Found problems: 3632
2020 AMC 12/AHSME, 15
In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. What is the greatest distance between a point of $A$ and a point of $B?$
$\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9 + \sqrt{3}$
2007 AMC 10, 24
Circles centered at $ A$ and $ B$ each have radius $ 2$, as shown. Point $ O$ is the midpoint of $ \overline{AB}$, and $ OA \equal{} 2\sqrt {2}$. Segments $ OC$ and $ OD$ are tangent to the circles centered at $ A$ and $ B$, respectively, and $ EF$ is a common tangent. What is the area of the shaded region $ ECODF$?
[asy]unitsize(6mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=3;
pair O=(0,0);
pair A=(-2*sqrt(2),0);
pair B=(2*sqrt(2),0);
pair G=shift(0,2)*A;
pair F=shift(0,2)*B;
pair C=shift(A)*scale(2)*dir(45);
pair D=shift(B)*scale(2)*dir(135);
pair X=A+2*dir(-60);
pair Y=B+2*dir(240);
path P=C--O--D--Arc(B,2,135,90)--G--Arc(A,2,90,45)--cycle;
fill(P,gray);
draw(Circle(A,2));
draw(Circle(B,2));
dot(A);
label("$A$",A,W);
dot(B);
label("$B$",B,E);
dot(C);
label("$C$",C,W);
dot(D);
label("$D$",D,E);
dot(G);
label("$E$",G,N);
dot(F);
label("$F$",F,N);
dot(O);
label("$O$",O,S);
draw(G--F);
draw(C--O--D);
draw(A--B);
draw(A--X);
draw(B--Y);
label("$2$",midpoint(A--X),SW);
label("$2$",midpoint(B--Y),SE);[/asy]$ \textbf{(A)}\ \frac {8\sqrt {2}}{3}\qquad \textbf{(B)}\ 8\sqrt {2} \minus{} 4 \minus{} \pi \qquad \textbf{(C)}\ 4\sqrt {2}$
$ \textbf{(D)}\ 4\sqrt {2} \plus{} \frac {\pi}{8}\qquad \textbf{(E)}\ 8\sqrt {2} \minus{} 2 \minus{} \frac {\pi}{2}$
2000 AMC 8, 14
What is the units digit of $19^{19} + 99^{99}$?
$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$
2022 AMC 12/AHSME, 18
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself?
$\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$
1999 AIME Problems, 6
A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the greatest integer that does not exceed $k.$
1992 AMC 12/AHSME, 17
The two digit integers from $19$ to $92$ are written consecutively to form the larger integer $N = 19202122\ldots909192$. If $3^{k}$ is the highest power of $3$ that is a factor of $N$, then $k =$
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \text{more than 3} $
1997 AMC 8, 3
Which of the following numbers is the largest?
$\textbf{(A)}\ 0.97 \qquad \textbf{(B)}\ 0.979 \qquad \textbf{(C)}\ 0.9709 \qquad \textbf{(D)}\ 0.907 \qquad \textbf{(E)}\ 0.9089$
2014 AMC 12/AHSME, 23
The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$?
$\textbf{(A) }874\qquad
\textbf{(B) }883\qquad
\textbf{(C) }887\qquad
\textbf{(D) }891\qquad
\textbf{(E) }892\qquad$
1975 AMC 12/AHSME, 24
In triangle $ABC$, $\measuredangle C=\theta$ and $\measuredangle B=2\theta$, where $0^{\circ} <\theta < 60^{\circ}$. The circle with center $A$ and radius $AB$ intersects $AC$ at $D$ and intersects $BC$, extended if necessary, at $B$ and at $E$ ($E$ may coincide with $B$). Then $EC=AD$
$ \textbf{(A)}\ \text{for no values of}\ \theta \qquad\textbf{(B)}\ \text{only if}\ \theta=45^{\circ} \qquad\textbf{(C)}\ \text{only if}\ 0^{\circ} < \theta \le 45^{\circ} \\ \qquad\textbf{(D)}\ \text{only if}\ 45^{\circ} \le \theta < 60^{\circ} \qquad\textbf{(E)}\ \text{for all}\ \theta \ \text{such that}\ 0^{\circ} <\theta < 60^{\circ} $
2011 AMC 10, 9
The area of $\triangle EBD$ is one third of the area of $3-4-5$ $ \triangle ABC$. Segment $DE$ is perpendicular to segment $AB$. What is $BD$?
[asy]
unitsize(10mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair A=(0,0), B=(5,0), C=(1.8,2.4), D=(5-4sqrt(3)/3,0), E=(5-4sqrt(3)/3,sqrt(3));
pair[] ps={A,B,C,D,E};
draw(A--B--C--cycle);
draw(E--D);
draw(rightanglemark(E,D,B));
dot(ps);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$D$",D,S);
label("$E$",E,NE);
label("$3$",midpoint(A--C),NW);
label("$4$",midpoint(C--B),NE);
label("$5$",midpoint(A--B),SW);[/asy]
$ \textbf{(A)}\ \frac{4}{3} \qquad
\textbf{(B)}\ \sqrt{5} \qquad
\textbf{(C)}\ \frac{9}{4} \qquad
\textbf{(D)}\ \frac{4\sqrt{3}}{3} \qquad
\textbf{(E)}\ \frac{5}{2} $
2011 AIME Problems, 5
The sum of the first 2011 terms of a geometric series is 200. The sum of the first 4022 terms of the same series is 380. Find the sum of the first 6033 terms of the series.
2004 India IMO Training Camp, 2
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
2007 AIME Problems, 6
An integer is called [i]parity-monotonic[/i] if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ is $a_{i}$ is even. How many four-digit parity-monotonic integers are there?
1996 AMC 12/AHSME, 26
An urn contains marbles of four colors: red, white, blue, and green. When
four marbles are drawn without replacement, the following events are equally
likely:
(a) the selection of four red marbles;
(b) the selection of one white and three red marbles;
(c) the selection of one white, one blue, and two red marbles; and
(d) the selection of one marble of each color.
What is the smallest number of marbles satisfying the given condition?
$\text{(A)}\ 19 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 46 \qquad \text{(D)}\ 69\qquad \text{(E)}\ \text{more than 69}$
1986 USAMO, 3
What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be
\[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]
2019 AMC 10, 25
How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?
$\textbf{(A) }55\qquad\textbf{(B) }60\qquad\textbf{(C) }65\qquad\textbf{(D) }70\qquad\textbf{(E) }75$
2011 AIME Problems, 9
Let $x_1,x_2,\dots ,x_6$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5+x_6=1$, and $x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1x_2x_3+x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2$. Find $p+q$.
2025 AIME, 6
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and $\overline{BC} \perp \overline{AD}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such that $\overline{EF} \perp \overline{BC}$, $C$ is closer to $\overline{GH}$ than to $\overline{EF}$, and $D$ is closer to $\overline{FG}$ than to $\overline{EH}$, as shown. Triangles $\triangle DGF$ and $\triangle CHG$ have equal areas. The area of rectangle $EFGH$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(5cm);
defaultpen(fontsize(10pt));
pair A = (9, 0), B = (15, 0), C = (-15, 0), D = (9, 12), E = (9+12/sqrt(5), -6/sqrt(5)), F = (9+12/sqrt(5), 6/sqrt(5)), G = (9-12/sqrt(5), 6/sqrt(5)), H = (9-12/sqrt(5), -6/sqrt(5));
filldraw(G--H--C--cycle, lightgray);
filldraw(D--G--F--cycle, lightgray);
draw(B--C);
draw(A--D);
draw(E--F--G--H--cycle);
draw(circle(origin, 15));
draw(circle(A, 6));
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
label("$A$", A, (.8, -.8));
label("$B$", B, (.8, 0));
label("$C$", C, (-.8, 0));
label("$D$", D, (.4, .8));
label("$E$", E, (.8, -.8));
label("$F$", F, (.8, .8));
label("$G$", G, (-.8, .8));
label("$H$", H, (-.8, -.8));
label("$\omega_1$", (9, -5));
label("$\omega_2$", (-1, -13.5));
[/asy]
2012 AMC 12/AHSME, 18
Let $(a_1,a_2, \dots ,a_{10})$ be a list of the first $10$ positive integers such that for each $2 \le i \le 10$ either $a_i+1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?
$ \textbf{(A)}\ 120\qquad\textbf{(B)}\ 512\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 181,440\qquad\textbf{(E)}\ 362,880 $
1996 AMC 8, 7
Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has $4$ goldfish at the same time that Gretel has $128$ goldfish, then in how many months from that time will they have the same number of goldfish?
$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$
2010 AMC 8, 4
What is the sum of the mean, median, and mode of the numbers, $2,3,0,3,1,4,0,3$?
$ \textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 7.5\qquad\textbf{(D)}\ 8.5\qquad\textbf{(E)}\ 9 $
1969 AMC 12/AHSME, 15
In a circle with center at $O$ and radius $r$, chord $AB$ is drawn with length equal to $r$ (units). From $O$ a perpendicular to $AB$ meets $AB$ at $M$. From $M$ a perpendicular to $OA$ meets $OA$ at $D$. In terms of $r$ the area of triangle $MDA$, in appropriate square units, is:
$\textbf{(A) }\dfrac{3r^2}{16}\qquad
\textbf{(B) }\dfrac{\pi r^2}{16}\qquad
\textbf{(C) }\dfrac{\pi r^2\sqrt2}{8}\qquad
\textbf{(D) }\dfrac{r^2\sqrt3}{32}\qquad
\textbf{(E) }\dfrac{r^2\sqrt6}{48}$
2020 CHMMC Winter (2020-21), 3
For two base-10 positive integers $a$ and $b$, we say $a \sim b$ if we can rearrange the digits of $a$ in some way to obtain $b$, where the leading digit of both $a$ and $b$ is nonzero. For instance, $463 \sim 463$ and $634 \sim 463$. Find the number of $11$-digit positive integers $K$ such that $K$ is divisible by $2$, $3$, and $5$, and there is some positive integer $K'$ such that $K' \sim K$ and $K'$ is divisible by $7$, $11$, $13$, $17$, $101$, and $9901$.
2013 AMC 12/AHSME, 21
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with a and b integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
${ \textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D}}\ 840\qquad\textbf{(E)}\ 870 $
1970 AMC 12/AHSME, 13
Given the binary operation $\ast$ defined by $a\ast b=a^b$ for all positive numbers $a$ and $b$. The for all positive $a,b,c,n,$ we have
$\textbf{(A) }a\ast b=b\ast a\qquad\textbf{(B) }a\ast (b\ast c)=(a\ast b)\ast c\qquad$
$\textbf{(C) }(a\ast b^n)=(a\ast n)\ast b\qquad\textbf{(D) }(a\ast b)^n=a\ast (bn)\qquad \textbf{(E) }\text{None of these}$