Found problems: 3632
1968 AMC 12/AHSME, 28
If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $\frac{a}{b}$, to the nearest integer, is
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ \text{none of these} $
2008 AMC 12/AHSME, 3
Suppose that $ \frac{2}{3}$ of $ 10$ bananas are worth as much as $ 8$ oranges. How many oranges are worth as much is $ \frac{1}{2}$ of $ 5$ bananas?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ \frac{5}{2} \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ \frac{7}{2} \qquad
\textbf{(E)}\ 4$
2006 AMC 10, 11
What is the tens digit in the sum $ 7! \plus{} 8! \plus{} 9! \plus{} \cdots \plus{} 2006!$?
$ \textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 9$
1961 AMC 12/AHSME, 23
Points $P$ and $Q$ are both in the line segment $AB$ and on the same side of its midpoint. $P$ divides $AB$ in the ratio $2:3$, and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of $AB$ is:
${{ \textbf{(A)}\ 60\qquad\textbf{(B)}\ 70\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 80}\qquad\textbf{(E)}\ 85} $
2023 AMC 12/AHSME, 10
Positive real numbers $x$ and $y$ satisfy $y^3 = x^2$ and $(y-x)^2 = 4y^2$. What is $x+y$?
$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 42$
2023 USAMO, 3
Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find all possible values of $k(C)$ as a function of $n$.
[i]Proposed by Holden Mui[/i]
1971 AMC 12/AHSME, 11
The numeral $47$ in base $a$ represents the same number as $74$ in base $b$. Assuming that both bases are positive integers, the least possible value of $a+b$ written as a Roman numeral, is
$\textbf{(A) }\mathrm{XIII}\qquad\textbf{(B) }\mathrm{XV}\qquad\textbf{(C) }\mathrm{XXI}\qquad\textbf{(D) }\mathrm{XXIV}\qquad \textbf{(E) }\mathrm{XVI}$
2000 AIME Problems, 15
Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \]
2002 AMC 12/AHSME, 24
Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$.
$ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$
1991 AMC 12/AHSME, 17
A positive integer $N$ is a [i]palindrome[/i] if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following two properties:
(a) It is a palindrome
(b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome.
How many years in the millennium between 1000 and 2000 (including the year 1991) have properties (a) and (b)?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $
2015 AMC 12/AHSME, 9
A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last two marbles. What is the probability that Cheryl gets $2$ marbles of the same color?
$\textbf{(A) }\dfrac1{10}\qquad\textbf{(B) }\dfrac16\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac13\qquad\textbf{(E) }\dfrac12$
1974 AMC 12/AHSME, 25
In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the triangle $QPO$ is equal to
[asy]
size((400));
draw((0,0)--(5,0)--(6,3)--(1,3)--cycle);
draw((6,3)--(-5,0)--(10,0)--(1,3));
label("A", (0,0), S);
label("B", (5,0), S);
label("C", (6,3), NE);
label("D", (1,3), NW);
label("P", (10,0), E);
label("Q", (-5,0), W);
label("M", (.5,1.5), NW);
label("N", (5.65, 1.5), NE);
label("O", (3.4,1.75));
[/asy]
$ \textbf{(A)}\ k \qquad\textbf{(B)}\ \frac{6k}{5} \qquad\textbf{(C)}\ \frac{9k}{8} \qquad\textbf{(D)}\ \frac{5k}{4} \qquad\textbf{(E)}\ 2k $
2017 USAJMO, 2
Consider the equation
\[(3x^3+xy^2)(x^2y+3y^3)=(x-y)^7\]
(a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation.
(b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.
1959 AMC 12/AHSME, 9
A farmer divides his herd of $n$ cows among his four sons so that one son gets one-half the herd, a second son, one-fourth, a third son, one-fifth, and the fourth son, 7 cows. Then $n$ is:
$ \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 180\qquad\textbf{(E)}\ 240 $
1979 AMC 12/AHSME, 16
A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is $3$, and if $A_1 , A_2, A_1 + A_2$ is an arithmetic progression, then the radius of the smaller circle is
$\textbf{(A) }\frac{\sqrt{3}}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{2}{\sqrt{3}}\qquad\textbf{(D) }\frac{3}{2}\qquad\textbf{(E) }\sqrt{3}$
2024 AMC 8 -, 11
The coordinates of $\triangle ABC$ are $A(5, 7)$, $B(11, 7)$, $C(3, y)$, with $y > 7$. The area of $\triangle ABC$ is $12$. What is the value of $y$?
[asy]
size(10cm);
draw((5,7)--(11,7)--(3,11)--cycle);
label("$A(5,7)$", (5,7),S);
label("$B(11,7)$", (11,7),S);
label("$C(3,y)$", (3,11),W);
[/asy]
$\textbf{(A) } 8\qquad\textbf{(B) } 9\qquad\textbf{(C) } 10\qquad\textbf{(D) } 11\qquad\textbf{(E) } 12$
1971 AMC 12/AHSME, 2
If $b$ men take $c$ days to lay $f$ bricks, then the number of days it will take $c$ men working at the same rate to lay $b$ bricks, is
$\textbf{(A) }fb^2\qquad\textbf{(B) }b/f^2\qquad\textbf{(C) }f^2/b\qquad\textbf{(D) }b^2/f\qquad \textbf{(E) }f/b^2$
2023 AIME, 14
A cube-shaped container has vertices $A$, $B$, $C$, and $D$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of the faces of the cube. Vertex $A$ of the cube is set on a horizontal plane $\mathcal P$ so that the plane of the rectangle $ABCD$ is perpendicular to $\mathcal P$, vertex $B$ is $2$ meters above $\mathcal P$, vertex $C$ is $8$ meters above $\mathcal P$, and vertex $D$ is $10$ meters above $\mathcal P$. The cube contains water whose surface is $7$ meters above $\mathcal P$. The volume of the water is $\tfrac mn$ cubic meters, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(250);
defaultpen(linewidth(0.6));
pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y;
pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W;
pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8);
filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2));
fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9));
draw(A--B--Z--X--A--Y--C--X^^C--D--Z);
draw(P1--P2--P3--P4--cycle^^D--P4);
dot("$A$",A,S);
dot("$B$",B,S);
dot("$C$",C,N);
dot("$D$",D,N);
label("$\mathcal P$",(-13,4.5));
[/asy]
2024 AMC 12/AHSME, 23
A right pyramid has regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V.$ Segments $\overline{AV}$ and $\overline{DV}$ are perpendicular. What is the square of the height of the pyramid?
$
\textbf{(A) }1 \qquad
\textbf{(B) }\frac{1+\sqrt2}{2} \qquad
\textbf{(C) }\sqrt2 \qquad
\textbf{(D) }\frac32 \qquad
\textbf{(E) }\frac{2+\sqrt2}{3} \qquad
$
1961 AMC 12/AHSME, 37
In racing over a distance $d$ at uniform speed, $A$ can beat $B$ by $20$ yards, $B$ can beat $C$ by $10$ yards, and $A$ can beat $C$ by $28$ yards. Then $d$, in yards, equals:
${{ \textbf{(A)}\ \text{Not determined by the given information} \qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 100 \qquad\textbf{(D)}\ 116}\qquad\textbf{(E)}\ 120} $
2021 AMC 12/AHSME Spring, 21
The five solutions to the equation $$(z-1)(z^2+2z+4)(z^2+4z+6)=0$$ may be written in the form $x_k+y_ki$ for $1\le k\le 5,$ where $x_k$ and $y_k$ are real. Let $\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The eccentricity of $\mathcal E$ can be written in the form $\sqrt{\frac mn}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? (Recall that the [i]eccentricity[/i] of an ellipse $\mathcal E$ is the ratio $\frac ca$, where $2a$ is the length of the major axis of $E$ and $2c$ is the is the distence between its two foci.)
$\textbf{(A) }7 \qquad \textbf{(B) }9 \qquad \textbf{(C) }11 \qquad \textbf{(D) }13\qquad \textbf{(E) }15$
Proposed by [b]djmathman[/b]
1987 AMC 12/AHSME, 19
Which of the following is closest to $\sqrt{65}-\sqrt{63}$?
$ \textbf{(A)}\ .12 \qquad\textbf{(B)}\ .13 \qquad\textbf{(C)}\ .14 \qquad\textbf{(D)}\ .15 \qquad\textbf{(E)}\ .16 $
2018 AMC 12/AHSME, 3
A line with slope $2$ intersects a line with slope $6$ at the point $(40, 30)$. What is the distance between the $x$-intercepts of these two lines?
$\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$
2020 AMC 12/AHSME, 18
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$?
$\textbf{(A) } 330 \qquad\textbf{(B) } 340 \qquad\textbf{(C) } 350 \qquad\textbf{(D) } 360 \qquad\textbf{(E) } 370$
2013 AMC 8, 20
A $1\times 2$ rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?
$\textbf{(A)}\ \frac\pi2 \qquad \textbf{(B)}\ \frac{2\pi}3 \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}3 \qquad \textbf{(E)}\ \frac{5\pi}3$