Found problems: 26
2017 AMC 10, 22
The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$?
$\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$
2017 AMC 12/AHSME, 18
The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$?
$\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$
2017 AMC 12/AHSME, 25
A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are among those $9$ people is equal to the reciprocal of the average, over all subsets of size $8$ of the set of $n$ participants, of the number of complete teams whose members are among those $8$ people. How many values $n$, $9\leq n\leq 2017$, can be the number of participants?
$\textbf{(A) } 477 \qquad \textbf{(B) } 482 \qquad \textbf{(C) } 487 \qquad \textbf{(D) } 557 \qquad \textbf{(E) } 562$
2017 AMC 12/AHSME, 1
Kymbrea's comic book collection currently has $30$ comic books in it, and she is adding to her collection at the rate of $2$ comic books per month. LaShawn's comic book collection currently has $10$ comic books in it, and he is adding to his collection at the rate of $6$ comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 25$
2017 AMC 12/AHSME, 23
The graph of $y=f(x)$, where $f(x)$ is a polynomial of degree $3$, contains points $A(2,4)$, $B(3,9)$, and $C(4,16)$. Lines $AB$, $AC$, and $BC$ intersect the graph again at points $D$, $E$, and $F$, respectively, and the sum of the $x$-coordinates of $D$, $E$, and $F$ is $24$. What is $f(0)$?
$\textbf{(A) } -2 \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } \frac{24}{5} \qquad \textbf{(E) } 8$
2017 AMC 12/AHSME, 9
A circle has center $ (-10,-4) $ and radius $13$. Another circle has center $(3,9) $ and radius $\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x+y=c$. What is $c$?
$\textbf{(A)} \text{ 3} \qquad \textbf{(B)} \text{ } 3 \sqrt{3} \qquad \textbf{(C)} \text{ } 4\sqrt{2} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ }\frac{13}{2}$
2017 AMC 10, 18
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(100);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy]
$\textbf{(A) } 6 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15$
2017 AMC 10, 19
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB' = 3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC' = 3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA' = 3CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?
$\textbf{(A) }9:1\qquad\textbf{(B) }16:1\qquad\textbf{(C) }25:1\qquad\textbf{(D) }36:1\qquad\textbf{(E) }37:1$
2017 AMC 12/AHSME, 22
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn - one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins?
$\textbf{(A)} \dfrac{7}{576} \qquad \textbf{(B)} \dfrac{5}{192} \qquad \textbf{(C)} \dfrac{1}{36} \qquad \textbf{(D)} \dfrac{5}{144} \qquad \textbf{(E)}\dfrac{7}{48}$
2017 AMC 12/AHSME, 3
Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\]
$\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$
2017 AMC 12/AHSME, 16
The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
$\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}$
2017 AMC 12/AHSME, 15
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB' = 3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC' = 3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA' = 3CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?
$\textbf{(A) }9:1\qquad\textbf{(B) }16:1\qquad\textbf{(C) }25:1\qquad\textbf{(D) }36:1\qquad\textbf{(E) }37:1$
2017 AMC 12/AHSME, 12
What is the sum of the roots of $z^{12} = 64$ that have a positive real part?
$\textbf{(A) }2 \qquad\textbf{(B) }4 \qquad\textbf{(C) }\sqrt{2} +2\sqrt{3}\qquad\textbf{(D) }2\sqrt{2}+ \sqrt{6} \qquad\textbf{(E) }(1 + \sqrt{3}) + (1+\sqrt{3})i$
2017 AMC 12/AHSME, 13
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(100);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy]
$\textbf{(A) } 6 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15$
2017 AMC 10, 23
Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 44$
2017 AMC 10, 7
Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
$\textbf{(A)}\ 2.0 \qquad \textbf{(B)}\ 2.2\qquad \textbf{(C)}\ 2.8 \qquad \textbf{(D)}\ 3.4 \qquad \textbf{(E)}\ 4.4$
2017 AMC 12/AHSME, 20
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. What is the probability that $\lfloor \log_2{x} \rfloor=\lfloor \log_2{y} \rfloor$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$?
$\textbf{(A)}\ \frac{1}{8}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}$
2017 AMC 12/AHSME, 4
Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
$\textbf{(A)}\ 2.0 \qquad \textbf{(B)}\ 2.2\qquad \textbf{(C)}\ 2.8 \qquad \textbf{(D)}\ 3.4 \qquad \textbf{(E)}\ 4.4$
2017 AMC 12/AHSME, 19
Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 44$
2017 AMC 10, 20
The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
$\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}$
2017 AMC 12/AHSME, 5
The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2=40$, first quartile $Q_1=33$, and third quartile $Q_3=43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartile ($Q_1$) or more than $1.5$ times the interquartile range above the third quartile ($Q_3$), where the interquartile range is defined as $Q_3-Q_1$. How many outliers does this data set have?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$
2017 AMC 10, 4
Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\]
$\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$
2017 AMC 12/AHSME, 6
The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point?
$\textbf{(A)}\ 4\sqrt2 \qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 5\sqrt2 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 6\sqrt2$
2017 AMC 12/AHSME, 17
A coin is biased in such a way that on each toss the probability of heads is $\frac{2}{3}$ and the probability of tails is $\frac{1}{3}$. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B?
$\textbf{(A)} \text{ The probability of winning Game A is }\frac{4}{81}\text{ less than the probability of winning Game B.} $
$\textbf{(B)} \text{ The probability of winning Game A is }\frac{2}{81}\text{ less than the probability of winning Game B.}$
$\textbf{(C)} \text{ The probabilities are the same.}$
$\textbf{(D)} \text{ The probability of winning Game A is }\frac{2}{81}\text{ greater than the probability of winning Game B.}$
$\textbf{(E)} \text{ The probability of winning Game A is }\frac{4}{81}\text{ greater than the probability of winning Game B.}$
2017 AMC 12/AHSME, 7
The functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$. What is the least period of the function $\cos(\sin(x))$?
$\textbf{(A)}\ \frac{\pi}{2}\qquad\textbf{(B)}\ \pi\qquad\textbf{(C)}\ 2\pi\qquad\textbf{(D)}\ 4\pi\qquad\textbf{(E)}$ It's not periodic.