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

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?

$\textbf{Connect the Lights}$

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?

$\textbf{Unique Number}$

What makes the number $8549176320$ unique?

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?

$\textbf{Coin Flipping}$

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?

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?

$\textbf{Child's Name}$

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?

$\textbf{Defective Machine}$

A company has $10$ machines that produce gold coins. One of the machines is producing coins that are one gram lighter. What is the minimum number of weighs you will need in order to find out the defective machine?

$\textbf{Tiger and Sheep}$

One hundred tigers and one sheep are put on a magic island where there is only grass. Tigers on this magic land can eat grass, but they would rather eat the sheep. However, upon having eaten the sheep, the tiger will become a sheep itself. If only one tiger can eat the sheep at any moment, what will happen? The assumption is that all the tigers are intelligent enough to secure their survival first and, if possible, eat the sheep.

$\textbf{Burning Ropes}$

Two ropes have different densities at different points, but both take exactly an hour to burn. Is it possible to use these two ropes to measure $45$ minutes? If so, how? If not, please explain.

$\textbf{Heavier Ball}$

There are $12$ balls of which $11$ weigh the same and the other one is heavier. What is the minimum number of weighs required to find this heavier ball using a balance?

$\textbf{Six Glasses}$

There are six identical glasses of which three are empty and three contain water. They are currently lined up in an alternating fashion, i.e. the $1^{st}$, $3^{rd}$ and $5^{th}$ are empty, and the others are full. Is it possible to move just one glass so that the three glasses containing water are next to each other without any empty one in between?

$\textbf{Coin and Cork}$

A coin is put into a bottle of wine and then the bottle is corked. Is it possible to take out the coin without taking out the cork or breaking the bottle?

$\textbf{Great Pyramid}$

Joe visited the Great Pyramid of Egypt in $1995$. He was so impressed that he vowed to visit this wonder again with his children. In $1975$, Joe brought his son there and fulfilled his vow. How was this possible?

$\textbf{Silver Link}$

Joe plans to hire an assistant for a week and pay this person exactly one silver link per day. The wage will be settled daily. Joe thinks of using a chain of seven links to finance this arrangement. What is the minimum number of chain cuts Joe needs?

$\textbf{Mountain Hiker}$

John starts to hike up a mountain at $7:00$ am and reaches the top at $7:00$ pm. He stays at the top overnight. On the next day, he starts to hike down at $7:00$ am along the same route and reaches his starting point at $7:00$ pm. His speed during the two-day hiking varies from time to time. What is the probability that there exists one spot he passes at the exactly same time during the two days?

$\textbf{Mixed Pills}$

John must take exactly one $A$ pill and one $B$ pill each day. These two types of pills look exactly the same and cannot be distinguished in any way. One day, while he has one $A$ pill in his hand, he accidentally gets two $B$ pills out of the bottle. Now he has three indistinguishable pills in his hand. As both medicines are quite expensive, John does not want to waste any pill. What can he do?