Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20, 20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k$?

There are integers $a$, $b$, and $c$, each greater than 1, such that $$\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}$$ for all $N > 1$. What is $b$?

Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\tfrac{m}{n}?$

In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. What is the greatest distance between a point of $A$ and a point of $B?$

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth?

The vertices of a quadrilateral lie on the graph of $y = \ln x$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln \frac{91}{90}$. What is the $x$-coordinate of the leftmost vertex?

Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$ ?

There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that $$\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.$$ What is $k?$

Let $T$ be the triangle in the coordinate plane with vertices $\left(0,0\right)$, $\left(4,0\right)$, and $\left(0,3\right)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)

How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$?

Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that $$(2 + i)^n = a_n + b_ni$$ for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is $$\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?$$

Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s?$

Find the number of real number solutions to the equation: $8^x +4=4^x + 2^{x+2}$.

Let $f_n (x) = (2 + (−2)^n ) x^2 + (n + 3) x + n^2$.

1. Write down $f_3(x)$ and find its maximum value. Also determine for what value of $n$ does the function $f_n(x)$ have a maximum value (as $x$ varies). You do not need to compute this maximum value.
2. Write down $f_1(x)$. Calculate $f_1(f_1(x))$ and $f_1(f_1(f_1(x)))$. Find an expression, simplified as much as possible, for $$\underbrace{f_1(f_1(\cdots f_1(x)))}_{k}$$
3. Write down $f_2(x)$. Find the degree of the function $$\underbrace{f_2(f_2(\cdots f_2(x)))}_{k}$$

Let $f(c)=\int_0^1\left( (x-c)^2 + c^2\right)dx$ where $c$ is a real number. Find the minimal value of $f(c)$ as $c$ varies and the maximum value of $f(\sin\theta)$ as $\theta$ varies.

Find the number of $k$ such that the function $y=e^{kx}$ satisfies the equation $$\left(\frac{d^2y}{dx^2}+\frac{dy}{dx}\right)\left(\frac{dy}{dx}-y\right)=y\frac{dy}{dx}$$

A circle of radius $2$, center on the origin, is drawn on a grid of points with integer coordinates. Let $n$ be the grid points that lie within or on the circle. What is the smallest amount of radius needs to increase by for there to be $(2n-5)$ grid points within or on the circle?

A particle moves in the $xy$-plane, starting at the origin $(0, 0)$. At each turn, the particle may move in one of the two ways:

• it may move two to the right and one up
• it may move one to the right and two up

What is the closet distance the particle may come to the point $(25, 75)$?

Let $n \ge k$ are two positive integers. Given function $x_1+x_2+\cdots + x_k =n$,

1. Find the number of positive integer solutions to this equation.
2. Find the number of non-negative integer solutions to this equation.
3. Explain the relation between these two cases. i.e. is it possible to derive (2) from (1), and vice versa?

Explain why the count of positive / non-negative integer solutions to the equation $x_1 + x_2 + \cdots + x_k=n$ is equivalent to the case of putting $n$ indistinguishable balls into $k$ distinguishable boxes.

Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} + \sqrt{x} - 2x - 1=0$$

Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} - 2\sqrt{x} + x - 1=0$$

Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} - 2\sqrt{x} - x + 1=0$$

Prove that, if $|\alpha| < 2\sqrt{2}$, then there is no value of $x$ for which $$x^2-\alpha|x| + 2 < 0\qquad\qquad(*)$$

Find the solution set of (*)  for $\alpha=3$.

For $\alpha > 2\sqrt{2}$, then the sum of the lengths of the intervals in which $x$ satisfies (*) is denoted by $S$. Find $S$ in terns of $\alpha$ and deduce that $S < 2\alpha$.

Which number is larger: $5^{4321}$ or $4^{5321}$?