Found problems: 3632
2011 AMC 10, 20
Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect?
$ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2} $
2018 AMC 12/AHSME, 5
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?
$\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$
2006 AMC 10, 25
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
$ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
2016 AIME Problems, 2
There is a $40\%$ chance of rain on Saturday and a $30\%$ of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
2024 AIME, 4
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
$$\log_2\left({x \over yz}\right) = {1 \over 2}$$
$$\log_2\left({y \over xz}\right) = {1 \over 3}$$
$$\log_2\left({z \over xy}\right) = {1 \over 4}$$
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is ${m \over n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$
2019 AMC 12/AHSME, 8
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?
$\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$
2008 AMC 12/AHSME, 14
A circle has a radius of $ \log_{10}(a^2)$ and a circumference of $ \log_{10}(b^4)$. What is $ \log_ab$?
$ \textbf{(A)}\ \frac {1}{4\pi} \qquad \textbf{(B)}\ \frac {1}{\pi} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \textbf{(E)}\ 10^{2\pi}$
2022 AMC 10, 22
Suppose that 13 cards numbered $1, 2, 3, \dots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes?
[asy]
size(11cm);
draw((0,0)--(2,0)--(2,3)--(0,3)--cycle);
label("7", (1,1.5));
draw((3,0)--(5,0)--(5,3)--(3,3)--cycle);
label("11", (4,1.5));
draw((6,0)--(8,0)--(8,3)--(6,3)--cycle);
label("8", (7,1.5));
draw((9,0)--(11,0)--(11,3)--(9,3)--cycle);
label("6", (10,1.5));
draw((12,0)--(14,0)--(14,3)--(12,3)--cycle);
label("4", (13,1.5));
draw((15,0)--(17,0)--(17,3)--(15,3)--cycle);
label("5", (16,1.5));
draw((18,0)--(20,0)--(20,3)--(18,3)--cycle);
label("9", (19,1.5));
draw((21,0)--(23,0)--(23,3)--(21,3)--cycle);
label("12", (22,1.5));
draw((24,0)--(26,0)--(26,3)--(24,3)--cycle);
label("1", (25,1.5));
draw((27,0)--(29,0)--(29,3)--(27,3)--cycle);
label("13", (28,1.5));
draw((30,0)--(32,0)--(32,3)--(30,3)--cycle);
label("10", (31,1.5));
draw((33,0)--(35,0)--(35,3)--(33,3)--cycle);
label("2", (34,1.5));
draw((36,0)--(38,0)--(38,3)--(36,3)--cycle);
label("3", (37,1.5));
[/asy]
$\textbf{(A) }4082\qquad\textbf{(B) }4095\qquad\textbf{(C) }4096\qquad\textbf{(D) }8178\qquad\textbf{(E) }8191$
1972 AMC 12/AHSME, 28
A circular disc with diameter $D$ is placed on an $8\times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is
$\textbf{(A) }48\qquad\textbf{(B) }44\qquad\textbf{(C) }40\qquad\textbf{(D) }36\qquad \textbf{(E) }32$
1987 AMC 12/AHSME, 10
How many ordered triples $(a, b, c)$ of non-zero real numbers have the property that each number is the product of the other two?
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $
1975 AMC 12/AHSME, 27
If $p, q$ and $r$ are distinct roots of $x^3-x^2+x-2=0$, then $p^3+q^3+r^3$ equals
$ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{none of these} $
2014 AMC 12/AHSME, 3
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
${ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6$
2008 AIME Problems, 3
Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $ 74$ kilometers after biking for $ 2$ hours, jogging for $ 3$ hours, and swimming for $ 4$ hours, while Sue covers $ 91$ kilometers after jogging for $ 2$ hours, swimming for $ 3$ hours, and biking for $ 4$ hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.
2008 AMC 12/AHSME, 4
Which of the following is equal to the product
\[ \frac {8}{4}\cdot\frac {12}{8}\cdot\frac {16}{12}\cdots\frac {4n \plus{} 4}{4n}\cdots\frac {2008}{2004}?
\]$ \textbf{(A)}\ 251 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 2008 \qquad \textbf{(E)}\ 4016$
2016 AMC 10, 10
A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle?
[asy]
size(6cm);
defaultpen(fontsize(9pt));
path rectangle(pair X, pair Y){
return X--(X.x,Y.y)--Y--(Y.x,X.y)--cycle;
}
filldraw(rectangle((0,0),(7,5)),gray(0.5));
filldraw(rectangle((1,1),(6,4)),gray(0.75));
filldraw(rectangle((2,2),(5,3)),white);
label("$1$",(0.5,2.5));
draw((0.3,2.5)--(0,2.5),EndArrow(TeXHead));
draw((0.7,2.5)--(1,2.5),EndArrow(TeXHead));
label("$1$",(1.5,2.5));
draw((1.3,2.5)--(1,2.5),EndArrow(TeXHead));
draw((1.7,2.5)--(2,2.5),EndArrow(TeXHead));
label("$1$",(4.5,2.5));
draw((4.5,2.7)--(4.5,3),EndArrow(TeXHead));
draw((4.5,2.3)--(4.5,2),EndArrow(TeXHead));
label("$1$",(4.1,1.5));
draw((4.1,1.7)--(4.1,2),EndArrow(TeXHead));
draw((4.1,1.3)--(4.1,1),EndArrow(TeXHead));
label("$1$",(3.7,0.5));
draw((3.7,0.7)--(3.7,1),EndArrow(TeXHead));
draw((3.7,0.3)--(3.7,0),EndArrow(TeXHead));
[/asy]
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) }8$
2024 AMC 10, 16
Jerry likes to play with numbers. One day, he wrote all the integers from $1$ to $2024$ on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them with either their sum or their product. (For example, Jerry's first step might have been to erase $1, 2, 3$, and $5$, and then write either $11$, their sum, or $30$, their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the board were odd. What is the maximum possible number of integers on the board at that time?
$
\textbf{(A) }1010 \qquad
\textbf{(B) }1011 \qquad
\textbf{(C) }1012 \qquad
\textbf{(D) }1013 \qquad
\textbf{(E) }1014 \qquad
$
1977 AMC 12/AHSME, 15
[asy]
size(120);
real t = 2/sqrt(3);
real x = 1 + sqrt(3);
pair A = t*dir(90), D = x*A;
pair B = t*dir(210), E = x*B;
pair C = t*dir(330), F = x*C;
draw(D--E--F--cycle);
draw(Circle(A, 1));
draw(Circle(B, 1));
draw(Circle(C, 1));
//Credit to MSTang for the diagram[/asy]
Each of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is
$\textbf{(A) }36+9\sqrt{2}\qquad\textbf{(B) }36+6\sqrt{3}\qquad\textbf{(C) }36+9\sqrt{3}\qquad\textbf{(D) }18+18\sqrt{3}\qquad \textbf{(E) }45$
2015 AMC 12/AHSME, 8
The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?
$\textbf{(A) }\dfrac27\qquad\textbf{(B) }\dfrac37\qquad\textbf{(C) }\dfrac{12}{25}\qquad\textbf{(D) }\dfrac{16}{25}\qquad\textbf{(E) }\dfrac34$
2014 AMC 12/AHSME, 15
When $p = \sum_{k=1}^{6} k \ln{k}$, the number $e^p$ is an integer. What is the largest power of $2$ that is a factor of $e^p$?
${\textbf{(A)}\ 2^{12}\qquad\textbf{(B)}\ 2^{14}\qquad\textbf{(C)}\ 2^{16}\qquad\textbf{(D)}}\ 2^{18}\qquad\textbf{(E)}\ 2^{20} $
1989 AMC 12/AHSME, 25
In a certain cross-country meet between two teams of five runners each, a runner who finishes in the $n^{th}$ position contributes $n$ to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 27 \qquad\textbf{(D)}\ 120 \qquad\textbf{(E)}\ 126 $
2010 AIME Problems, 11
Let $ \mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $ |8 \minus{} x| \plus{} y \le 10$ and $ 3y \minus{} x \ge 15$. When $ \mathcal{R}$ is revolved around the line whose equation is $ 3y \minus{} x \equal{} 15$, the volume of the resulting solid is $ \frac {m\pi}{n\sqrt {p}}$, where $ m$, $ n$, and $ p$ are positive integers, $ m$ and $ n$ are relatively prime, and $ p$ is not divisible by the square of any prime. Find $ m \plus{} n \plus{} p$.
2025 AIME, 11
A piecewise linear function is defined by \[f(x) = \begin{cases} x & \text{if } x \in [-1, 1) \\ 2 - x & \text{if } x \in [1, 3)\end{cases}\] and $f(x + 4) = f(x)$ for all real numbers $x.$ The graph of $f(x)$ has the sawtooth pattern depicted below. [color=transparent]Diagram from RandomMath.[/color]
[center][img width=45]https://i.ibb.co/JW8jH2Dr/image.png[/img][/center]
The parabola $x = 34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-coordinates of these intersection points can be expressed in the form $\tfrac{a + b\sqrt c}d,$ where $a, b, c$ and $d$ are positive integers, $a, b,$ and $d$ has greatest common divisor equal to $1,$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d.$
2009 AMC 10, 7
By inserting parentheses, it is possible to give the expression
\[ 2\times3\plus{}4\times5
\]several values. How many different values can be obtained?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 6$
2019 AMC 12/AHSME, 11
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$?
$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17$
1994 AMC 12/AHSME, 12
If $i^2=-1$, then $(i-i^{-1})^{-1}=$
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ -2i \qquad\textbf{(C)}\ 2i \qquad\textbf{(D)}\ -\frac{i}{2} \qquad\textbf{(E)}\ \frac{i}{2}$