Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i^{th}$ digit, then $d_i\le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the numbers $1337$ and $123$. How many $5$-digit numbers does Kelvin like?

Emily starts with an empty bucket. Every second, she either adds a stone to the bucket or removes a stone from the bucket, each with probability $\frac{1}{2}$ . If she wants to remove a stone from the bucket and the bucket is currently empty, she merely does nothing for that second (still with probability $\frac{1}{2}$). What is the probability that after $2017$ seconds her bucket contains exactly $1337$ stones? 

There are $2017$ frogs and $2017$ toads in a room. Each frog is friends with exactly $2$ distinct toads. Let $N$ be the number of ways to pair every frog with a toad who is its friend, so that no toad is paired with more than one frog. Let $D$ be the number of distinct possible values of $N$, and let $S$ be the sum of all possible values of $N$. Find the ordered pair $(D,\ S)$.

Kelvin and $15$ other frogs are in a meeting, for a total of $16$ frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$ . Kelvin thinks the situation after the meeting is cool if for each of the $16$ frogs, the number of friends they made during the meeting is a multiple of $4$. Find the probability of the situation being cool.

Let m be a positive integer, and let $\mathbb{T}$ denote the set of all subsets of $\{1,\ 2,\ \cdots,\ m\}$. Call a subset $\mathbb{S}$ of $\mathbb{T}$ ${\delta}-good$ if for all $s_1,\ s_2\in\mathbb{S}$, $s_1\ne s_2$, $\mid\Delta(s_1,\ s_2)\mid\ge{\delta}m$, where $\Delta$ denotes symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer $s$ such that there exists an integer $m$ and a $\frac{1024}{2047}-good$ set of size $s$.

Compute the number of possible words $w = w_1w_2\cdots w_{100}$ satisfying:

• $w$ has exactly $50$ $A$’s and $50$ $B$’s (and no other letters).
• For $i = 1,\ 2,\ \cdots,\ 100$, the number of $A$’s among $w_1$, $w_2$, $\cdots$, $w_i$ is at most the number of $B$’s among $w_1$, $w_2$, $\cdots$ , $w_i$.
• For all $i = 44,\ 45,\ \cdots ,\ 57$, if $w_i$ is an $B$, then $w_{i+1}$ must be an $B$
  

Solve the recursion $$a_n=\sum^{n-1}_{k=0}a_{k}a_{n-k-1}=a_0a_{n-1}+a_1a_{n-2}+\cdots+a_{n-1}a_0$$

where $a_0=a_1=1$.

Show that the limit of $f(n)=\left(1+\frac{1}{n}\right)^n$ exits when $n$ becomes infinitely large.

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?

Joe and Mary flip a coin ($n+1$) and $n$ times, respectively. What is the probability that Joe gets more heads than Mary does?

$\textbf{Animal Kingdom}$

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?

$\textbf{Seat on a Flight}$

There are $100$ airline passengers waiting in line to board a $100$-seat plane. For convenience, let the $n^{th}$ passenger in line hold a ticket for the $n^{th}$ seat. For some reasons, the first passenger decides to pick a random seat instead of his assigned seat (it is still possible that he or she picks the $1^{st}$ seat). Everybody will sit on his or her assigned seat unless this seat is occupied. In the latter case, that passenger will pick a random seat for himself or herself. Find the probability that the last passenger will sit on his or her assigned seat.

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?

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

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?

$\textbf{Offer Letter}$

After a whole day of interviews, a HR manager comes with three sealed envelopes. One of them contains an offer letter, and the other two contain rejection letters. You can select one of them and will be hired if you get the offer letter. After you pick one envelope, the HR manager opens one of the other two which contains a rejection letter and offers you a chance to change your mind. Should you change your selection? Explain.

$\textbf{Mafia}$

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?

$\textbf{Red Cards}$

There are $7$ cards. Two of them have both sides red, two of them have both sides black, the rest three have one side red and one side black. Joe draws one card randomly and finds one side is red, what is the probability that the other side is red too?

$\textbf{Birthday Problem}$

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 $99.9\%$?

$\textbf{Number of Routes}$

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

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

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?

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth?