The set of points $(x, y)$ in the plane satisfying $x^{2/5} + |y| = 1$ form a curve enclosing a region. Compute the area of this region.

Compute the value of $$\int_0^2\sqrt{\frac{4-x}{x}}-\sqrt{\frac{x}{4-x}}dx$$

Compute $$\lim_{x\to\infty}\left[x-x^2\ln\left(\frac{1+x}{x}\right)\right]$$

For a given $x > 0$, let $a_n$ be the sequence defined by $a_1=x$ for $n = 1$ and $a_n = x^{a_{n−1}}$ for $n\ge 2$. Find the largest $x$ for which the limit $\displaystyle\lim_{n\to\infty} a_n$ converges.

Determine whether or not these two series converge: $$(A)\ \ \sum_{n=1}^{\infty}\sin\left(\frac{\cos{n}}{n^2}\right)\qquad (B)\ \  \sum_{n=1}^{\infty}\cos\left(\frac{\sin{n}}{n^2}\right)$$

The equation $x^y=y^x$ describes a curve in the first quadrant of the plane containing the point $P=(4, 2)$. Compute the slope of the line that is tangent to this curve at $P$.

Circle $\omega$ is inscribed in unit square $PLUM$ and poins $I$ and $E$ lie on $\omega$ such that $U$, $I$, and $E$ are collinear. Find, with proof, the greatest possible area for $\triangle{PIE}$.

A group of $100$ friends stands in a circle. Initially, one person has $2019$ mangos, and no one else has mangos. The friends split the mangos according to the following rules:

• sharing: to share, a friend passes two mangos to the left and one mango to the right.
• eating: the mangos must also be eaten and enjoyed. However, no friend wants to be selfish and eat too many mangos. Every time a person eats a mango, they must also pass another mango to the right.

A person may only share if they have at least three mangos, and they may only eat if they have at least two mangos. The friends continue sharing and eating, until so many mangos have been eaten that no one is able to share or eat anymore. Show that there are exactly eight people stuck with mangos, which can no longer be shared or eaten.

$\textbf{Who Finishes the Second}$

Adam, Bob, and Charlie are the only three athletes who are competing in a series of track and field events. The first, second and third places in each event are awarded $X$, $Y$ and $Z$ points respectively, where $X > Y > Z$ and all are integers. It is known that

• Adam finishes first with $22$ points overall
• Bob wins the javelin event and finishes with $9$ points overall.
• Charlie also finishes $9$ points overall.

Who finishes second in the $100$-meter dash and why?

$\textbf{Cookies}$

Steve, Tony, and Bruce have a plate of $1,000$ cookies to share according to the following rules. Beginning with Steve, each of them in turn takes as many cookies as he likes (but must be at least $1$ if there are still cookies on the plate), and then passes the plate to the next person (Steve to Tony to Bruce to Steve and so on). They all want to appear to be modest, but at the same time, want to have as many cookies as possible. This means that they all try to achieve:

1. Have one person get more cookies than himself, and one person get fewer cookies than himself.
2. Have as many cookies as possible.

The first objective takes infinite priority over the second one. If all of them are sufficiently intelligent and can choose the best strategy for themselves, what will be the end result?

$\textbf{Lucky Seven}$

Two non-identical dice both have six faces but do not necessarily have one to six dots on each face. Some numbers are missing and some have more than six dots. These two dice roll every number from $2$ to $12$. What is the largest possible probability of rolling a $7$?

$\textbf{Heist}$

1. Are you guilty?
2. How many of the seven of you are guilty?
3. How many of the seven of you tell the truth?

Here are their responses:

• Person $1$: Yes; $1$; $1$
• Person $2$: Yes; $3$; $3$
• Person $3$: No; $2$; $2$
• Person $4$: No; $4$; $1$
• Person $5$: No; $3$; $3$
• Person $6$: No; $3$; $3$
• Person $7$: Yes; $2$; $2$

After looking these answers over, the chief correctly arrests those responsible gang members. Who out of these seven are arrested?

$\textbf{Guess the Card}$

At a work picnic, Todd invites his coworkers, Ava and Bruce, to play a game. Ava and Bruce will each draw a random card from a standard $52$-card deck and place it on their own forehead. So they can see the other's card, but not his or her own. Meanwhile, they cannot communicate in any way. Then they will each write down a guess of his or her own card's color, i.e. red or black. If at least one of them guesses correctly, Todd will pay them $\$50$ each. If both guesses are incorrect, they shall each pay Todd $\$50$. If Ava and Bruce are given a chance to discuss a strategy before the game starts, can they guarantee to win?

After this game, Todd invites two more colleagues, Charlie and Doug, to join a new game. These four players will each draw a card and place it on their own foreheads so only others can see. What is different this time is that instead of color, they should guess the suite, i.e. spade, heart, club, and diamond. If at least one of them makes a correct guess, Todd will pay each of them $\$50$. Otherwise, they should each pay Todd $\$50$. Can these four co-workers guarantee to win if they are given a chance to discuss a strategy before the game starts?

$\textbf{Class Substitute}$

The possible candidates for Prof Kurt's class are list below.

• Calc $1$ at $9$ in North Hall
• Calc $2$ at noon in West Hall
• Calc $1$ at $3$ in West Hall
• Calc $1$ at $10$ in East Hall
• Calc $2$ at $10$ in North Hall
• Calc $1$ at $10$ in South Hall
• Calc $1$ at $10$ in North Hall
• Calc $2$ at $11$ in East Hall
• Calc $3$ at noon in West Hall
• Calc $2$ at noon in South Hall

After looking over the list, Julia says, "Does anyone know which class it is?" Michael and Mary Ellen immediately respond, "Well, you don't." Julia asks, "Do you?" Michael and Mary Ellen both shake their heads. Julia then smiles and says, "I now know." Which class does Kurt need a substitute for?

A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?

In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?

Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?

What is the median of the following list of $4040$ numbers?

$$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$

How many solutions does the equation $\tan{(2x)} = \cos{(\tfrac{x}{2})}$ have on the interval $[0, 2\pi]?$

There is a unique positive integer $n$ such that $$\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}$$ What is the sum of the digits of $n?$

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 $k$?