Found problems: 110
2018 AMC 12/AHSME, 4
A circle has a chord of length $10$, and the distance from the center of the circle to the chord is $5$. What is the area of the circle?
$\textbf{(A) }25\pi\qquad\textbf{(B) }50\pi\qquad\textbf{(C) }75\pi\qquad\textbf{(D) }100\pi\qquad\textbf{(E) }125\pi$
2019 AMC 12/AHSME, 17
Let $s_k$ denote the sum of the $\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$?
$\textbf{(A)} \; -6 \qquad \textbf{(B)} \; 0 \qquad \textbf{(C)} \; 6 \qquad \textbf{(D)} \; 10 \qquad \textbf{(E)} \; 26$
2018 AMC 12/AHSME, 5
What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common?
$
\textbf{(A) }3 \qquad
\textbf{(B) }4 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }10 \qquad
$
2019 AMC 12/AHSME, 15
Positive real numbers $a$ and $b$ have the property that
\[
\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100
\] and all four terms on the left are positive integers, where $\text{log}$ denotes the base 10 logarithm. What is $ab$?
$\textbf{(A) } 10^{52} \qquad \textbf{(B) } 10^{100} \qquad \textbf{(C) } 10^{144} \qquad \textbf{(D) } 10^{164} \qquad \textbf{(E) } 10^{200} $
2018 AMC 12/AHSME, 22
The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt 3i,$ where $i=\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q-r\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p+q+r+s?$
$\textbf{(A) } 20 \qquad
\textbf{(B) } 21 \qquad
\textbf{(C) } 22 \qquad
\textbf{(D) } 23 \qquad
\textbf{(E) } 24 $
2012 AMC 10, 20
A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?
$ \textbf{(A)}\ \dfrac{49}{512}
\qquad\textbf{(B)}\ \dfrac{7}{64}
\qquad\textbf{(C)}\ \dfrac{121}{1024}
\qquad\textbf{(D)}\ \dfrac{81}{512}
\qquad\textbf{(E)}\ \dfrac{9}{32}
$
2021 AMC 12/AHSME Spring, 25
Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the [i]divisor function[/i].) Let \[f(n)=\frac{d(n)}{\sqrt[3]{n}}.\] There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N?$
$\textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8\qquad \textbf{(E) }9$
2020 AMC 12/AHSME, 16
A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth$?$
$\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$
2012 AMC 10, 3
A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether?
$ \textbf{(A)}\ 9
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 13
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 15
$
2018 AMC 10, 9
All of the triangles in the diagram below are similar to iscoceles triangle $ABC$, in which $AB=AC$. Each of the 7 smallest triangles has area 1, and $\triangle ABC$ has area 40. What is the area of trapezoid $DBCE$?
[asy]
unitsize(5);
dot((0,0));
dot((60,0));
dot((50,10));
dot((10,10));
dot((30,30));
draw((0,0)--(60,0)--(50,10)--(30,30)--(10,10)--(0,0));
draw((10,10)--(50,10));
label("$B$",(0,0),SW);
label("$C$",(60,0),SE);
label("$E$",(50,10),E);
label("$D$",(10,10),W);
label("$A$",(30,30),N);
draw((10,10)--(15,15)--(20,10)--(25,15)--(30,10)--(35,15)--(40,10)--(45,15)--(50,10));
draw((15,15)--(45,15));
[/asy]
$\textbf{(A) } 16 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 22 \qquad \textbf{(E) } 24 $
2021 AMC 10 Fall, 20
How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions?
$\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 12 \qquad$
2020 AMC 12/AHSME, 22
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[
(2 + i)^n = a_n + b_ni
\]
for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is \[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]
$\textbf{(A) }\frac 38\qquad\textbf{(B) }\frac7{16}\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac9{16}\qquad\textbf{(E) }\frac47$
2020 AMC 12/AHSME, 9
How many solutions does the equation $\tan{(2x)} = \cos{(\tfrac{x}{2})}$ have on the interval $[0, 2\pi]?$
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$
2017 AMC 12/AHSME, 19
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$?
$\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$
2016 AMC 12/AHSME, 24
There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$?
$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$
2017 AMC 12/AHSME, 9
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2019 AMC 12/AHSME, 13
How many ways are there to paint each of the integers $2, 3, \dots, 9$ either red, green, or blue so that each number has a different color from each of its proper divisors?
$\textbf{(A)}\ 144\qquad\textbf{(B)}\ 216\qquad\textbf{(C)}\ 256\qquad\textbf{(D)}\ 384\qquad\textbf{(E)}\ 432$
2012 AMC 10, 7
In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?
$ \textbf{(A)}\ \dfrac{2}{5}
\qquad\textbf{(B)}\ \dfrac{3}{7}
\qquad\textbf{(C)}\ \dfrac{4}{7}
\qquad\textbf{(D)}\ \dfrac{3}{5}
\qquad\textbf{(E)}\ \dfrac{4}{5}
$
2017 AMC 12/AHSME, 14
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?
$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$
2012 AMC 10, 23
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
$ \textbf{(A)}\ 60
\qquad\textbf{(B)}\ 170
\qquad\textbf{(C)}\ 290
\qquad\textbf{(D)}\ 320
\qquad\textbf{(E)}\ 660
$
2021 AMC 10 Spring, 9
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$
2019 AMC 12/AHSME, 25
Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$ define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse?
$\phantom{}$
$\textbf{(A) } 10 \qquad \textbf{(B) }11 \qquad \textbf{(C) } 13\qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$
2021 AMC 10 Fall, 6
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?
$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15$
2017 AMC 10, 20
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$
2007 AMC 10, 4
The larger of two consecutive odd integers is three times the smaller. What is their sum?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 20$