AMC 12 2022b

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 \textbf{(B)}\ {-}1 \textbf{(C)}\ 0 \textbf{(D)}\ 1 \textbf{(E)}\ 2

In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ so that $\overline{BP}$ $\perp$ $\overline{AD}$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$? (Note: The figure is not drawn to scale.)

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);

$\textbf{(A) }3\sqrt 5 \textbf{(B) }10 \textbf{(C) }6\sqrt 5 \textbf{(D) }20 \textbf{(E) }25$

How many of the first ten numbers of the sequence $121, 11211, 1112111, \ldots$ are prime numbers?

$\textbf{(A) } 0 \textbf{(B) }1 \textbf{(C) }2 \textbf{(D) }3 \textbf{(E) }4$

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?

$\textbf{(A) }6 \textbf{(B) }8 \textbf{(C) }9 \textbf{(D) }14 \textbf{(E) }16$

The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position?

$\textbf{(A) }\ (-3, -4) \textbf{(B) }\ (0,5) \textbf{(C) }\ (2,-1) \textbf{(D) }\ (4,3) \textbf{(E) }\ (6,-3)$

Consider the following $100$ sets of $10$ elements each: $$\begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*}$$ How many of these sets contain exactly two multiples of $7$?

\textbf{(A)}\ 40 \textbf{(B)}\ 42 \textbf{(C)}\ 43 \textbf{(D)}\ 49 \textbf{(E)}\ 50

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?

$\textbf{(A) }5 \textbf{(B) }7 \textbf{(C) }9 \textbf{(D) }11 \textbf{(E) }13$

What is the graph of $y^4+1=x^4+2y^2$ in the coordinate plane?

$\textbf{(A) }\ \text{two intersecting parabolas} \textbf{(B) }\ \text{two nonintersecting parabolas} \textbf{(C) }\ \text{two intersecting circles}$

$\textbf{(D) }\ \text{a circle and a hyperbola} \textbf{(E) }\ \text{a circle and two parabolas}$

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 $a_2$?

$\textbf{(A) }\ 8 \textbf{(B) }\ 12 \textbf{(C) }\ 16 \textbf{(D) }\ 17 \textbf{(E) }\ 22$

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 $GCHF$?

$\textbf{(A) }\ 4\sqrt3 \textbf{(B) }\ 8 \textbf{(C) }\ 4\sqrt5 \textbf{(D) }\ 4\sqrt7 \textbf{(E) }\ 12$

Let $f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n$, where $i = \sqrt{-1}$. What is $f(2022)$?

$\textbf{(A) }\ -2 \textbf{(B) }\ -1 \textbf{(C) }\ 0 \textbf{(D) }\ \sqrt{3} \textbf{(E) }\ 2$

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 $2$?

$\textbf{(A) }\frac{2}{3} \textbf{(B) }\frac{19}{27} \textbf{(C) }\frac{59}{81} \textbf{(D) }\frac{61}{81} \textbf{(E) }\frac{7}{9}$

The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?

size(5cm);
filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray);
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1));
draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1));
label("$4$", (8,2), E);
label("$8$", (4,0), S);
label("$5$", (3,11/2), NW);
draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(1.1));

$\textbf{(A) }15\dfrac{1}{8} \textbf{(B) }15\dfrac{3}{8} \textbf{(C) }15\dfrac{1}{2} \textbf{(D) }15\dfrac{5}{8} \textbf{(E) }15\dfrac{7}{8}$

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 $\tan(\angle ABC)$?

$\textbf{(A) }\frac{1}{7} \textbf{(B) }\frac{1}{4} \textbf{(C) }\frac{3}{7} \textbf{(D) }\frac{1}{2} \textbf{(E) }\frac{4}{7}$

One of the following numbers is not divisible by any prime number less than $10.$ Which is it?

$\textbf{(A) } 2^{606}-1 \textbf{(B) } 2^{606}+1 \textbf{(C) } 2^{607}-1 \textbf{(D) } 2^{607}+1 \textbf{(E) } 2^{607}+3^{607}$

Suppose $x$ and $y$ are positive real numbers such that $$x^y=2^{64}\text{ and }(\log_2{x})^{\log_2{y}}=2^{7}.$$ What is the greatest possible value of $\log_2{y}$?

$\textbf{(A) }3 \textbf{(B) }4 \textbf{(C) }3+\sqrt{2} \textbf{(D) }4+\sqrt{3} \textbf{(E) }7$

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 $$\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} \right]$$ satisfies the condition.

$\textbf{(A) }144 \textbf{(B) }240 \textbf{(C) }336 \textbf{(D) }576 \textbf{(E) }624$

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:

• 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.

A sample transformation is shown in the figure below.

import geometry;
unitsize(0.6cm);

void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,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));

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.)

import geometry;
unitsize(0.6cm);

void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,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));

\textbf{(A)}\ 14 \textbf{(B)}\ 18 \textbf{(C)}\ 22 \textbf{(D)}\ 26 \textbf{(E)}\ 30

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 $m+n+p?$

$\textbf{(A) }44 \textbf{(B) }48 \textbf{(C) }52 \textbf{(D) }56 \textbf{(E) }60$

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?

$\textbf{(A) } 10 \textbf{(B) } 13 \textbf{(C) } 19 \textbf{(D) } 20 \textbf{(E) } 23$

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 $S$?

$\textbf{(A) } 48 \pi \textbf{(B) } 68 \pi \textbf{(C) } 96 \pi \textbf{(D) } 102 \pi \textbf{(E) } 136 \pi$

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 $1$?

$\textbf{(A) }\frac{1}{3} \textbf{(B) }\frac{1}{2} \textbf{(C) }\frac{2}{3} \textbf{(D) }\frac{3}{4} \textbf{(E) }\frac{5}{6}$

Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define $$S_n = \sum_{k=0}^{n-1} x_k 2^k$$ Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$. What is the value of the sum $$x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?$$ $\textbf{(A) } 6 \textbf{(B) } 7 \textbf{(C) }12 \textbf{(D) } 14 \textbf{(E) }15$

The figure below depicts a regular $7$-gon inscribed in a unit circle.

import geometry;
unitsize(3cm);
draw(circle((0,0),1),linewidth(1.5));
for (int i = 0; i < 7; ++i) {
for (int j = 0; j < i; ++j) {
draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5));
}
}
for(int i = 0; i < 7; ++i) {
dot(dir(i * 360/7),5+black);
}

What is the sum of the $4$th powers of the lengths of all $21$ of its edges and diagonals?

$\textbf{(A) }49 \textbf{(B) }98 \textbf{(C) }147 \textbf{(D) }168 \textbf{(E) }196$

Four regular hexagons surround a square with a side length $1$, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as $m\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime. What is $m + n + p$?

import geometry;
unitsize(3cm);
draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle);
draw(shift((1/2,1-sqrt(3)/2))*polygon(6));
draw(shift((1/2,sqrt(3)/2))*polygon(6));
draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6));
draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*polygon(6));
draw((0,1-sqrt(3))--(1,1-sqrt(3))--(3-sqrt(3),sqrt(3)-2)--(sqrt(3),0)--(sqrt(3),1)--(3-sqrt(3),3-sqrt(3))--(1,sqrt(3))--(0,sqrt(3))--(sqrt(3)-2,3-sqrt(3))--(1-sqrt(3),1)--(1-sqrt(3),0)--(sqrt(3)-2,sqrt(3)-2)--cycle,linewidth(2));

$\textbf{(A) } -12 \textbf{(B) }-4 \textbf{(C) } 4 \textbf{(D) }24 \textbf{(E) }32$