Difference between revisions of "2018 AMC 12A Problems"

Revision as of 00:07, 9 February 2018

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

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

Solution

Problem 8

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? $x+3y=3$ $\big||x|-|y|\big|=1$ $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 8$

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 solutions to the equation $\log_{3x} 4 = \log_{2x} 8$ , where $x$ is a positive real number other than $\tfrac{1}{3}$ or $\tfrac{1}{2}$ , can be written as $\tfrac {p}{q}$ 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

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] draw((0,0)--(4,0)--(0,3)--(0,0)); draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0)); fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray); label("$4$", (2,0), N); label("$3$", (0,1.5), E); label("$2$", (.8,1), E); label("$S$", (0,0), NE); draw((0.3,0.3)--(1.4,1.9), 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 $m+n$ ?

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

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

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

@@ Line 92: / Line 92: @@
 ==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?</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 solutions to the equation <math>\log_{3x} 4 = \log_{2x} 8</math>, where <math>x</math> is a positive real number other than <math>\tfrac{1}{3}</math> or <math>\tfrac{1}{2}</math>, can be written as <math>\tfrac {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]]
@@ Line 104: / Line 125: @@
 [[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 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?
+<asy>
+draw((0,0)--(4,0)--(0,3)--(0,0));
+draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0));
+fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray);
+label("$4$", (2,0), N);
+label("$3$", (0,1.5), E);
+label("$2$", (.8,1), E);
+label("$S$", (0,0), NE);
+draw((0.3,0.3)--(1.4,1.9), 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)} \text{ 16} \qquad \textbf{(B)} \text{ 17} \qquad \textbf{(C)} \text{ 19} \qquad \textbf{(D)} \text{ 23} \qquad \textbf{(E)} \text{ 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==
 [[2018 AMC 12A  Problems/Problem 21|Solution]]
 ==Problem 22==
 [[2018 AMC 12A  Problems/Problem 22|Solution]]
 ==Problem 23==
 [[2018 AMC 12A  Problems/Problem 23|Solution]]
 ==Problem 24==
 [[2018 AMC 12A  Problems/Problem 24|Solution]]
 ==Problem 25==
 [[2018 AMC 12A  Problems/Problem 25|Solution]]

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Difference between revisions of "2018 AMC 12A Problems"

Revision as of 00:07, 9 February 2018

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