2021 Fall AMC 12A

What is the value of $$\frac{(2112 - 2021)^2}{169}?$$

Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?

Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\tfrac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A?

The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$

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?

As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ.$ Point $F$ lies on $\overline{AD}$ so that $DE = DF,$ and $ABCD$ is a square. What is the degree measure of $\angle AFE?$

A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50,$ $20,$ $20,$ $5,$ and $5.$ Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t - s?$

Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,$ $32,$ $33,$ $34,$ $35,$ $36,$ $37,$ $38,$ $39,$ and $40.$ What is the value of $\dfrac{N}{M}?$

A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\log_2 x,$ $\log_3 x,$ and $\log_4 x.$ What is $x?$

The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_9.$ What is the remainder when $N$ is divided by $5?$

Consider two concentric circles of radius $17$ and $19.$ The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle?

What is the number of terms with rational coefficients among the $1001$ terms in the expansion of $$\left(x\sqrt[3]{2} + y\sqrt{3}\right)^{1000}?$$

The angle bisector of the acute angle formed at the origin by the graphs of the lines $y = x$ and $y = 3x$ has equation $y = kx.$ What is $k?$

In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ.$ The enclosed area of the hexagon is $6\sqrt{3}.$ What is the perimeter of the hexagon?

Recall that the conjugate of the complex number $w = a + bi,$ where $a$ and $b$ are real numbers and $i = \sqrt{-1},$ is the complex number $\overline{w} = a - bi.$ For any complex number $z,$ let $f(z) = 4i\overline{z}.$ The polynomial $$P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$$ has four complex roots: $z_1,$ $z_2,$ $z_3,$ and $z_4.$ Let $$\begin{aligned} &Q(z) = z^4 + Az^3 + Bz^2 \\ &\quad {}+ Cz + D \end{aligned}$$ be the polynomial whose roots are $f(z_1),$ $f(z_2),$ $f(z_3),$ and $f(z_4),$ where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. What is $B + D?$

An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?

For how many ordered pairs $(b, c)$ of positive integers does neither $x^2 + bx + c = 0$ nor $x^2 + cx + b = 0$ have two distinct real solutions?

Each of $20$ balls is tossed independently and at random into one of $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\dfrac{p}{q}?$

Let $x$ be the least real number greater than $1$ such that $\sin(x) = \sin(x^2),$ where the arguments are in degrees. What is $x$ rounded up to the closest integer?

For each positive integer $n,$ let $f_1(n)$ be twice the number of positive integer divisors of $n,$ and for $j \ge 2,$ let $f_j(n) = f_1(f_{j-1}(n)).$ For how many values of $n \le 50$ is $f_{50}(n) = 12?$

Let $ABCD$ be an isosceles trapezoid with $\overline{BC} \parallel \overline{AD}$ and $AB = CD.$ Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y,$ as shown in the figure. Suppose $\angle AXD = \angle BYC = 90^\circ,$ $AX = 3,$ $XY = 1,$ and $YC = 2.$ What is the area of $ABCD?$

Azar and Carl play a game of tic-tac-toe. Azar places an $X$ in one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all $9$ boxes are filled or one of the players has $3$ of their symbols in a row — horizontal, vertical, or diagonal — whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third $O.$ How many ways can the board look after the game is over?

A quadratic polynomial with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)?$

Convex quadrilateral $ABCD$ has $AB = 18,$ $\angle A = 60^\circ,$ and $\overline{AB} \parallel \overline{CD}.$ In some order, the lengths of the four sides form an arithmetic progression, and side $AB$ is a side of maximum length. The length of another side is $a.$ What is the sum of all possible values of $a?$

Let $m \ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1 \le a_i \le m$ for all $i$ such that $m$ divides $a_1 + a_2 + a_3 + a_4.$ There is a polynomial $$q(x) = c_3x^3 + c_2x^2 + c_1x + c_0$$ such that $D(m) = q(m)$ for all odd integers $m \ge 5.$ What is $c_1?$