so that <math>AI>AE</math> and <math>AIME</math> is a cyclic quadrilateral. Given that triangle <math>EMI</math> has area <math>2</math>, the length <math>CI</math> can be written as <math>\frac{a-\sqrt{b}}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers and <math>b</math> is not divisible by the square of any prime. What is the value of <math>a+b+c</math>?
+<math>
+\textbf{(A) }9 \qquad
+\textbf{(B) }10 \qquad
+\textbf{(C) }11 \qquad
+\textbf{(D) }12 \qquad
+\textbf{(E) }13 \qquad
+</math>
+[[2018 AMC 12A  Problems/Problem 20|Solution]]
+==Problem 21==
+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>
+[[2018 AMC 12A  Problems/Problem 21|Solution]]
+==Problem 22==
+The solutions to the equations <math>z^2=4+4\sqrt{15}i</math> and <math>z^2=2+2\sqrt 3i,</math> where <math>i=\sqrt{-1},</math> form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form <math>p\sqrt q-r\sqrt s,</math> where <math>p,</math> <math>q,</math> <math>r,</math> and <math>s</math> are positive integers and neither <math>q</math> nor <math>s</math> is divisible by the square of any prime number. What is <math>p+q+r+s?</math>
+<math>\textbf{(A) } 20 \qquad
+\textbf{(B) } 21 \qquad
+\textbf{(C) } 22 \qquad
+\textbf{(D) } 23 \qquad
+\textbf{(E) } 24 </math>
+[[2018 AMC 12A  Problems/Problem 22|Solution]]
+==Problem 23==
+In <math>\triangle PAT,</math> <math>\angle P=36^{\circ},</math> <math>\angle A=56^{\circ},</math> and <math>PA=10.</math> Points <math>U</math> and <math>G</math> lie on sides <math>\overline{TP}</math> and <math>\overline{TA},</math> respectively, so that <math>PU=AG=1.</math> Let <math>M</math> and <math>N</math> be the midpoints of segments <math>\overline{PA}</math> and <math>\overline{UG},</math> respectively. What is the degree measure of the acute angle formed by lines <math>MN</math> and <math>PA?</math>
+<math>\textbf{(A) } 76 \qquad
+\textbf{(B) } 77 \qquad
+\textbf{(C) } 78 \qquad
+\textbf{(D) } 79 \qquad
+\textbf{(E) } 80 </math>
+[[2018 AMC 12A  Problems/Problem 23|Solution]]
+==Problem 24==
+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>
+\textbf{(A) }\frac{1}{2}\qquad
+\textbf{(B) }\frac{13}{24} \qquad
+\textbf{(C) }\frac{7}{12} \qquad
+\textbf{(D) }\frac{5}{8} \qquad
+\textbf{(E) }\frac{2}{3}\qquad
+</math>
+[[2018 AMC 12A  Problems/Problem 24|Solution]]
+==Problem 25==
+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) } 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]]
+==See also==
+{{AMC12 box|year=2018|ab=A|before=[[2017 AMC 12B Problems]]|after=[[2018 AMC 12B Problems]]}}
+{{MAA Notice}}