Found problems: 38
2022 AMC 10, 6
How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers?
$\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$
2022 AMC 10, 22
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 $S$?
$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$
2022 AMC 10, 1
Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]
$ \textbf{(A)}\ -2 \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ 2$
2022 AMC 12/AHSME, 5
The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position?
$\textbf{(A)}\ (-3, -4) \qquad \textbf{(B)}\ (0,5) \qquad \textbf{(C)}\ (2,-1) \qquad \textbf{(D)}\ (4,3) \qquad \textbf{(E)}\ (6,-3)$
2022 AMC 12/AHSME, 17
How many $4 \times 4$ arrays whose entries are $0$s and $1$s are there such that the row sums (the sum of the entries in each row) are $1,2,3,$ and $4,$ in some order, and the column sums (the sum of the entries in each column) are also $1,2,3,$ and $4,$ in some order? For example, the array
$\begin{bmatrix}
1 & 1 & 1 & 0\\
0 & 1 & 1 & 0\\
1 & 1 & 1 & 1\\
0 & 1 & 0 & 0
\end{bmatrix}$
satisfies the condition.
$\textbf{(A)}144~\textbf{(B)}240~\textbf{(C)}336~\textbf{(D)}576~\textbf{(E)}624$
2022 AMC 12/AHSME, 7
Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?
$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad\textbf{(E) }13$
2022 AMC 12/AHSME, 22
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 $1$?
$\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}$
2022 AMC 10, 2
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$?
[asy]
import olympiad;
size(180);
real r = 3, s = 5, t = sqrt(r*r+s*s);
defaultpen(linewidth(0.6) + fontsize(10));
pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0);
draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D));
label("$A$",A,SW);
label("$B$", B, NW);
label("$C$",C,NE);
label("$D$",D,SE);
label("$P$",P,S);
[/asy]
$\textbf{(A) }3\sqrt 5 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }6\sqrt 5 \qquad
\textbf{(D) }20\qquad
\textbf{(E) }25$
2022 AMC 12/AHSME, 8
What is the graph of $y^4+1=x^4+2y^2$ in the coordinate plane?
$ \textbf{(A)}\ \textbf{Two intersecting parabolas} \qquad
\textbf{(B)}\ \textbf{Two nonintersecting parabolas} \qquad
\textbf{(C)}\ \textbf{Two intersecting circles} \qquad
\textbf{(D)}\ \textbf{A circle and a hyperbola} \qquad
\textbf{(E)}\ \textbf{A circle and two parabolas}$
2022 AMC 10, 10
Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?
$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad\textbf{(E) }13$
2022 AMC 12/AHSME, 1
Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]
$ \textbf{(A)}\ -2 \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ 2$
2022 AMC 10, 7
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$
2022 AMC 12/AHSME, 23
Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define
\[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\]
Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum
\[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\]
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 15$
2022 AMC 10, 21
Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
$\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23$
2022 AMC 10, 8
Consider the following $100$ sets of $10$ elements each:
\begin{align*}
&\{1,2,3,\cdots,10\}, \\
&\{11,12,13,\cdots,20\},\\
&\{21,22,23,\cdots,30\},\\
&\vdots\\
&\{991,992,993,\cdots,1000\}.
\end{align*}
How many of these sets contain exactly two multiples of $7$?
$\textbf{(A)} 40\qquad\textbf{(B)} 42\qquad\textbf{(C)} 43\qquad\textbf{(D)} 49\qquad\textbf{(E)} 50$
2022 AMC 12/AHSME, 10
Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$?
$ \textbf{(A)}\ 4\sqrt3 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 4\sqrt5 \qquad
\textbf{(D)}\ 4\sqrt7 \qquad
\textbf{(E)}\ 12$
2022 AMC 12/AHSME, 18
Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:
[list]
[*] Any filled square with two or three filled neighbors remains filled.
[*] Any empty square with exactly three filled neighbors becomes a filled square.
[*] All other squares remain empty or become empty.
[/list]
A sample transformation is shown in the figure below.
[asy]
import geometry;
unitsize(0.6cm);
void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible);
}
ds((1,1));
ds((2,1));
ds((3,1));
ds((1,3));
for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
}
label("Initial", (2.5,-1));
draw((6,2.5)--(8,2.5),Arrow);
ds((10,2));
ds((11,1));
ds((11,0));
for (int i = 0; i <= 5; ++i) {
draw((9,i)--(14,i));
draw((i+9,0)--(i+9,5));
}
label("Transformed", (11.5,-1));
[/asy]
Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
[asy]
import geometry;
unitsize(0.6cm);
void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible);
}
for (int i = 1; i < 4; ++ i) {
for (int j = 1; j < 4; ++j) {
label("?",(i + 0.5, j + 0.5));
}
}
for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
}
label("Initial", (2.5,-1));
draw((6,2.5)--(8,2.5),Arrow);
ds((11,2));
for (int i = 0; i <= 5; ++i) {
draw((9,i)--(14,i));
draw((i+9,0)--(i+9,5));
}
label("Transformed", (11.5,-1));
[/asy]
$$\textbf{(A) 14}~\textbf{(B) 18}~\textbf{(C) 22}~\textbf{(D) 26}~\textbf{(E) 30}$$
2022 AMC 12/AHSME, 14
The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is $\tan(\angle ABC)$?
$\textbf{(A)}\frac{1}{7}~\textbf{(B)}\frac{1}{4}~\textbf{(C)}\frac{3}{7}~\textbf{(D)}\frac{1}{2}~\textbf{(E)}\frac{4}{7}$
2022 AMC 10, 25
Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define
\[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\]
Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum
\[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\]
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 15$
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 $S$?
$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$
2022 AMC 12/AHSME, 12
Kayla rolls four fair $6$-sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than $4$ and at least two of the numbers she rolls are greater than $2$?
$\textbf{(A)}\frac{2}{3}~\textbf{(B)}\frac{19}{27}~\textbf{(C)}\frac{59}{81}~\textbf{(D)}\frac{61}{81}~\textbf{(E)}\frac{7}{9}$
2022 AMC 12/AHSME, 6
Consider the following $100$ sets of $10$ elements each:
\begin{align*}
&\{1,2,3,\cdots,10\}, \\
&\{11,12,13,\cdots,20\},\\
&\{21,22,23,\cdots,30\},\\
&\vdots\\
&\{991,992,993,\cdots,1000\}.
\end{align*}
How many of these sets contain exactly two multiples of $7$?
$\textbf{(A)} 40\qquad\textbf{(B)} 42\qquad\textbf{(C)} 43\qquad\textbf{(D)} 49\qquad\textbf{(E)} 50$
2022 AMC 12/AHSME, 15
One of the following numbers is not divisible by any prime number less than 10. Which is it?
(A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $
2022 AMC 12/AHSME, 16
Suppose $x$ and $y$ are positive real numbers such that
$x^y=2^{64}$ and $(\log_2{x})^{\log_2{y}}=2^{7}$.
What is the greatest possible value of $\log_2{y}$?
$\textbf{(A)}3~\textbf{(B)}4~\textbf{(C)}3+\sqrt{2}~\textbf{(D)}4+\sqrt{3}~\textbf{(E)}7$
2022 AMC 12/AHSME, 9
The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that
\[2^{a_7}=2^{27} \cdot a_7.\]
What is the minimum possible value of $a_2$?
$\textbf{(A)}8~\textbf{(B)}12~\textbf{(C)}16~\textbf{(D)}17~\textbf{(E)}22$