A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square?

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\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]

Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?

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 $s?$

Solve $x^2 - x -1=0$.

Solve $x^4-x^2-1=0$.

Let $r$ and $s$ be integers. Find the condition such that the expression $\frac{6^{r+s}\times 12^{r-s}}{8^r\times 9^{r+2s}}$ is an integer.

Find the number of solutions to the equation $7\sin x + 2\cos^2 x = 5$ for $0\le x < 2\pi$.

Find the point on the circle $(x − 5)^2 + (y − 4)^2 = 4$ which is closest to the circle $(x − 1)^2 + (y − 1)^2 = 1$.

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

Let function $f(x)$ satisfy:  $$\int^1_0 3f (x) dx +\int^2_1 2f (x) dx = 7$$

and $$\int^2_0 f (x) dx + \int^2_1 f (x) dx = 1$$

Find the value of $$\int^2_0 f (x) dx$$

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.

In the diagram below, a line is tangent to a unit circle centered at $Q (1, 1)$ and intersects the two axes at $P$ and $R$, respectively. The angle $\angle{OPR}=\theta$. The area bounded by the circle and the $x-$axis is $A(\theta)$ and the are bounded by the circle and the $y-$axis is $B(\theta)$.

1. Show the coordinates of the point $Q$ is $(1+\sin\theta, 1+\cos\theta)$. Find the equation of line $PQR$ and determine the coordinates of $P$.
2. Explain why $A(\theta)=B\left(\frac{\pi}{2}-\theta\right)$ always holds and calculates $A\left(\frac{\pi}{2}\right)$.
3. Show that $A\left(\frac{\pi}{3}\right)=\sqrt{3}-\frac{\pi}{3}$.

Find the area of the region bounded by the curve $y=\sqrt{x}$, the line line $y=x-2$, and the $x-$ axis.