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?

$\textbf{Coin Flipping}$

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.

$\textbf{How Far Can You Go}$

There are $50$ motorcycles with a tank that has the capacity to go $100$ km. Using these $50$ motorcycles, what is the maximum distance that you can go?

$\textbf{Shatter the Ball}$

You are in a $100$-story building with two identical bowling balls. You want to find the lowest floor at which the ball will shatter when dropped to the ground. What is the minimum number of drops you need in order to find the answer?

$\textbf{Make Four Liters}$

If you have an infinite supply of water, a $5$-liter bucket, and a $3$-liter bucket, how would you measure exactly $4$ liters of water? The buckets do not have any intermediate scales.

The probability of a specific parking slot gets occupied is $\frac{1}{3}$ on any single day. If you find this slot vacant for $9$ consecutive days, what is the probability that it will be vacant on the $10^{th}$ day?

$\textbf{Coin Toss}$

Joe tosses a coin. If he gets heads, he stops, otherwise he tosses again. If the second toss is heads, he stops. Otherwise, he tosses the coin again. The process continues until either he gets heads or $100$ tosses have been done. What is the ratio of heads to tails in all the possible scenarios?

$\textbf{Three Switches}$

There are three switches in the control room. Two of them are disconnected and the other one is connected to a light in another room. Upon leaving the control room, you will not be permitted to return again. How can you determine which switch is connected to the light?

$\textbf{Bitter Water}$

There are $1000$ bottles of water. All of them are tasteless except one which tastes bitter. How do you find the bottle of bitter water in the smallest number of sips?

$\textbf{Missing Number}$

An $99$-element array contains all but one integer between $1$ and $100$. Find the missing number.

$\textbf{Pirates and Gold}$

Five pirates are trying to split up $1000$ gold pieces according to the following rules

• The $1^{st}$ pirate can propose a plan. If his proposal wins the majority's support (including himself), then it is done. Otherwise, he will be instantly killed.
• If the $1^{st}$ pirate is killed, then the $2^{nd}$ pirate will make a proposal. If his proposal wins the majority's support (a tie does not suffice, i.e. he needs at least $3$ votes), it is done. Otherwise, he will be instantly killed.
• The process continues until a proposal is agreed by a majority.

Assuming all these five pirates are intelligent (i.e. always choose the optimal strategy for himself), greedy (i.e. get as much as gold for himself) and ruthless (i.e. the more pirates dead, the better), what will be the final distribution of the gold?

$\textbf{Child's Name}$

Tracy's mother has four children. The first one is called April, the second is called May, and the third is called June. What is the name of her fourth child?