$\textbf{Missing Number}$

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

$\textbf{Snail in Well}$

At dawn on Monday, a snail falls into a $12$-inch deep well. During the day, it can climb up $3$ inches. However, during the night, it will fall back 2 inches. On what day can the snail finally manage to get out of the well?

$\textbf{Hole in a Sheet}$

There is a circular hole in the middle of a metal sheet. Will the hole become bigger, smaller or stay the same when this metal sheet is heated?

$\textbf{Tournament}$

A tournament has $2020$ participants. In each round, two people are paired and the winner advances to the next round. No game will end up with a tie. If in any round, there is an odd number of participants, the one without a paired opponent will automatically advance to the next round. The tournament will continue till the champion is declared. How many matches will this tournament have?

$\textbf{Mixed Tea and Coffee}$

Joe has two identical cups, one filled up with tea and the other filled up with coffee. He pours half of tea and a quarter of coffee into a third cup, mixes them evenly. Then, he fills up both cups using this mixed tea and coffee. Afterwards, he pours half of liquid in the first cup (originally containing the tea) and a quarter of the liquid in the second cup, and mixes them up before pouring them back. After three such iterations, is there more coffee in the first cup than the tea in the second cup?

Three ants sit at the three vertices of an equilateral triangle. At the same moment, they all start moving along the edge of the triangle at the same speed but each of them randomly chooses a direction independently. What is the probability that none of the ants collides?

$\textbf{Running Dog}$

Joe and Mary walk towards each other from $1000$m apart at speeds of $1.20$m/s and $0.80$m/s, respectively. Joe's dog, starting at the same time as Joe, runs toward Mary at a speed of $2.50$m/s. When it meets Mary, the dog immediately turns back and run towards Joe at the same speed. When the dog meets Joe the next time, it turns back and run towards Mary at the same speed again. In another word, the dog runs between Joe and Mary back and forth until the two people meet. What is the total distance run by the dog? 

$\textbf{Number of People}$

There are $10$ people dancing in a room with only one door. A terrorist busts in and opens fire. Eight people are killed immediately. How many people are there inside the room now? 

$\textbf{Three Coins}$

With an infinite supply of coins, what is the minimum number of coins required for each and every coin to touch exactly three other coins.

$\textbf{Butcher}$

A clerk at a butcher shop is five feet ten inches tall and wears size $13$ sneakers. What does he weigh?

$\textbf{Mountain}$

Before Mt. Everest was discovered, what was the highest mountain on earth?

$\textbf{Interesting Word}$

Which word in English is always spelled incorrectly?

$\textbf{Race}$

Joe has just passed the person in the second place of a marathon. What position is he in now?

$\textbf{Yolk}$

$\textbf{Son}$

$\textbf{The Pet Inc}$

The Pet Inc is owned by three gentlemen pets: a dog, a cat, and a pig. One day, while they are chatting to each other. Mr. Pig says: "Isn't it a bit odd that our surnames match our species, but none of our surnames matches our own species?" The dog replies: "Yes, but does it matter?" Can you relate their surnames and species?

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?

Solve $x^2 - x -1=0$.

Solve $x^4-x^2-1=0$.

Let $r$ and $s$ be integers. Find the condition such that the expression $\frac{6^{r+s}\times 12^{r-s}}{8^r\times 9^{r+2s}}$ is an integer.