Found problems: 3632
1991 AMC 12/AHSME, 28
Initially an urn contains 100 black marbles and 100 white marbles. Repeatedly, three marbles are removed from the urn and replaced from a pile outside the urn as follows:
\[
\begin{tabular}{ccc}
\textbf{\underline{MARBLES REMOVED}} & & \textbf{\underline{REPLACED WITH}} \\
3 black & & 1 black \\
2 black, 1 white & &1 black, 1 white\\
1 black, 2 white & & 2 white \\
3 white & & 1 black, 1 white
\end{tabular}
\]
Which of the following sets of marbles could be the contents of the urn after repeated applications of this procedure?
$ \textbf{(A)}\ \text{2 black marbles} $
$\textbf{(B)}\ \text{2 white marbles} $
$\textbf{(C)}\ \text{1 black marble} $
$\textbf{(D)}\ \text{1 black and 1 white marble} $
$\textbf{(E)}\ \text{1 white marble} $
2002 AMC 10, 22
A sit of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
$ \text{(A)}\ 10 \qquad
\text{(B)}\ 11 \qquad
\text{(C)}\ 18 \qquad
\text{(D)}\ 19 \qquad
\text{(E)}\ 20$
2021 AMC 10 Fall, 3
The expression $\frac{2021}{2020} - \frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$, where $p$ and $q$ are positive integers whose greatest common divisor is $1$. What is $p$?
$\textbf{(A) }1\qquad\textbf{(B) }9\qquad\textbf{(C) }2020\qquad\textbf{(D) }2021\qquad\textbf{(E) }4041$
2015 AMC 10, 11
Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probablility that the selected number is prime?
$\textbf{(A) } \dfrac{8}{99}
\qquad\textbf{(B) } \dfrac{2}{5}
\qquad\textbf{(C) } \dfrac{9}{20}
\qquad\textbf{(D) } \dfrac{1}{2}
\qquad\textbf{(E) } \dfrac{9}{16}
$
2001 AMC 12/AHSME, 21
Four positive integers $ a,b,c,$ and $ d$ have a product of 8! and satisfy\begin{align*}ab \plus{} a \plus{} b &\equal{} 524\\
bc \plus{} b \plus{} c &\equal{} 146\\
cd \plus{} c \plus{} d &\equal{} 104.\end{align*} What is $ a \minus{} d$?
$ \textbf{(A)} \ 4 \qquad \textbf{(B)} \ 6 \qquad \textbf{(C)} \ 8 \qquad \textbf{(D)} \ 10 \qquad \textbf{(E)} \ 12$
2015 AMC 10, 5
David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place?
$\textbf{(A) } \text{David}
\qquad\textbf{(B) } \text{Hikmet}
\qquad\textbf{(C) } \text{Jack}
\qquad\textbf{(D) } \text{Rand}
\qquad\textbf{(E) } \text{Todd}
$
2010 AMC 12/AHSME, 10
The first four terms of an arithmetic sequence are $ p,9,3p\minus{}q,$ and $ 3p\plus{}q$. What is the $ 2010^{\text{th}}$ term of the sequence?
$ \textbf{(A)}\ 8041\qquad \textbf{(B)}\ 8043\qquad \textbf{(C)}\ 8045\qquad \textbf{(D)}\ 8047\qquad \textbf{(E)}\ 8049$
2016 AMC 12/AHSME, 4
The ratio of the measures of two acute angles is $5:4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
$\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$
1993 AMC 12/AHSME, 11
If $\log_2(\log_2(\log_2(x)))=2$, then how many digits are in the base-ten representation for $x$?
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 13 $
2018 AMC 10, 14
A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list?
$\textbf{(A)}\ 202\qquad\textbf{(B)}\ 223\qquad\textbf{(C)}\ 224\qquad\textbf{(D)}\ 225\qquad\textbf{(E)}\ 234$
2012 AIME Problems, 10
Find the number of positive integers $n$ less than $1000$ for which there exists a positive real number $x$ such that $n = x \lfloor x \rfloor$.
[b]Note[/b]: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.
2011 AMC 10, 18
Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$?
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 60 \qquad
\textbf{(E)}\ 75 $
1978 AMC 12/AHSME, 17
If $k$ is a positive number and $f$ is a function such that, for every positive number $x$, \[\left[f(x^2+1)\right]^{\sqrt{x}}=k;\] then, for every positive number $y$, \[\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}\] is equal to
$\textbf{(A) }\sqrt{k}\qquad\textbf{(B) }2k\qquad\textbf{(C) }k\sqrt{k}\qquad\textbf{(D) }k^2\qquad \textbf{(E) }y\sqrt{k}$
2013 AIME Problems, 12
Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.
1991 AIME Problems, 11
Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from $\pi(a-b\sqrt{c})$, where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.
[asy]
real r=2-sqrt(3);
draw(Circle(origin, 1));
int i;
for(i=0; i<12; i=i+1) {
draw(Circle(dir(30*i), r));
dot(dir(30*i));
}
draw(origin--(1,0)--dir(30)--cycle);
label("1", (0.5,0), S);[/asy]
2006 AIME Problems, 3
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $\frac{1}{29}$ of the original integer.
2023 AMC 12/AHSME, 13
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$
2014 AMC 10, 2
What is $\frac{2^3+2^3}{2^{-3}+2^{-3}}?$
${ \textbf{(A)}\ \ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}}\ 48\qquad\textbf{(E)}\ 64 $
2000 AMC 8, 8
Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is
[asy]
draw((0,0)--(2,0)--(3,1)--(3,7)--(1,7)--(0,6)--cycle);
draw((3,7)--(2,6)--(0,6));
draw((3,5)--(2,4)--(0,4));
draw((3,3)--(2,2)--(0,2));
draw((2,0)--(2,6));
dot((1,1)); dot((.5,.5)); dot((1.5,.5)); dot((1.5,1.5)); dot((.5,1.5));
dot((2.5,1.5));
dot((.5,2.5)); dot((1.5,2.5)); dot((1.5,3.5)); dot((.5,3.5));
dot((2.25,2.75)); dot((2.5,3)); dot((2.75,3.25)); dot((2.25,3.75)); dot((2.5,4)); dot((2.75,4.25));
dot((.5,5.5)); dot((1.5,4.5));
dot((2.25,4.75)); dot((2.5,5.5)); dot((2.75,6.25));
dot((1.5,6.5));
[/asy]
$\text{(A)}\ 21 \qquad \text{(B)}\ 22 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 41 \qquad \text{(E)}\ 53$
2021 AMC 10 Fall, 18
Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$
[asy]
size(160);
defaultpen(linewidth(1.1));
path square = (1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle;
filldraw(square,white);
filldraw(rotate(30)*square,white);
filldraw(rotate(60)*square,white);
dot((0,0),linewidth(7));
[/asy]
$\textbf{(A)}\: 75\qquad\textbf{(B)} \: 93\qquad\textbf{(C)} \: 96\qquad\textbf{(D)} \: 129\qquad\textbf{(E)} \: 147$
1969 AMC 12/AHSME, 25
If it is known that $\log_2a+\log_2b\geq 6$, then the least value that can be taken on by $a+b$ is:
$\textbf{(A) }2\sqrt6\qquad
\textbf{(B) }6\qquad
\textbf{(C) }8\sqrt2\qquad
\textbf{(D) }16\qquad
\textbf{(E) }\text{none of these.}$
2023 AIME, 7
Each vertex of a regular dodecagon (12-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.
1978 AMC 12/AHSME, 6
The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations: \begin{align*}x&=x^2+y^2, \\ y&=2xy\end{align*} is
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$
1963 AMC 12/AHSME, 14
Given the equations $x^2+kx+6=0$ and $x^2-kx+6=0$. If, when the roots of the equation are suitably listed, each root of the second equation is $5$ more than the corresponding root of the first equation, then $k$ equals:
$\textbf{(A)}\ 5 \qquad
\textbf{(B)}\ -5 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ -7 \qquad
\textbf{(E)}\ \text{none of these}$
2020 AMC 12/AHSME, 14
Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\tfrac{m}{n}?$
$\textbf{(A) } \frac{\sqrt{2}}{4} \qquad \textbf{(B) } \frac{\sqrt{2}}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{3\sqrt{2}}{5} \qquad \textbf{(E) } \frac{2\sqrt{2}}{3}$