Let $N$ be the value of the following expression. $$\sum_{k=0}^{n-1}\left(\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{k}\right)\left(\binom{n}{k+1}+\binom{n}{k+2}+\cdots+\binom{n}{n}\right)$$

Show $$N=\frac{n}{2}\binom{2n}{n}$$

Show that $$\sum_{k=1}^{n}(-1)^k\binom{n}{k}\left(1+\frac{1}{2}+\cdots+\frac{1}{k}\right)=-\frac{1}{n}$$

Prove $$\sum_{k=0}^{n}(-1)^k\frac{{n \choose k}}{\binom{m+k}{k}}=\frac{m}{m+n}$$

How many $4$-digit integers are there whose sum of all digits equals $12$?

A deck of poker has three different colors each of which contains $10$ cards numbered from $1$ to $10$, respectively. In addition, there are two jokers both of which are numbered as $0$. A card with number $k$ is valued as $2^k$ points. How many different ways are there to draw several cards from this deck so that their total value equals $2004$?

Let $n$ be a positive integer. Find the number $a_n$ of polynomials $f(x)$ with coefficients in $\{0, 1, 2, 3\}$ such that $f(2)=n$.

Let $a_0$, $a_1$, $a_2$, $\cdots$ be an increasing sequence of non-negative integers such that every non-negative integer can be expressed uniquely in the form of $(a_i + 2a_j+4a_k)$ where $i$, $j$, and $k$ are not necessarily distinct. Determine $a_{1998}$.

Let $\mathbb{N}$ be the set containing all positive integers. Is it possible to partition $\mathbb{N}$ to more than one but still a finite number of arithmetic sequences with no two having the same common difference?

Let $p$ be an odd prime number. Find the number of subsets $\mathbb{A}$ of the set $\{1, 2, \cdots, 2p\}$ such that $\mathbb{A}$ has exactly $p$ elements and the sum of all elements in $\mathbb{A}$ is divisible by $p$.

Let $a$, $b$, $p$, and $q$ be fixed positive integers. If an $a\times b$ grid can be tiled using some $1\times p$ and $q\times 1$ pieces, show that either $a$ is divisible by $p$ or $b$ is divisible by $q$. Here, a $1\times k$ and $k\times 1$ grids are treated as different.

Find the number of subsets of $\{1,\ 2,\ ,3\ ,\cdots,\ 2021\}$ the sum of whose elements is divisible by $101$. (Empty subset is permitted).

Let $n$, $m$ and $k$ be three positive integers satisfying $m(k-1) < n$. Find the number of ways to select $k$ items from $\{1,\ 2,\ \cdots,\ n\}$ for form a strict increasing sequence and the difference between adjacent terms is no more than $m$.

As shown, an isosceles trapezoid is obtained by removing the top part of an equilateral triangle. The lengths of its two bases are $a$ and $b$, respectively, which are both integers. Both bases and sides are equally divided into unit-length parts. Their ending points are then connected to create several segments which are parallel to either two bases or one side. Find the number of equilateral triangles in this diagram.

Contessa is taking a random lattice walk in the plane, starting at $(1, 1)$. (In a random lattice walk, one moves up, down, left, or right $1$ unit with equal probability at each step.) If she lands on a point of the form $(6m, 6n)$ for $m$, $n\in\mathbb{Z}$, she ascends to heaven, but if she lands on a point of the form $(6m+ 3, 6n+ 3)$ for $m,\ n\in\mathbb{Z}$, she descends to hell. What is the probability that she ascends to heaven? 

In an election for the Peer Pressure High School student council president, there are $2019$ voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice’s boyfriend Bob votes for Alice as well. Then one by one, each of the remaining $2016$ voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate’s votes. For example, the first undecided voter David has a $2/3$ probability of voting for Alice and a $1/3$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?

Given that the value $\ln(2)$ is not the root of any polynomial with rational coefficients. For any nonnegative integer $n$, let $p_n(x)$ be the unique polynomial with integer coefficients such that $$p_n(\ln(2)) =\int_1^2 (ln(x))^n dx$$

Compute the value of the $$\sum_{n=0}^{\infty}\frac{1}{p_n(0)}$$

$\textbf{Outlier Ball}$

There are $12$ balls of which $11$ weigh the same and an outlier that weighs differently. We do not know whether the outlier weighs more or less than the other balls. What is the minimum number of weighs required to find this outlier using a balance? (This problem is similar to <myProblem>GetLink/4663</myProblem> except that the outlier ball can be either heavier or lighter.)

$\textbf{Seat on a Flight}$

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.

$\textbf{Poisonous Wine}$

A king has $1000$ bottles of expensive wine. One assassin is just able to poison one bottle of wine before being killed. The king does not want to discard all the bottles, so he decides to force some prisoners to taste these wines in order to find out the poisonous one. While he is ruthless, the king is also intelligent. He figures that it is not necessary to use $1000$ prisoners because it is known that this type poison will take effect and kill an in-taker after $24$ hours. Is it possible for the king to use no more than $10$ prisoners to identify the poisonous bottle?

$\textbf{Prisoners in Solitary Cells}$

There are $100$ prisoners locked up in solitary cells. The king gets bored and offers them a challenge. Everyday, he will randomly select and put one prisoner into a special room. (A prisoner may be selected more than once.) This special room has a light and its controlling switch. The prisoner inside the special room can turn on, turn off, or do nothing with the switch. But no other prisoner can see or control the light. On any day, the prisoners can stop this process by declaring that every one of them has been in the special room at least once. If that happens to be true, then all the prisoners will be freed. Otherwise, they will all be executed. Before starting the challenge, the prisoners are given some time to discuss. Is there a strategy to free themselves?

$\textbf{Children's Age}$

Joe and John meet at a bar and become acquainted. John tells Joe that he has three children. So Joe asks John "How old are they?". "Well, the product of their ages is $72$.", John replies, "and the sum of their ages is the street number of this bar." Joe walks out the door, checks the number, and comes back "The information is not sufficient." John says "I agree. Here is another piece of information: my youngest child likes ice cream." Joe thinks for a minute and says "Now I know your children's ages."

What are the ages of John's children?

$\textbf{Tricky Sequence}$

Find the next term in the sequence: $F21$, $S23$, $T25$, $T27$, $S29$, $M31$.

$\textbf{Circular Killing}$

One hundred prisoners on the death row are ordered to stand in a circle and are numbered from $1$ to $100$ in sequence. The king then gives a sword to No $1$. No 1 kills the No $2$ and passes the sword to No $3$. The No $3$ then kills No $4$ and passes the sword to the next person alive, i.e. No $5$. All people continuously does the same until only one person survives. Who is the last survivor?

$\textbf{Who Finishes the Second}$

Adam, Bob, and Charlie are the only three athletes who are competing in a series of track and field events. The first, second and third places in each event are awarded $X$, $Y$ and $Z$ points respectively, where $X > Y > Z$ and all are integers. It is known that

• Adam finishes first with $22$ points overall
• Bob wins the javelin event and finishes with $9$ points overall.
• Charlie also finishes $9$ points overall.

Who finishes second in the $100$-meter dash and why?