Found problems: 3632
2013 AMC 10, 22
The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In how many ways can this be done?
$\textbf{(A) }384\qquad
\textbf{(B) }576\qquad
\textbf{(C) }1152\qquad
\textbf{(D) }1680\qquad
\textbf{(E) }3546\qquad$
[asy]
size(175);
defaultpen(linewidth(0.8));
path octagon;
string labels[]={"A","B","C","D","E","F","G","H","I"};
for(int i=0;i<=7;i=i+1)
{
pair vertex=dir(135-45/2-45*i);
octagon=octagon--vertex;
label("$"+labels[i]+"$",vertex,dir(origin--vertex));
}
draw(octagon--cycle);
dot(origin);
label("$J$",origin,dir(0));
[/asy]
2023 USAJMO, 6
Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$.
[i]Proposed by Anton Trygub[/i]
1993 AMC 12/AHSME, 4
Define the operation "$\circ$" by $x \circ y=4x-3y+xy$, for all real numbers $x$ and $y$. For how many real numbers $y$ does $3 \circ y=12$?
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{more than}\ 4 $
2010 AMC 12/AHSME, 9
Let $ n$ be the smallest positive integer such that $ n$ is divisible by $ 20$, $ n^2$ is a perfect cube, and $ n^3$ is a perfect square. What is the number of digits of $ n$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
1998 AMC 8, 6
Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is
[asy]
for(int a=0; a<4; ++a)
{
for(int b=0; b<4; ++b)
{
dot((a,b));
}
}
draw((0,0)--(0,2)--(1,2)--(2,3)--(2,2)--(3,2)--(3,0)--(2,0)--(2,1)--(1,0)--cycle);[/asy]
$ \text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 $
1992 AMC 12/AHSME, 30
Let $ABCD$ be an isosceles trapezoid with bases $AB = 92$ and $CD = 19$. Suppose $AD = BC = x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2 = $
$ \textbf{(A)}\ 1369\qquad\textbf{(B)}\ 1679\qquad\textbf{(C)}\ 1748\qquad\textbf{(D)}\ 2109\qquad\textbf{(E)}\ 8825 $
2012 AIME Problems, 6
Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$.
Find $c+d$.
1974 AMC 12/AHSME, 19
In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is
[asy]
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((.82,0)--(1,1)--(0,.76)--cycle);
label("A", (0,0), S);
label("B", (1,0), S);
label("C", (1,1), N);
label("D", (0,1), N);
label("M", (0,.76), W);
label("N", (.82,0), S);
[/asy]
$ \textbf{(A)}\ 2\sqrt{3}-3 \qquad\textbf{(B)}\ 1-\frac{\sqrt{3}}{3} \qquad\textbf{(C)}\ \frac{\sqrt{3}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{3} \qquad\textbf{(E)}\ 4-2\sqrt{3} $
2018 AIME Problems, 8
A frog is positioned at the origin in the coordinate plane. From the point $(x,y)$, the frog can jump to any of the points $(x+1, y), (x+2, y), (x, y+1),$ or $(x, y+2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0,0)$ and ends at $(4,4)$.
1994 AMC 12/AHSME, 13
In triangle $ABC$, $AB=AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, then $\angle A =$
[asy]
draw((0,0)--(8,0)--(4,12)--cycle);
draw((8,0)--(1.6,4.8));
label("A", (4,12), N);
label("B", (0,0), W);
label("C", (8,0), E);
label("P", (1.6,4.8), NW);
dot((0,0));
dot((4,12));
dot((8,0));
dot((1.6,4.8));
[/asy]
$ \textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 36^{\circ} \qquad\textbf{(C)}\ 48^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 72^{\circ} $
1959 AMC 12/AHSME, 38
If $4x+\sqrt{2x}=1$, then $x$:
$ \textbf{(A)}\ \text{is an integer} \qquad\textbf{(B)}\ \text{is fractional}\qquad\textbf{(C)}\ \text{is irrational}\qquad\textbf{(D)}\ \text{is imaginary}\qquad\textbf{(E)}\ \text{may have two different values} $
2015 AMC 10, 2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
$ \textbf{(A) }\text{3:10 PM}\qquad\textbf{(B) }\text{3:30 PM}\qquad\textbf{(C) }\text{4:00 PM}\qquad\textbf{(D) }\text{4:10 PM}\qquad\textbf{(E) }\text{4:30 PM} $
2014 AMC 12/AHSME, 7
For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer?
${ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}}\ 7\qquad\textbf{(E)}\ 8 $
2012 AMC 12/AHSME, 12
A square region $ABCD$ is externally tangent to the circle with equation $x^2+y^2=1$ at the point $(0,1)$ on the side $CD$. Vertices $A$ and $B$ are on the circle with equation $x^2+y^2=4$. What is the side length of this square?
$ \textbf{(A)}\ \frac{\sqrt{10}+5}{10}\qquad\textbf{(B)}\ \frac{2\sqrt{5}}{5}\qquad\textbf{(C)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(D)}\ \frac{2\sqrt{19}-4}{5}\qquad\textbf{(E)}\ \frac{9-\sqrt{17}}{5} $
1997 AMC 8, 16
Penni Precisely buys $\$100$ worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up $20\%$, BB was down $25\%$, and CC was unchanged. For the second year, AA was down $20\%$ from the previous year, BB was up $25\%$ from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then
$\textbf{(A)}\ A=B=C \qquad \textbf{(B)}\ A=B<C \qquad \textbf{(C)}\ C<B=A$
$\textbf{(D)}\ A<B<C \qquad \textbf{(E)}\ B<A<C$
2018 AIME Problems, 2
The number \(n\) can be written in base \(14\) as \(\underline{a}\) \(\underline{b}\) \(\underline{c}\), can be written in base \(15\) as \(\underline{a}\) \(\underline{c}\) \(\underline{b}\), and can be written in base \(6\) as \(\underline{a}\) \(\underline{c}\) \(\underline{a}\) \(\underline{c}\), where \(a > 0\). Find the base-\(10\) representation of \(n\).
2022 AMC 12/AHSME, 11
What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$?
$\textbf{(A) }10\qquad\textbf{(B) }18\qquad\textbf{(C) }25\qquad\textbf{(D) }36\qquad\textbf{(E) }81$
2014 AMC 10, 6
Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days?
${ \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}$
1979 AMC 12/AHSME, 21
The length of the hypotenuse of a right triangle is $h$ , and the radius of the inscribed circle is $r$. The ratio of the area of the
circle to the area of the triangle is
$\textbf{(A) }\frac{\pi r}{h+2r}\qquad\textbf{(B) }\frac{\pi r}{h+r}\qquad\textbf{(C) }\frac{\pi}{2h+r}\qquad\textbf{(D) }\frac{\pi r^2}{r^2+h^2}\qquad\textbf{(E) }\text{none of these}$
2009 AMC 10, 3
Which of the following is equal to $ 1\plus{}\frac{1}{1\plus{}\frac{1}{1\plus{}1}}$?
$ \textbf{(A)}\ \frac{5}{4} \qquad
\textbf{(B)}\ \frac{3}{2} \qquad
\textbf{(C)}\ \frac{5}{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 3$
2022 AMC 12/AHSME, 19
Don't have original wording:
In $\triangle{ABC}$ medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$ and $\triangle{AGE}$ is equilateral. Then $\cos(C)$ can be written as $\frac{m\sqrt p}n$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the square of any prime. What is $m+n+p?$
[asy]
import geometry;
unitsize(2cm);
real arg(pair p) {
return atan2(p.y, p.x) * 180/pi;
}
pair G=(0,0),E=(1,0),A=(1/2,sqrt(3)/2),D=1.5*G-0.5*A,C=2*E-A,B=2*D-C;
pair t(pair p) {
return rotate(-arg(dir(B--C)))*p;
}
path t(path p) {
return rotate(-arg(dir(B--C)))*p;
}
void d(path p, pen q = black+linewidth(1.5)) {
draw(t(p),q);
}
void o(pair p, pen q = 5+black) {
dot(t(p),q);
}
void l(string s, pair p, pair d) {
label(s, t(p),d);
}
d(A--B--C--cycle);
d(A--D);
d(B--E);
o(A);
o(B);
o(C);
o(D);
o(E);
o(G);
l("$A$",A,N);
l("$B$",B,SW);
l("$C$",C,SE);
l("$D$",D,S);
l("$E$",E,NE);
l("$G$",G,NW);
[/asy]
$\textbf{(A)}44~\textbf{(B)}48~\textbf{(C)}52~\textbf{(D)}56~\textbf{(E)}60$
2004 AIME Problems, 1
The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37$?
2015 AMC 12/AHSME, 7
A regular $15$-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$?
$\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }32\qquad\textbf{(D) }39\qquad\textbf{(E) }54$
2024 AMC 10, 7
The product of three integers is $60$. What is the least possible positive sum of the three integers?
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$
2009 AMC 12/AHSME, 10
A particular $ 12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $ 1$, it mistakenly displays a $ 9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
$ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac58\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac56\qquad \textbf{(E)}\ \frac {9}{10}$