Found problems: 35
2017 AMC 10, 11
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$
2017 AMC 12/AHSME, 9
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2017 AMC 12/AHSME, 14
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?
$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$
2017 AMC 12/AHSME, 7
Define a function on the positive integers recursively by $f(1) = 2$, $f(n) = f(n-1) + 1$ if $n$ is even, and $f(n) = f(n-2) + 2$ if $n$ is odd and greater than $1$. What is $f(2017)$?
$\textbf{(A) } 2017 \qquad \textbf{(B) } 2018 \qquad \textbf{(C) } 4034 \qquad \textbf{(D) } 4035 \qquad \textbf{(E) } 4036$
2017 AMC 10, 7
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
$\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$
2017 AMC 10, 20
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$
2017 AMC 12/AHSME, 15
Let $f(x)=\sin x+2\cos x+3\tan x$, using radian measure for the variable $x$. In what interval does the smallest positive value of $x$ for which $f(x)=0$ lie?
$\textbf{(A) } (0,1) \qquad \textbf{(B) } (1,2) \qquad \textbf{(C) } (2,3) \qquad \textbf{(D) } (3,4) \qquad \textbf{(E) } (4,5)$
2017 AMC 12/AHSME, 24
Quadrilateral $ABCD$ is inscribed in circle $O$ and has sides $AB = 3$, $BC = 2$, $CD = 6$, and $DA = 8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that
\[\frac{DX}{BD} = \frac{1}{4} \quad \text{and} \quad \frac{BY}{BD} = \frac{11}{36}.\]
Let $E$ be the intersection of intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF \cdot XG$?
$\textbf{(A) }17\qquad\textbf{(B) }\frac{59 - 5\sqrt{2}}{3}\qquad\textbf{(C) }\frac{91 - 12\sqrt{3}}{4}\qquad\textbf{(D) }\frac{67 - 10\sqrt{2}}{3}\qquad\textbf{(E) }18$
2017 AMC 10, 19
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?
$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$
2017 AMC 12/AHSME, 13
Driving at a constant speed, Sharon usually takes $180$ minutes to drive from her house to her mother's house. One day Sharon begins the drive at her usual speed, but after driving $\frac{1}{3}$ of the way, she hits a bad snowstorm and reduces her speed by $20$ miles per hour. This time the trip takes her a total of $276$ minutes. How many miles is it from Sharon's house to her mother's house?
$\textbf{(A)}\ 132\qquad\textbf{(B)}\ 135\qquad\textbf{(C)}\ 138\qquad\textbf{(D)}\ 141\qquad\textbf{(E)}\ 144$
2017 AMC 12/AHSME, 4
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
$\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$
2017 AMC 12/AHSME, 6
Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$
2017 AMC 12/AHSME, 8
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$
2017 AMC 12/AHSME, 2
The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$
2017 AMC 10, 24
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$?
$\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$
2017 AMC 10, 12
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2017 AMC 12/AHSME, 17
There are 24 different complex numbers $z$ such that $z^{24} = 1$. For how many of these is $z^6$ a real number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }24$
2017 AMC 10, 6
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically?
$\textbf{(A)}$ If Lewis did not receive an A, then he got all of the multiple choice questions wrong. \\
$\textbf{(B)}$ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. \\
$\textbf{(C)}$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. \\
$\textbf{(D)}$ If Lewis received an A, then he got all of the multiple choice questions right. \\
$\textbf{(E)}$ If Lewis received an A, then he got at least one of the multiple choice questions right.
2017 AMC 12/AHSME, 16
In the figure below, semicircles with centers at $A$ and $B$ and with radii $2$ and $1$, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter $\overline{JK}$. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at $P$ is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at $P$?
[asy]
size(8cm);
draw(arc((0,0),3,0,180));
draw(arc((2,0),1,0,180));
draw(arc((-1,0),2,0,180));
draw((-3,0)--(3,0));
pair P = (-1,0)+(2+6/7)*dir(36.86989);
draw(circle(P,6/7));
dot((-1,0)); dot((2,0)); dot((-3,0)); dot((3,0)); dot(P);
label("$J$",(-3,0),W);
label("$A$",(-1,0),NW);
label("$B$",(2,0),NE);
label("$K$",(3,0),E);
label("$P$",P,NW);
[/asy]
$ \textbf{(A)}\ \frac{3}{4}
\qquad \textbf{(B)}\ \frac{6}{7}
\qquad\textbf{(C)}\ \frac{1}{2}\sqrt{3}
\qquad\textbf{(D)}\ \frac{5}{8}\sqrt{2}
\qquad\textbf{(E)}\ \frac{11}{12} $
2017 AMC 12/AHSME, 18
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$
2017 AMC 10, 5
The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$
2017 AMC 10, 8
At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
$\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$
2017 AMC 12/AHSME, 25
The vertices $V$ of a centrally symmetric hexagon in the complex plane are given by
$$V=\left\{ \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.$$
For each $j$, $1\leq j\leq 12$, an element $z_j$ is chosen from $V$ at random, independently of the other choices. Let $P={\prod}_{j=1}^{12}z_j$ be the product of the $12$ numbers selected. What is the probability that $P=-1$?
$\textbf{(A) } \dfrac{5\cdot11}{3^{10}} \qquad \textbf{(B) } \dfrac{5^2\cdot11}{2\cdot3^{10}} \qquad \textbf{(C) } \dfrac{5\cdot11}{3^{9}} \qquad \textbf{(D) } \dfrac{5\cdot7\cdot11}{2\cdot3^{10}} \qquad \textbf{(E) } \dfrac{2^2\cdot5\cdot11}{3^{10}}$
2017 AMC 12/AHSME, 3
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically?
$\textbf{(A)}$ If Lewis did not receive an A, then he got all of the multiple choice questions wrong. \\
$\textbf{(B)}$ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. \\
$\textbf{(C)}$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. \\
$\textbf{(D)}$ If Lewis received an A, then he got all of the multiple choice questions right. \\
$\textbf{(E)}$ If Lewis received an A, then he got at least one of the multiple choice questions right.
2017 AMC 10, 16
There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$