$\textbf{Animal Kingdom}$

In an animal kingdom, there are $n$ carnivores and $m$ herbivores. When two herbivores meet, nothing will happen. When two carnivores meet, both will die. If one herbivore meets one carnivore, the herbivore will die. All such meets can only happen between two animals. All living animals will meet another one sooner or later. If a new animal, either a carnivore or a herbivore, enters this kingdom, what is its probability of survival?

$\textbf{Overlapping Clock Hands}$

How many times in a day do the minute and hour hands of a clock overlap?

$\textbf{Camel and Bananas}$

Joe, the owner of a banana farm, has a camel. He wants to transport his $3000$ bananas to the market which is located at the other side of the desert. The distance between his banana farm and the market is $1000$ kilometers. The camel can carry at most $1000$ bananas at a time, and also eats one banana for every kilometer traveled. What is the maximum number of bananas Joe can bring to the market?

$\textbf{Interesting Locations}$

$\textbf{Lighting Bulb}$

There are $100$ bulbs, all are off, each of which is controlled by a switch. Joe was playing with them in the following way:

• In the first round, he toggled every switch. So, all the lights are on now.
• In the second round, he toggled switches $2$, $4$, $6$, $\cdots$, $100$. Now half are on and half are off.
• In the third round, he toggled switches $3$, $6$, $9$, $\cdots$, $99$,
• $\cdots$
• In the $10^{th}$ round, he toggled switch $10$, $20$, $\cdots$, $100$
• $\cdots$
• In the $100^{th}$ round, he toggled the switch $100$

Now, the question is, how many bulbs are on at the end?

$\textbf{Boys v.s. Girls}$

In a remote town, people generally prefer boys over girls. Therefore, every married couple will continue giving birth to a baby until they have a son. Assuming there is fifty-fifty chance for a couple to give birth to a boy or a girl, what is the ratio of boys to girls in this town over many years?

$\textbf{Bus Direction}$

Which way is this moving bus going?

$\textbf{Average Speed}$

Joe travels at an average of $30$ miles per hour from home to visit a friend who lives $60$ miles away. How fast should he drive on his way straight back to home so that his average speed is $60$ miles per hour for this entire trip?

$\textbf{Mafia}$

You are captured by a mafia. He puts two bullets in adjacent chambers of a standard $6$-chamber revolver. Then he points the gun at your head, and pulls the trigger. You survives. He thinks you may be a lucky man and thus promises to free you if you can survive the second shot. Meanwhile, he also gives you the option to re-spin the revolver before he pulls the trigger again. Should you accept his offer?

$\textbf{Stack of Coins}$

Can you fit a stack of quarter coins as high as the Empire State Building into a normal room?

$\textbf{Medalists}$

Five runners, $A$, $B$, $C$, $D$, and $E$, enter the final. The fastest three win a gold, silver, and bronze medal, respectively. The other two get nothing. Who are the three medalists if all of the following statements are false?

• $A$ does not win the gold and $B$ does not get the silver.
• $B$ does not get the bronze and $D$ does not win silver.
• $C$ wins a medal, but $D$ does not.
• $A$ wins a medal, but $C$ does not.
• Both $D$ and $E$ win a medal.

$\textbf{Orange}$

I am an honest man. I can tell you that I love and hate orange at the same time. Do you know why?

$\textbf{Birthday Problem}$

Statistically what is the minimum number of people among which the probability of two people having the same birthday exceeds $50\%$? How about if this probability needs to exceed $99.9\%$?

$\textbf{Cookies}$

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?

$\textbf{Coins on a Table}$

$\textbf{Picture}$

$\textbf{Haystacks}$

$\textbf{Someone's Name}$

Someone's mother has four sons: North, West and South. What is the name of the fourth son. You are asked to write down the name of the fourth son. What will you write?

$\textbf{Number of Routes}$

The shortest route from point $A$ to $B$ takes $10$ steps. How many such routes are there that do not pass point $C$?

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?