Found problems: 3632
1960 AMC 12/AHSME, 21
The diagonal of square I is $a+b$. The perimeter of square II with [i]twice[/i] the area of I is:
$ \textbf{(A)}\ (a+b)^2\qquad\textbf{(B)}\ \sqrt{2}(a+b)^2\qquad\textbf{(C)}\ 2(a+b)\qquad\textbf{(D)}\ \sqrt{8}(a+b) \qquad$
$\textbf{(E)}\ 4(a+b) $
2024 AMC 12/AHSME, 8
How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0$?
$
\textbf{(A) }0 \qquad
\textbf{(B) }1 \qquad
\textbf{(C) }2 \qquad
\textbf{(D) }3 \qquad
\textbf{(E) }4 \qquad
$
1979 AMC 12/AHSME, 20
If $a=\tfrac{1}{2}$ and $(a+1)(b+1)=2$ then the radian measure of $\arctan a + \arctan b$ equals
$\textbf{(A) }\frac{\pi}{2}\qquad\textbf{(B) }\frac{\pi}{3}\qquad\textbf{(C) }\frac{\pi}{4}\qquad\textbf{(D) }\frac{\pi}{5}\qquad\textbf{(E) }\frac{\pi}{6}$
1978 AMC 12/AHSME, 24
If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation
$\textbf{(A) }r^2+r+1=0\qquad\textbf{(B) }r^2-r+1=0\qquad\textbf{(C) }r^4+r^2-1=0$
$\qquad\textbf{(D) }(r+1)^4+r=0\qquad \textbf{(E) }(r-1)^4+r=0$
1976 AMC 12/AHSME, 13
If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk?
$\textbf{(A) }\frac{x(x+2)(x+5)}{(x+1)(x+3)}\qquad\textbf{(B) }\frac{x(x+1)(x+5)}{(x+2)(x+3)}\qquad$
$\textbf{(C) }\frac{(x+1)(x+3)(x+5)}{x(x+2)}\qquad\textbf{(D) }\frac{(x+1)(x+3)}{x(x+2)(x+5)}\qquad \textbf{(E) }\text{none of these}$
2022 AMC 10, 23
Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $1$?
$\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}$
2024 AMC 10, 22
Let $\mathcal K$ be the kite formed by joining two right triangles with legs $1$ and $\sqrt3$ along a common hypotenuse. Eight copies of $\mathcal K$ are used to form the polygon shown below. What is the area of triangle $\Delta ABC$?
[img]https://cdn.artofproblemsolving.com/attachments/1/3/03abbd4df2932f4a1d16a34c2b9e15b683dedb.png[/img]
$\textbf{(A) }2+3\sqrt3\qquad\textbf{(B) }\dfrac92\sqrt3\qquad\textbf{(C) }\dfrac{10+8\sqrt3}{3}\qquad\textbf{(D) }8\qquad\textbf{(E) }5\sqrt3$
1963 AMC 12/AHSME, 31
The number of solutions in positive integers of $2x+3y=763$ is:
$\textbf{(A)}\ 255 \qquad
\textbf{(B)}\ 254\qquad
\textbf{(C)}\ 128 \qquad
\textbf{(D)}\ 127 \qquad
\textbf{(E)}\ 0$
2023 AMC 10, 2
The weight of $\frac 13$ of a large pizza together with $3 \frac 12$ cups of orange slices is the same as the weight of $\frac 34$ of a large pizza together with $\frac 12$ cup of orange slices. A cup of orange slices weighs $\frac 14$ of a pound. What is the weight, in pounds, of a large pizza?
$\textbf{(A)}~1\frac45\qquad\textbf{(B)}~2\qquad\textbf{(C)}~2\frac25\qquad\textbf{(D)}~3\qquad\textbf{(E)}~3\frac35$
2007 AIME Problems, 12
In isosceles triangle $ABC$, $A$ is located at the origin and $B$ is located at $(20, 0)$. Point $C$ is in the first quadrant with $AC = BC$ and $\angle BAC = 75^\circ$. If $\triangle ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive y-axis, the area of the region common to the original and the rotated triangle is in the form $p\sqrt{2}+q\sqrt{3}+r\sqrt{6}+s$ where $p$, $q$, $r$, $s$ are integers. Find $(p-q+r-s)/2$.
2010 Contests, 1
What is $ (20\minus{}(2010\minus{}201)) \plus{} (2010\minus{}(201\minus{}20))$?
$ \textbf{(A)}\ \minus{}4020\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 401\qquad \textbf{(E)}\ 4020$
2018 AMC 12/AHSME, 23
Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degree measure of $\angle ACB$?
$
\textbf{(A) }105 \qquad
\textbf{(B) }112\frac{1}{2} \qquad
\textbf{(C) }120 \qquad
\textbf{(D) }135 \qquad
\textbf{(E) }150 \qquad
$
2006 AMC 12/AHSME, 12
A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?
[asy]
size(200);
defaultpen(linewidth(3));
real[] inrad = {40,34,28,21};
real[] outrad = {55,49,37,30};
real[] center;
path[][] quad = new path[4][4];
center[0] = 0;
for(int i=0;i<=3;i=i+1) {
if(i != 0) {
center[i] = center[i-1] - inrad[i-1] - inrad[i]+3.5;
}
quad[0][i] = arc((0,center[i]),inrad[i],0,90)--arc((0,center[i]),outrad[i],90,0)--cycle;
quad[1][i] = arc((0,center[i]),inrad[i],90,180)--arc((0,center[i]),outrad[i],180,90)--cycle;
quad[2][i] = arc((0,center[i]),inrad[i],180,270)--arc((0,center[i]),outrad[i],270,180)--cycle;
quad[3][i] = arc((0,center[i]),inrad[i],270,360)--arc((0,center[i]),outrad[i],360,270)--cycle;
draw(circle((0,center[i]),inrad[i])^^circle((0,center[i]),outrad[i]));
}
void fillring(int i,int j) {
if ((j % 2) == 0) {
fill(quad[i][j],white);
} else {
filldraw(quad[i][j],black);
} }
for(int i=0;i<=3;i=i+1) {
for(int j=0;j<=3;j=j+1) {
fillring(((2-i) % 4),j);
} }
for(int k=0;k<=2;k=k+1) {
filldraw(circle((0,-228 - 25 * k),3),black);
}
real r = 130, s = -90;
draw((0,57)--(r,57)^^(0,-57)--(r,-57),linewidth(0.7));
draw((2*r/3,56)--(2*r/3,-56),linewidth(0.7),Arrows(size=3));
label("$20$",(2*r/3,-10),E);
draw((0,39)--(s,39)^^(0,-39)--(s,-39),linewidth(0.7));
draw((9*s/10,38)--(9*s/10,-38),linewidth(0.7),Arrows(size=3));
label("$18$",(9*s/10,0),W);
[/asy]
$ \textbf{(A) } 171\qquad \textbf{(B) } 173\qquad \textbf{(C) } 182\qquad \textbf{(D) } 188\qquad \textbf{(E) } 210$
1984 AIME Problems, 1
Find the value of $a_2 + a_4 + a_6 + \dots + a_{98}$ if $a_1$, $a_2$, $a_3$, $\dots$ is an arithmetic progression with common difference 1, and $a_1 + a_2 + a_3 + \dots + a_{98} = 137$.
2011 AMC 10, 11
There are $52$ people in a room. What is the largest value of $n$ such that the statement "At least $n$ people in this room have birthdays falling in the same month" is always true?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 12 $
2013 AMC 8, 13
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
$\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 47 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 49$
2018 AIME Problems, 5
For each ordered pair of real numbers $(x,y)$ satisfying
\[ \log_2(2x+y) = \log_4(x^2+xy+7y^2) \]
there is a real number $K$ such that
\[ \log_3(3x+y) = \log_9(3x^2+4xy+Ky^2). \]
Find the product of all possible values of $K$.
1972 AMC 12/AHSME, 3
If $x=\dfrac{1-i\sqrt{3}}{2}$ where $i=\sqrt{-1}$, then $\dfrac{1}{x^2-x}$ is equal to
$\textbf{(A) }-2\qquad\textbf{(B) }-1\qquad\textbf{(C) }1+i\sqrt{3}\qquad\textbf{(D) }1\qquad \textbf{(E) }2$
1968 AMC 12/AHSME, 17
Let $f(n)=\dfrac{x_1+x_2+...+x_n}{n}$, where $n$ is a positive integer. If $x_k=(-1)^k,k=1,2,...,n$, the set of possible values of $f(n)$ is:
$\textbf{(A)}\ \{0\} \qquad
\textbf{(B)}\ \{\dfrac{1}{n}\} \qquad
\textbf{(C)}\ \{0,-\dfrac{1}{n}\} \qquad
\textbf{(D)}\ \{0,\dfrac{1}{n}\} \qquad
\textbf{(E)}\ \{1,\dfrac{1}{n}\}$
2020 AMC 12/AHSME, 7
Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
$\textbf{(A) } 644 \qquad \textbf{(B) } 658 \qquad \textbf{(C) } 664 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 749$
1992 AMC 12/AHSME, 21
For a finite sequence $A = (a_1, a_2,\ldots,a_n)$ of numbers, the [i]Cesaro sum[/i] of $A$ is defined to be \[\frac{S_1 + S_2 + \cdots + S_n}{n}\] where $S_k = a_1 + a_2 + \cdots + a_k\ \ \ \ (1 \le k \le n)$. If the Cesaro sum of the 99-term sequence $(a_1, a_2, \ldots, a_{99})$ is $1000$, what is the Cesaro sum of the 100-term sequence $(1,a_1,a_2,\ldots,a_{99})$?
$ \textbf{(A)}\ 991\qquad\textbf{(B)}\ 999\qquad\textbf{(C)}\ 1000\qquad\textbf{(D)}\ 1001\qquad\textbf{(E)}\ 1009 $
1998 AMC 8, 14
An Annville Junior High School, $30\%$ of the students in the Math Club are in the Science Club, and $80\%$ of the students in the Science Club are in the Math Club. There are $15$ students in the Science Club. How many students are in the Math Club?
$ \text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 30\qquad\text{(D)}\ 36\qquad\text{(E)}\ 40 $
1960 AMC 12/AHSME, 20
The coefficient of $x^7$ in the expansion of $(\frac{x^2}{2}-\frac{2}{x})^8$ is:
$ \textbf{(A)}\ 56\qquad\textbf{(B)}\ -56\qquad\textbf{(C)}\ 14\qquad\textbf{(D)}\ -14\qquad\textbf{(E)}\ 0 $
2003 AIME Problems, 5
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $(m + n \pi)/p$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p$.
2009 AMC 12/AHSME, 25
The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\le|x|\le7$, $ 3\le|y|\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$?
[asy]defaultpen(black+0.75bp+fontsize(8pt));
size(5cm);
path p = scale(.15)*unitcircle;
draw((-8,0)--(8.5,0),Arrow(HookHead,1mm));
draw((0,-8)--(0,8.5),Arrow(HookHead,1mm));
int i,j;
for (i=-7;i<8;++i) {
for (j=-7;j<8;++j) {
if (((-7 <= i) && (i <= -3)) || ((3 <= i) && (i<= 7))) { if (((-7 <= j) && (j <= -3)) || ((3 <= j) && (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp);
draw((-3,-.2)--(-3,.2),black+0.5bp);
draw((3,-.2)--(3,.2),black+0.5bp);
draw((7,-.2)--(7,.2),black+0.5bp);
draw((-.2,-7)--(.2,-7),black+0.5bp);
draw((-.2,-3)--(.2,-3),black+0.5bp);
draw((-.2,3)--(.2,3),black+0.5bp);
draw((-.2,7)--(.2,7),black+0.5bp);
label("$-7$",(-7,0),S);
label("$-3$",(-3,0),S);
label("$3$",(3,0),S);
label("$7$",(7,0),S);
label("$-7$",(0,-7),W);
label("$-3$",(0,-3),W);
label("$3$",(0,3),W);
label("$7$",(0,7),W);[/asy]$ \textbf{(A)}\ 125\qquad \textbf{(B)}\ 150\qquad \textbf{(C)}\ 175\qquad \textbf{(D)}\ 200\qquad \textbf{(E)}\ 225$