Found problems: 730
2024 AMC 10, 12
A group of $100$ students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?
$
\textbf{(A) }9 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }12 \qquad
\textbf{(D) }51 \qquad
\textbf{(E) }100 \qquad
$
2018 AMC 12/AHSME, 11
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper?
[asy]size(270pt);
defaultpen(fontsize(10pt));
filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey);
dot((-3,3));
label("$A$",(-3,3),NW);
draw((1,3)--(-3,-1),dashed+linewidth(.5));
draw((-1,3)--(3,-1),dashed+linewidth(.5));
draw((-1,-3)--(3,1),dashed+linewidth(.5));
draw((1,-3)--(-3,1),dashed+linewidth(.5));
draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5));
draw((0,3)--(0,-3),linetype("2.5 2.5")+linewidth(.5));
draw((3,0)--(-3,0),linetype("2.5 2.5")+linewidth(.5));
label('$w$',(-1,-1),SW);
label('$w$',(1,-1),SE);
draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle);
draw((4.5,0)--(8.5,0));
draw((6.5,2)--(6.5,-2));
label("$A$",(6.5,0),NW);
dot((6.5,0));
[/asy]
$\textbf{(A) } 2(w+h)^2 \qquad \textbf{(B) } \frac{(w+h)^2}2 \qquad \textbf{(C) } 2w^2+4wh \qquad \textbf{(D) } 2w^2 \qquad \textbf{(E) } w^2h $
2022 AMC 10, 11
Ted mistakenly wrote $2^m \cdot \sqrt{\frac{1}{4096}}$ as $2\cdot \sqrt[m]{\frac{1}{4096}}$. What is the sum of all real numbers $m$ for which these two expressions have the same value?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2017 AMC 10, 1
What is the value of $2(2(2(2(2(2+1)+1)+1)+1)+1)+1$?
$\textbf{(A) } 70 \qquad
\textbf{(B) } 97 \qquad
\textbf{(C) } 127 \qquad
\textbf{(D) } 159 \qquad
\textbf{(E) } 729 $
2022 AMC 12/AHSME, 21
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?
$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$
2021 AMC 10 Fall, 14
How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations?
\begin{align*}
x^2+3y&=9\\
(|x|+|y|-4)^2&=1\\
\end{align*}
$\textbf{(A)}\: 1\qquad\textbf{(B)} \: 2\qquad\textbf{(C)} \: 3\qquad\textbf{(D)} \: 5\qquad\textbf{(E)} \: 7$
2021 AMC 10 Fall, 9
The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical?
$\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{3}{13}\qquad\textbf{(C) }\frac{7}{27}\qquad\textbf{(D) }\frac{2}{7}\qquad\textbf{(E) }\frac{1}{3}$
2021 AMC 12/AHSME Fall, 7
Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation
$$x(x-y)+y(y-z)+z(z-x) = 1?$$$\textbf{(A)}\: x>y$ and $y=z$
$\textbf{(B)}\: x=y-1$ and $y=z-1$
$\textbf{(C)} \: x=z+1$ and $y=x+1$
$\textbf{(D)} \: x=z$ and $y-1=x$
$\textbf{(E)} \: x+y+z=1$
2017 AMC 12/AHSME, 8
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$
2016 AMC 12/AHSME, 3
Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$?
$\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$
2023 AMC 10, 9
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?
$\textbf{(A)}~5\qquad\textbf{(B)}~6\qquad\textbf{(C)}~7\qquad\textbf{(D)}~8\qquad\textbf{(E)}~9$
2017 AMC 12/AHSME, 2
The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$
2023 AMC 10, 4
A quadrilateral has all integer sides lengths, a perimeter of $26$, and one side of length $4$. What is the greatest possible length of one side of this quadrilateral?
$\textbf{(A)}~9\qquad\textbf{(B)}~10\qquad\textbf{(C)}~11\qquad\textbf{(D)}~12\qquad\textbf{(E)}~13$
2017 AMC 12/AHSME, 13
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(100);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy]
$\textbf{(A) } 6 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15$
2021 AMC 10 Spring, 14
All the roots of polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers. What is the value of $B$?
$\textbf{(A)}\ -88 \qquad\textbf{(B)}\ -80 \qquad\textbf{(C)}\ -64\qquad\textbf{(D)}\ -41 \qquad\textbf{(E)}\ -40$
2020 AMC 10, 17
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?
$\textbf{(A) } 11 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$
2019 AMC 10, 15
Two right triangles, $T_1$ and $T_2$, have areas of 1 and 2, respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the square of the product of the third side lengths of $T_1$ and $T_2$?
$\textbf{(A) }\frac{28}3\qquad\textbf{(B) }10\qquad\textbf{(C) }\frac{32}3\qquad\textbf{(D) }\frac{34}3\qquad\textbf{(E) }12$
2018 AMC 12/AHSME, 3
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$
2021 AMC 10 Spring, 21
A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C’$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Suppose that $C’D=\frac{1}{3}$. What is the perimeter of $\triangle AEC’$?
[asy]
//Diagram by Samrocksnature
pair A=(0,1);
pair CC=(0.666666666666,1);
pair D=(1,1);
pair F=(1,0.62);
pair C=(1,0);
pair B=(0,0);
pair G=(0,0.25);
pair H=(-0.13,0.41);
pair E=(0,0.5);
dot(A^^CC^^D^^C^^B^^E);
draw(E--A--D--F);
draw(G--B--C--F, dashed);
fill(E--CC--F--G--H--E--CC--cycle, gray);
draw(E--CC--F--G--H--E--CC);
label("A",A,NW);
label("B",B,SW);
label("C",C,SE);
label("D",D,NE);
label("E",E,NW);
label("C'",CC,N);
[/asy]
$\textbf{(A) }2 \qquad \textbf{(B) }1+\frac{2}{3}\sqrt{3} \qquad \textbf{(C) }\frac{13}{6} \qquad \textbf{(D) }1+\frac{3}{4}\sqrt{3} \qquad \textbf{(E) }\frac{7}{3}$
2024 AMC 12/AHSME, 18
The Fibonacci numbers are defined by $F_1=1,$ $F_2=1,$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3.$ What is $$\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?$$
$\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$
2017 AMC 10, 24
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$?
$\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$
2018 AMC 10, 6
A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required?
$\textbf{(A)} \frac{1}{15} \qquad \textbf{(B)} \frac{1}{10} \qquad \textbf{(C)} \frac{1}{6} \qquad \textbf{(D)} \frac{1}{5} \qquad \textbf{(E)} \frac{1}{4}$
2016 AMC 10, 13
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
$\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
2016 AMC 10, 18
Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
$\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$
2020 AMC 10, 11
What is the median of the following list of $4040$ numbers$?$
$$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$
$\textbf{(A) } 1974.5 \qquad \textbf{(B) } 1975.5 \qquad \textbf{(C) } 1976.5 \qquad \textbf{(D) } 1977.5 \qquad \textbf{(E) } 1978.5$