Compute $$\int\arctan{x}dx$$

Evaluate $$\int e^{ax}\cos(bx)d{x}\quad\text{and}\quad\int e^{ax}\sin(bx)d{x}$$

Evaluate $$\int x^2e^xd{x}$$

Evaluate $$\int x^2\sin{x}dx$$

Compute $$\int\sqrt{x^2+a^2}dx$$

Evaluate $$\int\frac{1}{\sqrt{x^2 + a^2}}dx$$

Compute $$\int\frac{x+1}{x^2+x+1}dx$$

Evaluate $$\int\frac{5x+6}{x^2+3x+1}dx$$

Evaluate $$\int\frac{5x+6}{(x^2+x+2)^2}dx$$

Evaluate $$\int\frac{x}{(x^2+1)(x-1)}dx$$

$\textbf{Two Doormen}$

Two doormen are guarding two rooms. One room contains tons of gold and the other is empty. Among these two doormen, one is honest who always tells the truth and the other is a liar who always gives false answers. While they know each other well, you do not know who is honest and who is not. If you are given just one chance to ask one question to one of them, what can you do in order to find out which room contains the gold?

$\textbf{Toggler's Problem}$

Among a group of $100$ people, only one is a truth teller and the rest $99$ are togglers. A truth teller always tells the truth. A toggler will tell the truth and a lie in an alternating fashion. That is, after he or she tells the truth the first time, this person will tell a lie next time. However, if his or her first answer is false, then the next answer will be true. It is unknown whether a toggler's first answer is the truth or a lie.

If all these people know who is the truth teller, how many questions do you need to ask in order to identify the truth teller?

$\textbf{Connect the Lights}$

You are in a control room which has three switches. Each switch controls one of three lights in another room. Once you leave the control room, you can not touch the switches again. How can you figure out which switch controls which light?

$\textbf{What Bear}$

Joe leaves his campsite and hikes south for $3$ miles. He then turns east and hikes for $3$ miles. Finally he turns north and hikes for $3$ miles. At this moment he sees a bear inside his tent eating his food! What color is the bear?

$\textbf{Unique Number}$

What makes the number $8549176320$ unique?

$\textbf{Right to Marry}$

Can a man legally marry his widow's sister in the state of California?

$\textbf{Color the Grid}$

Two geniuses are playing a game of coloring a $2\times n$ grid where $n$ is an odd integer. Each of them in turn picks a uncolored cell and colors it in either green or red until all the cells are filled. At the end of the game, if the number of adjacent pairs with the same color is greater than the number of adjacent pairs with different colors, then the person who picks and colors first wins the game. (An adjacent pair consists of two cells next to each other.) Otherwise, if there are more adjacent pairs with different colors than those with same color, the person who starts later wins. If these two numbers are the same, the result is a tie. Who will win if both players make no mistake?

$\textbf{Split the Coins}$

There are $100$ regular coins lying flat on a table. Among these coins, $10$ are heads up and $90$ are tails up. You are blindfolded and can not feel, see or in any other way to find out which $10$ are heads up. Is it possible to split the coins into two piles so there are equal numbers of heads-up coins in each pile?

$\textbf{Bottle of Bacteria}$

A scientist puts a bacteria in a bottle at exactly noon. Every minute the bacteria divides into two and doubles in size. At exactly $1$ PM the bottle is full. At what time is the bottle half full?

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 $\triangle{PIE}$.

$\textbf{Label the Boxes}$

There are three boxes, one contains only apples, one contains only oranges, and one contains both apples and oranges. The boxes are incorrectly labeled such that no label identifies the actual contents inside the box. Is it possible to correct all the labels by just randomly retrieving one fruit from one box? You cannot look inside the chosen box.

$\textbf{A Boat Full of People}$

You walk across a bridge and you see a boat full of people, yet there isn’t a single person on board. How is that possible?

$\textbf{Angle on a Clock}$

The time is $3:15$ now. What is the measurement of the angle between the hour and the minute hands?

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.

$\textbf{Coin Flipping}$

There are $9$ coins on the table, all heads up. In each operation, you can flip any two of them. Is it possible to make all of them heads down after a series of operations? If yes, please list a series of such operations. If no, please explain.