Found problems: 3632
1989 AMC 12/AHSME, 24
Five people are sitting at a round table. Let $f \ge 0$ be the number of people sitting next to at least one female and $m \ge 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is
$ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $
2012 AMC 8, 3
On February 13 [i]The Oshkosh Northwester[/i] listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57 \textsc{am}$, and the sunset as $8:15 \textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?
$\textbf{(A)}\hspace{.05in}5:10 \textsc{pm} \quad \textbf{(B)}\hspace{.05in}5:21 \textsc{pm} \quad \textbf{(C)}\hspace{.05in}5:41\textsc{pm} \quad \textbf{(D)}\hspace{.05in}5:57 \textsc{pm} \quad \textbf{(E)}\hspace{.05in}6:03 \textsc{pm} $
1997 AMC 8, 15
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2));
draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2));
draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8));
draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2));
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);
[/asy]
$\textbf{(A)}\ \dfrac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \dfrac{5}{9} \qquad \textbf{(C)}\ \dfrac{2}{3} \qquad \textbf{(D)}\ \dfrac{\sqrt{5}}{3} \qquad \textbf{(E)}\ \dfrac{7}{9}$
2021 AMC 12/AHSME Fall, 5
Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15$
2000 AMC 8, 1
Aunt Anna is $42$ years old. Caitlin is $5$ years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?
$\textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 17\qquad
\textbf{(D)}\ 21\qquad
\textbf{(E)}\ 37$
1988 AMC 12/AHSME, 6
A figure is an equiangular parallelogram if and only if it is a
$ \textbf{(A)}\ \text{rectangle}\qquad\textbf{(B)}\ \text{regular polygon}\qquad\textbf{(C)}\ \text{rhombus}\qquad\textbf{(D)}\ \text{square}\qquad\textbf{(E)}\ \text{trapezoid} $
2014 Moldova Team Selection Test, 4
Define $p(n)$ to be th product of all non-zero digits of $n$. For instance $p(5)=5$, $p(27)=14$, $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:
\[p(1)+p(2)+p(3)+...+p(999).\]
2020 AMC 12/AHSME, 1
Carlos took $70\%$ of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
$\textbf{(A)}\ 10\%\qquad\textbf{(B)}\ 15\%\qquad\textbf{(C)}\ 20\%\qquad\textbf{(D)}\ 30\%\qquad\textbf{(E)}\ 35\%$
2006 AMC 10, 24
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?
$ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 16 \qquad \textbf{(C) } \frac 14 \qquad \textbf{(D) } \frac 13 \qquad \textbf{(E) } \frac 12$
2013 AMC 12/AHSME, 8
Line $\ell_1$ has equation $3x-2y=1$ and goes through $A=(-1,-2)$. Line $\ell_2$ has equation $y=1$ and meets line $\ell_1$ at point $B$. Line $\ell_3$ has positive slope, goes through point $A$, and meets $\ell_2$ at point $C$. The area of $\triangle ABC$ is $3$. What is the slope of $\ell_3$?
$ \textbf{(A)}\ \frac{2}{3}\qquad\textbf{(B)}\ \frac{3}{4}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{4}{3}\qquad\textbf{(E)}\ \frac{3}{2} $
2019 AMC 12/AHSME, 19
Raashan, Sylvia, and Ted play the following game. Each starts with $\$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $\$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $\$1$? (For example, Raashan and Ted may each decide to give $\$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $\$0$, Sylvia would have $\$2$, and Ted would have $\$1$, and and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their $\$1$ to, and and the holdings will be the same as the end of the second [sic] round.
$\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}$
2021 AMC 10 Spring, 11
For which of the following integers $b$ is the base-$b$ number $2021_b - 221_b$ not divisible by $3$?
$\textbf{(A) } 3 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
2008 AMC 10, 18
A right triangle has perimeter $ 32$ and area $ 20$. What is the length of its hypotenuse?
$ \textbf{(A)}\ \frac{57}{4} \qquad
\textbf{(B)}\ \frac{59}{4} \qquad
\textbf{(C)}\ \frac{61}{4} \qquad
\textbf{(D)}\ \frac{63}{4} \qquad
\textbf{(E)}\ \frac{65}{4}$
1991 AMC 12/AHSME, 6
If $x \ge 0$, then $\sqrt{x \sqrt{x \sqrt{x}}} = $
$ \textbf{(A)}\ x\sqrt{x}\qquad\textbf{(B)}\ x\sqrt[4]{x}\qquad\textbf{(C)}\ \sqrt[8]{x}\qquad\textbf{(D)}\ \sqrt[8]{x^{3}}\qquad\textbf{(E)}\ \sqrt[8]{x^{7}} $
1971 AMC 12/AHSME, 8
The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that
$\textbf{(A) }\textstyle -2<x<1\qquad\textbf{(B) }-\frac{4}{3}<x<\frac{1}{2}\qquad\textbf{(C) }-\frac{1}{2}<x<\frac{4}{3}\qquad$
$\textbf{(D) }x<\textstyle\frac{1}{2}\text{ or }x>-\frac{4}{3}\qquad\textbf{(E) }x<-\frac{4}{3}\text{ or }x>\frac{1}{2}$
2019 AIME Problems, 7
Triangle $ABC$ has side lengths $AB=120$, $BC=220$, and $AC=180$. Lines $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ are drawn parallel to $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively, such that the intersection of $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ with the interior of $\triangle ABC$ are segments of length $55$, $45$, and $15$, respectively. Find the perimeter of the triangle whose sides lie on $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$.
1974 AMC 12/AHSME, 17
If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals
$ \textbf{(A)}\ -1024 \qquad\textbf{(B)}\ -1024i \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1024 \qquad\textbf{(E)}\ 1024i $
1995 AMC 12/AHSME, 28
Two parallel chords in a circle have lengths $10$ and $14$, and the distance between them is $6$. The chord parallel to these chords and midway between them is of length $\sqrt{a}$ where $a$ is
[asy]
// note: diagram deliberately not to scale -- azjps
void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); }
size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3);
real min = -0.6, step = 0.5;
pair[] A, B; D(unitcircle);
for(int i = 0; i < 3; ++i) {
A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]);
D(D(A[i])--D(B[i]));
}
MP("10",(A[0]+B[0])/2,N);
MP("\sqrt{a}",(A[1]+B[1])/2,N);
MP("14",(A[2]+B[2])/2,N);
htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);[/asy]
$\textbf{(A)}\ 144 \qquad
\textbf{(B)}\ 156 \qquad
\textbf{(C)}\ 168 \qquad
\textbf{(D)}\ 176 \qquad
\textbf{(E)}\ 184$
1983 AIME Problems, 13
For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique [b]alternating sum[/b] is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.) Find the sum of all such alternating sums for $n = 7$.
2009 AMC 10, 20
Triangle $ ABC$ has a right angle at $ B$, $ AB \equal{} 1$, and $ BC \equal{} 2$. The bisector of $ \angle BAC$ meets $ \overline{BC}$ at $ D$. What is $ BD$?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair A=(0,1), B=(0,0), C=(2,0);
pair D=extension(A,bisectorpoint(B,A,C),B,C);
pair[] ds={A,B,C,D};
dot(ds);
draw(A--B--C--A--D);
label("$1$",midpoint(A--B),W);
label("$B$",B,SW);
label("$D$",D,S);
label("$C$",C,SE);
label("$A$",A,NW);
draw(rightanglemark(C,B,A,2));[/asy]$ \textbf{(A)}\ \frac {\sqrt3 \minus{} 1}{2} \qquad \textbf{(B)}\ \frac {\sqrt5 \minus{} 1}{2} \qquad \textbf{(C)}\ \frac {\sqrt5 \plus{} 1}{2} \qquad \textbf{(D)}\ \frac {\sqrt6 \plus{} \sqrt2}{2}$
$ \textbf{(E)}\ 2\sqrt3 \minus{} 1$
1993 AMC 8, 21
If the length of a rectangle is increased by $20\% $ and its width is increased by $50\% $, then the area is increased by
$\text{(A)}\ 10\% \qquad \text{(B)}\ 30\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 80\% \qquad \text{(E)}\ 100\% $
1993 AMC 12/AHSME, 9
Country $\mathcal{A}$ has $c\%$ of the world's population and owns $d\%$ of the world's wealth. Country $\mathcal{B}$ has $e\%$ of the world's population and $f\%$ of its wealth. Assume that the citizens of $\mathcal{A}$ share the wealth of $\mathcal{A}$ equally, and assume that those of $\mathcal{B}$ share the wealth of $\mathcal{B}$ equally. Find the ratio of the wealth of a citizen of $\mathcal{A}$ to the wealth of a citizen of $\mathcal{B}$.
$ \textbf{(A)}\ \frac{cd}{ef} \qquad\textbf{(B)}\ \frac{ce}{df} \qquad\textbf{(C)}\ \frac{cf}{de} \qquad\textbf{(D)}\ \frac{de}{cf} \qquad\textbf{(E)}\ \frac{df}{ce} $
2015 AMC 12/AHSME, 7
Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?
$\textbf{(A) }\text{The second height is 10\% less than the first.}$
$\textbf{(B) }\text{The first height is 10\% more than the second.}$
$\textbf{(C) }\text{The second height is 21\% less than the first.}$
$\textbf{(D) }\text{The first height is 21\% more than the second.}$
$\textbf{(E) }\text{The second height is 80\% of the first.}$
2013 AMC 10, 24
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other's school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?
$\textbf{(A)} \ 540 \qquad \textbf{(B)} \ 600 \qquad \textbf{(C)} \ 720 \qquad \textbf{(D)} \ 810 \qquad \textbf{(E)} \ 900$
1993 AMC 12/AHSME, 10
Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$
$ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 3a^2 \qquad\textbf{(C)}\ 27a^2 \qquad\textbf{(D)}\ 2a^{3b} \qquad\textbf{(E)}\ 3a^{2b} $