AMC 12 2022a

What is the value of $$3+\frac{1}{3+\frac{1}{3+\frac13}}?$$

\textbf{(A)}\ \frac{31}{10} \textbf{(B)}\ \frac{49}{15} \textbf{(C)}\ \frac{33}{10} \textbf{(D)}\ \frac{109}{33} \textbf{(E)}\ \frac{15}{4}

The sum of three numbers is $96.$ The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers?

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

Five rectangles, $A$, $B$, $C$, $D$, and $E$, are arranged in a square as shown below. These rectangles have dimensions $1\times6$, $2\times4$, $5\times6$, $2\times7$, and $2\times3$, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?

size(150);
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draw((0,0)--(7,0)--(7,7)--(0,7)--(0,0));
draw((3,0)--(3,4.5));
draw((0,4.5)--(5.3,4.5));
draw((5.3,7)--(5.3,2.5));
draw((7,2.5)--(3,2.5));

$\textbf{(A) }A \textbf{(B) }B \textbf{(C) }C \textbf{(D) }D \textbf{(E) }E$

The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$?

$\textbf{(A) } 3 \textbf{(B) } 6 \textbf{(C) } 8 \textbf{(D) } 9 \textbf{(E) } 12$

The taxicab distance between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by $$|x_1 - x_2| + |y_1 - y_2|.$$ For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$?

\textbf{(A)} 441 \textbf{(B)} 761 \textbf{(C)} 841 \textbf{(D)} 921 \textbf{(E)} 924

A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all possible values of $X$?

$\textbf{(A) } 10 \textbf{(B) } 26 \textbf{(C) } 32 \textbf{(D) } 36 \textbf{(E) } 40$

A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?

size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle);

$\textbf{(A) }120 \textbf{(B) }270 \textbf{(C) }360 \textbf{(D) }540 \textbf{(E) }720$

The infinite product $$\sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \cdots$$ evaluates to a real number. What is that number?

$\textbf{(A) }\sqrt{10} \textbf{(B) }\sqrt[3]{100} \textbf{(C) }\sqrt[4]{1000} \textbf{(D) }10 \textbf{(E) }10\sqrt[3]{10}$

On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.

"Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes.

"Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes.

"Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?

$\textbf{(A) } 7 \textbf{(B) } 12 \textbf{(C) } 21 \textbf{(D) } 27 \textbf{(E) } 31$

How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser number?

$\textbf{(A) } 108 \textbf{(B) } 120 \textbf{(C) } 126 \textbf{(D) } 132 \textbf{(E) } 144$

What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$?

$\textbf{(A) } 10 \textbf{(B) } 18 \textbf{(C) } 25 \textbf{(D) } 36 \textbf{(E) } 81$

Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. What is $\cos(\angle CMD)$?

$\textbf{(A) } \frac14 \textbf{(B) } \frac13 \textbf{(C) } \frac25 \textbf{(D) } \frac12 \textbf{(E) } \frac{\sqrt{3}}{2}$

Let $\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. What integer is closest to the area of $\mathcal{R}$?

$\textbf{(A) } 13 \textbf{(B) } 14 \textbf{(C) } 15 \textbf{(D) } 16 \textbf{(E) } 17$

What is the value of $$(\log 5)^3+(\log 20)^3+(\log 8)(\log 0.25)$$ where $\log$ denotes the base-ten logarithm?

$\textbf{(A) } \frac{3}{2} \textbf{(B) } \frac{7}{4} \textbf{(C) } 2 \textbf{(D) } \frac{9}{4} \textbf{(E) } 3$

The roots of the polynomial $10x^3-39x^2+29x-6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?

$\textbf{(A) } \frac{24}{5} \textbf{(B) } \frac{42}{5} \textbf{(C) } \frac{81}{5} \textbf{(D) } 30 \textbf{(E) } 48$

A triangular number is a positive integer that can be expressed in the form $t_n=1+2+3+\cdots+n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1=1=1^2, t_8=36=6^2,$ and $t_{49}=1225=35^2$. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?

\textbf{(A)} ~6 \textbf{(B)} ~9 \textbf{(C)} ~12 \textbf{(D)} ~18 \textbf{(E)} ~27

Suppose $a$ is a real number such that the equation $$a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)}$$ has more than one solution in the interval $(0, \pi)$. The set of all such $a$ that can be written in the form $$(p,q) \cup (q,r),$$ where $p, q,$ and $r$ are real numbers with $p < q< r$. What is $p+q+r$?

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

Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself?

$\textbf{(A) } 359 \textbf{(B) } 360 \textbf{(C) } 719 \textbf{(D) } 720 \textbf{(E) } 721$

Suppose that $13$ cards numbered $1, 2, 3, \ldots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards $1, 2, 3$ are picked up on the first pass, $4$ and $5$ on the second pass, $6$ on the third pass, $7, 8, 9, 10$ on the fourth pass, and $11, 12, 13$ on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes?

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draw((6,0)--(8,0)--(8,3)--(6,3)--cycle);
label("8", (7,1.5));
draw((9,0)--(11,0)--(11,3)--(9,3)--cycle);
label("6", (10,1.5));
draw((12,0)--(14,0)--(14,3)--(12,3)--cycle);
label("4", (13,1.5));
draw((15,0)--(17,0)--(17,3)--(15,3)--cycle);
label("5", (16,1.5));
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label("12", (22,1.5));
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label("1", (25,1.5));
draw((27,0)--(29,0)--(29,3)--(27,3)--cycle);
label("13", (28,1.5));
draw((30,0)--(32,0)--(32,3)--(30,3)--cycle);
label("10", (31,1.5));
draw((33,0)--(35,0)--(35,3)--(33,3)--cycle);
label("2", (34,1.5));
draw((36,0)--(38,0)--(38,3)--(36,3)--cycle);
label("3", (37,1.5));

$\textbf{(A) } 4082 \textbf{(B) } 4095 \textbf{(C) } 4096 \textbf{(D) } 8178 \textbf{(E) } 8191$

Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$

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

Let $$P(x) = x^{2022} + x^{1011} + 1.$$ Which of the following polynomials is a factor of $P(x)$?

\textbf{(A)} x^2 -x + 1 \textbf{(B)} x^2 + x + 1 \textbf{(C)} x^4 + 1 \textbf{(D)} x^6 - x^3 + 1 \textbf{(E)} x^6 + x^3 + 1

Let $c$ be a real number, and let $z_1$ and $z_2$ be the two complex numbers satisfying the equation $z^2 - cz + 10 = 0$. Points $z_1$, $z_2$, $\frac{1}{z_1}$, and $\frac{1}{z_2}$ are the vertices of (convex) quadrilateral $\mathcal{Q}$ in the complex plane. When the area of $\mathcal{Q}$ obtains its maximum possible value, $c$ is closest to which of the following?

$\textbf{(A) }4.5 \textbf{(B) }5 \textbf{(C) }5.5 \textbf{(D) }6 \textbf{(E) }6.5$

Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that $$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\frac{h_n}{k_n}.$$ Let $L_n$ denote the least common multiple of the numbers $1, 2, 3, \ldots, n$. For how many integers with $1\le{n}\le{22}$ is $k_n<L_n$?

$\textbf{(A) }0 \textbf{(B) }3 \textbf{(C) }7 \textbf{(D) }8 \textbf{(E) }10$

How many strings of length $5$ formed from the digits $0$, $1$, $2$, $3$, $4$ are there such that for each $j \in \{1,2,3,4\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies this condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, at least $3$ digits less than $3$, and at least $4$ digits less than $4$. The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$.)

$\textbf{(A) }500 \textbf{(B) }625 \textbf{(C) }1089 \textbf{(D) }1199 \textbf{(E) }1296$

A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$. What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$?

\textbf{(A)} ~\frac{21}{5} \textbf{(B)} ~\frac{85}{13} \textbf{(C)} ~7 \textbf{(D)} ~\frac{39}{5} \textbf{(E)} ~17