Found problems: 133
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.
2016 AMC 12/AHSME, 24
There are exactly $77,000$ ordered quadruples $(a,b,c,d)$ such that $\gcd(a,b,c,d)=77$ and $\operatorname{lcm}(a,b,c,d)=n$. What is the smallest possible value of $n$?
$\textbf{(A)}\ 13,860 \qquad
\textbf{(B)}\ 20,790 \qquad
\textbf{(C)}\ 21,560 \qquad
\textbf{(D)}\ 27,720 \qquad
\textbf{(E)}\ 41,580$
2013 AMC 12/AHSME, 14
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of $N$?
${ \textbf{(A)}\ 55\qquad\textbf{(B)}\ 89\qquad\textbf{(C)}\ 104\qquad\textbf{(D}}\ 144\qquad\textbf{(E)}\ 273 $
2020 AMC 10, 10
A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
[asy]
draw(Arc((0,0), 4, 0, 270));
draw((0,-4)--(0,0)--(4,0));
label("$4$", (2,0), S);
[/asy]
$\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$
2021 AMC 12/AHSME Spring, 18
Let $z$ be a complex number satisfying $12\lvert z\rvert^2 = 2 \lvert z+2 \rvert ^2+\lvert z^2+1\rvert ^2+31.$ What is the value of $z+\frac{6}{z}?$
$\textbf{(A) }-2\qquad\textbf{(B) }-1\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }1\qquad\textbf{(E) }4$
2019 AMC 10, 20
As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\overarc{AEB},\overarc{BFC},$ and $\overarc{CGD},$ have their diameters on $\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F. $ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form
\[\frac{a}{b}\cdot\pi-\sqrt{c}+d,\]
where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$?
[asy]
size(6cm);
filldraw(circle((0,0),2), gray(0.7));
filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0));
filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0));
filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0));
dot((-3,-1));
label("$A$",(-3,-1),S);
dot((-2,0));
label("$E$",(-2,0),NW);
dot((-1,-1));
label("$B$",(-1,-1),S);
dot((0,0));
label("$F$",(0,0),N);
dot((1,-1));
label("$C$",(1,-1), S);
dot((2,0));
label("$G$", (2,0),NE);
dot((3,-1));
label("$D$", (3,-1), S);
[/asy]
$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17$
2021 AMC 12/AHSME Fall, 17
A bug starts at a vertex of a grid made of equilateral triangles of side length $1$. At each step the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. What is the probability that after $5$ moves the bug never will have been more than $1$ unit away from the starting position?
$\textbf{(A)}\ \frac{13}{108} \qquad\textbf{(B)}\ \frac{7}{54} \qquad\textbf{(C)}\ \frac{29}{216} \qquad\textbf{(D)}\
\frac{4}{27} \qquad\textbf{(E)}\ \frac{1}{16}$
2013 AMC 10, 21
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of $N$?
${ \textbf{(A)}\ 55\qquad\textbf{(B)}\ 89\qquad\textbf{(C)}\ 104\qquad\textbf{(D}}\ 144\qquad\textbf{(E)}\ 273 $
2012 AMC 12/AHSME, 21
Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square?
[asy]
size(200);
defaultpen(linewidth(1));
pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60);
pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A;
draw(A--B--C--D--E--F--cycle);
draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype("2 2"));
dot("$A$",A,W,linewidth(4));
dot("$B$",B,dir(0),linewidth(4));
dot("$C$",C,dir(0),linewidth(4));
dot("$D$",D,dir(20),linewidth(4));
dot("$E$",E,dir(100),linewidth(4));
dot("$F$",F,W,linewidth(4));
dot("$X$",X,dir(0),linewidth(4));
dot("$Y$",Y,N,linewidth(4));
dot("$Z$",Z,W,linewidth(4));
[/asy]
$ \textbf{(A)}\ 29\sqrt{3} \qquad\textbf{(B)}\ \frac{21}{2}\sqrt{2}+\frac{41}{2}\sqrt{3}\qquad\textbf{(C)}\ 20\sqrt{3}+16$
$\textbf{(D)}\ 20\sqrt{2}+13\sqrt{3}
\qquad\textbf{(E)}\ 21\sqrt{6}$
2021 AMC 12/AHSME Fall, 18
Set $u_0 = \frac{1}{4},$ and for $k \geq 0$ let $u_{k+1}$ be determined by the recurrence $u_{k+1} = 2u_k - 2u_k^2.$ This sequence tends to a limit, call it $L.$ What is the least value of $k$ such that $$|u_k - L| \leq \frac{1}{2^{1000}}?$$
$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 97 \qquad\textbf{(C)}\ 253 \qquad\textbf{(D)}\
329 \qquad\textbf{(E)}\ 401$
2020 AMC 12/AHSME, 10
In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$
$\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}$
2024 AMC 10, 3
For how many integer values of $x$ is $|2x|\leq 7\pi?$
$\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$
2024 AMC 12/AHSME, 6
The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base $5$? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem.)
$
\textbf{(A) }18 \qquad
\textbf{(B) }20 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }24 \qquad
\textbf{(E) }26 \qquad
$
2016 AMC 10, 14
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$
$\textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 41 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 57$
2024 AMC 12/AHSME, 3
For how many integer values of $x$ is $|2x|\leq 7\pi?$
$\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$
2021 AMC 10 Fall, 8
The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor of $16383$?
$\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }10\qquad\textbf{(D) }16\qquad\textbf{(E) }22$
2020 AMC 12/AHSME, 14
Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
$\textbf{(A) } \text{Bela will always win.}$
$\textbf{(B) } \text{Jenn will always win.} $
$\textbf{(C) } \text{Bela will win if and only if }n \text{ is odd.}$
$\textbf{(D) } \text{Jenn will win if and only if }n \text{ is odd.} $
$\textbf{(E) } \text{Jenn will win if and only if }n > 8.$
2016 AMC 12/AHSME, 19
Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?
$\textbf{(A)}\ \frac{1}{8} \qquad
\textbf{(B)}\ \frac{1}{7} \qquad
\textbf{(C)}\ \frac{1}{6} \qquad
\textbf{(D)}\ \frac{1}{4} \qquad
\textbf{(E)}\ \frac{1}{3}$
2024 AMC 12/AHSME, 12
Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?
$\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\displaystyle\frac{4}{3}\qquad\textbf{(D)}~\displaystyle\frac{3}{2}\qquad\textbf{(E)}~\displaystyle\frac{5}{3}$
2013 AMC 12/AHSME, 13
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA=\angle DCB$ and $\angle ADB=\angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?
${\textbf{(A)}\ 210\qquad\textbf{(B)}\ 220\qquad\textbf{(C)}\ 230\qquad\textbf{(D}}\ 240\qquad\textbf{(E)}\ 250$
2020 AMC 12/AHSME, 5
Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\tfrac{2}{3}$ of its games and team $B$ has won $\tfrac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ more games than team $A.$ How many games has team $A$ played?
$\textbf{(A) } 21 \qquad \textbf{(B) } 27 \qquad \textbf{(C) } 42 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 63$
2016 AMC 12/AHSME, 22
For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period $6$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie?
$\textbf{(A)}\ [1,200] \qquad
\textbf{(B)}\ [201,400] \qquad
\textbf{(C)}\ [401,600] \qquad
\textbf{(D)}\ [601,800] \qquad
\textbf{(E)}\ [801,999] $
2018 AMC 12/AHSME, 13
Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\triangle{ABP}$, $\triangle{BCP}$, $\triangle{CDP}$, and $\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral?
[asy]
unitsize(120);
pair B = (0, 0), A = (0, 1), D = (1, 1), C = (1, 0), P = (1/4, 2/3);
draw(A--B--C--D--cycle);
dot(P);
defaultpen(fontsize(10pt));
draw(A--P--B);
draw(C--P--D);
label("$A$", A, W);
label("$B$", B, W);
label("$C$", C, E);
label("$D$", D, E);
label("$P$", P, N*1.5+E*0.5);
dot(A);
dot(B);
dot(C);
dot(D);
[/asy]
$\textbf{(A) }100\sqrt{2}\qquad\textbf{(B) }100\sqrt{3}\qquad\textbf{(C) }200\qquad\textbf{(D) }200\sqrt{2}\qquad\textbf{(E) }200\sqrt{3}$
2017 AMC 12/AHSME, 24
Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. What is $\tfrac{AB}{BC}$?
$\textbf{(A) \ } 1+\sqrt{2} \qquad \textbf{(B) \ } 2+\sqrt{2}\qquad \textbf{(C) \ } \sqrt{17}\qquad \textbf{(D) \ } 2+\sqrt{5} \qquad \textbf{(E) \ } 1+2\sqrt{3}$