2024 AMC 12B

In a long line of people arranged left to right, the $1013$th person from the left is also the $1010$th person from the right. How many people are in the line?

What is $10! - 7! \cdot 6!?$

For how many integer values of $x$ is $|2x| \le 7\pi?$

Balls numbered $1,$ $2,$ $3,$ $\ldots$ are deposited in $5$ bins, labeled $A,$ $B,$ $C,$ $D,$ and $E,$ using the following procedure. Ball $1$ is deposited in bin $A,$ and balls $2$ and $3$ are deposited in $B.$ The next three balls are deposited in bin $C,$ the next $4$ in bin $D,$ and so on, cycling back to bin $A$ after balls are deposited in bin $E.$ (For example, $22,$ $23,$ $\ldots,$ $28$ are deposited in bin $B$ at step $7$ of this process.) In which bin is ball $2024$ deposited?

In the following expression, Melanie changed some of the plus signs to minus signs:

$$
1 + 3 + 5 + 7 + \cdots + 97 + 99
$$

When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?

The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by $2033.$ How many digits does this number of dollars have when written as a numeral in base $5?$ (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem.)

In the figure below $WXYZ$ is a rectangle with $WX = 4$ and $WZ = 8.$ Point $M$ lies on $\overline{XY},$ point $A$ lies on $\overline{YZ},$ and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA?$

What value of $x$ satisfies

$$
\frac{\log_2 x \cdot \log_3 x}{\log_2 x + \log_3 x} = 2?
$$

A dartboard is the region $B$ in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8.$ A target $T$ is the region where $(x^2 + y^2 - 25)^2 \le 49.$ A dart is thrown and lands at a random point in $B.$ The probability that the dart lands in $T$ can be expressed as $\dfrac{m}{n} \cdot \pi,$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$

A list of $9$ real numbers consists of $1,$ $2.2,$ $3.2,$ $5.2,$ $6.2,$ and $7,$ as well as $x,$ $y,$ $z$ with $x \le y \le z.$ The range of the list is $7,$ and the mean and median are both positive integers. How many ordered triples $(x, y, z)$ are possible?

Let $x_n = \sin^2(n^\circ).$ What is the mean of $x_1,$ $x_2,$ $x_3,$ $\ldots,$ $x_{90}?$

Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1,$ and with $|z| = 2.$ In the complex plane, the four points $0,$ $z,$ $z^2,$ and $z^3$ are the vertices of a quadrilateral with area $15.$ What is the imaginary part of $z?$

There are real numbers $x,$ $y,$ $h,$ and $k$ that satisfy the system of equations

$$
x^2 + y^2 - 6x - 8y = h
$$

$$
x^2 + y^2 - 10x + 4y = k.
$$

What is the minimum possible value of $h + k?$

How many different remainders can result when the $100$th power of an integer is divided by $125?$

A triangle in the coordinate plane has vertices $A(\log_2 1, \log_2 2),$ $B(\log_2 3, \log_2 4),$ and $C(\log_2 7, \log_2 8).$ What is the area of $\triangle ABC?$

A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^r M,$ where $r$ and $M$ are positive integers and $M$ is not divisible by $3.$ What is $r?$

Integers $a$ and $b$ are randomly chosen without replacement from the set of integers with absolute value not exceeding $10.$ What is the probability that the polynomial $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots?

The Fibonacci numbers are defined by $F_1 = 1,$ $F_2 = 1,$ and $F_n = F_{n-1} + F_{n-2}$ for $n \ge 3.$ What is

$$
\frac{F_2}{F_1} + \frac{F_4}{F_2} + \frac{F_6}{F_3} + \cdots + \frac{F_{20}}{F_{10}}?
$$

Equilateral $\triangle ABC$ with side length $14$ is rotated about its center by angle $\theta,$ where $0 \lt \theta \lt 60^\circ,$ to form $\triangle DEF.$ See the figure. The area of hexagon $ADBECF$ is $91\sqrt3.$ What is $\tan\theta?$

Suppose $A,$ $B,$ and $C$ are points in the plane with $AB = 40$ and $AC = 42,$ and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}.$ Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC.$ Then the domain of $f$ is an open interval $(p, q),$ and the maximum value $r$ of $f(x)$ occurs at $x = s.$ What is $p + q + r + s?$