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Difference between revisions of "2018 AMC 12A Problems"
Line 92: Line 92:
  
 
==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?</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==
 +
 +
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]]
 
[[2018 AMC 12A  Problems/Problem 13|Solution]]
 
==Problem 14==
 
==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]]
 
[[2018 AMC 12A  Problems/Problem 14|Solution]]
Line 104: Line 125:
 
[[2018 AMC 12A  Problems/Problem 15|Solution]]
 
[[2018 AMC 12A  Problems/Problem 15|Solution]]
 
==Problem 16==
 
==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]]
 
[[2018 AMC 12A  Problems/Problem 16|Solution]]
 
==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?
 +
 +
<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]]
 
[[2018 AMC 12A  Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==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]]
 
[[2018 AMC 12A  Problems/Problem 18|Solution]]
 
==Problem 19==
 
==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]]
 
[[2018 AMC 12A  Problems/Problem 19|Solution]]
 +
 
==Problem 20==
 
==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]]
 
[[2018 AMC 12A  Problems/Problem 20|Solution]]
 +
 
==Problem 21==
 
==Problem 21==
 +
  
 
[[2018 AMC 12A  Problems/Problem 21|Solution]]
 
[[2018 AMC 12A  Problems/Problem 21|Solution]]
 +
 
==Problem 22==
 
==Problem 22==
  
 
[[2018 AMC 12A  Problems/Problem 22|Solution]]
 
[[2018 AMC 12A  Problems/Problem 22|Solution]]
 +
 
==Problem 23==
 
==Problem 23==
  
 
[[2018 AMC 12A  Problems/Problem 23|Solution]]
 
[[2018 AMC 12A  Problems/Problem 23|Solution]]
 +
 
==Problem 24==
 
==Problem 24==
  
 
[[2018 AMC 12A  Problems/Problem 24|Solution]]
 
[[2018 AMC 12A  Problems/Problem 24|Solution]]
 +
 
==Problem 25==
 
==Problem 25==
  
 
[[2018 AMC 12A  Problems/Problem 25|Solution]]
 
[[2018 AMC 12A  Problems/Problem 25|Solution]]

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

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