Show that from any given $m$ integers, it is always possible to select one or more integers such that their sum is a multiple of $m$.

Let $n$ be a positive integer and $k$ be an odd positive integer, show $k^{2^n}\equiv 1\pmod{2^{n+2}}$.

Show that there exists an infinite number of squares in the form of $(n\cdot 2^k - 7)$ where $n$ and $k$ are both positive integers.

Let $a$, $b$, and $x_0$ all be positive integers. Sequence $\{x_n\}$ is defined as $x_{n+1}=ax_n + b$ where $n \ge 1$. Show that $x_1$, $x_2$, $\cdots$ cannot be all prime.

Let $N$ be the number of possible ways to pick up two adjacent squares in a $(n\times m)$ grid. Find $N$.

Show that $$\frac{1}{(1-x)^n}=\sum_{k=0}^{\infty}\binom{n-1+k}{n-1}x^k$$

Let $n$ be a positive integer greater than $1$. Show $$\sum_{k=1}^{n-1}\frac{1}{k(n-k)}\binom{2(k-1)}{k-1}\binom{2(n-k-1)}{n-k-1}=\frac{1}{n}\binom{2(n-1)}{n-1}$$

Show that $$\frac{1}{\sqrt{1-4x}}=\sum_{n=0}^{\infty}\binom{2n}{n}x^n$$

How many ordered integers $(x_1,\ x_2,\ x_3,\ x_4)$ are there such that $0 < x_1 \le x_2\le x_3\le x_4 < 7$?

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)?

$\textbf{Color the Grid}$

Two geniuses are playing a game of coloring a $2\times n$ grid where $n$ is an odd integer. Each of them in turn picks a uncolored cell and colors it in either green or red until all the cells are filled. At the end of the game, if the number of adjacent pairs with the same color is greater than the number of adjacent pairs with different colors, then the person who picks and colors first wins the game. (An adjacent pair consists of two cells next to each other.) Otherwise, if there are more adjacent pairs with different colors than those with same color, the person who starts later wins. If these two numbers are the same, the result is a tie. Who will win if both players make no mistake?

$\textbf{Coin Flipping}$

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.

$\textbf{Cheating Husbands}$

A remote town comprises of $100$ married couples. Everyone in the town lives by the following rule: If a husband cheats on his wife, the husband is executed at the night as soon as his wife finds out about it. All the women in the town only gossip about husbands of other women. No woman ever tells another woman if that woman's husband is cheating on her. So every woman in the town knows about all the cheating husbands in the town except her own. It can also be assumed that a husband remains silent about his infidelity. One day, the mayor of the town announces to the whole town that there is at least $1$ cheating husband in the town. What will happen afterwards?

$\textbf{Coins on a Table}$

Let $a$, $b$, $c$, $x$, $y$, and $z$ be positive numbers. Show that $$\sqrt{a^2+x^2} + \sqrt{b^2 + y^2}+\sqrt{c^2+z^2} \ge \sqrt{(a+b+c)^2 + (x+y+z)^2}$$

Let positive numbers $a$, $b$ and $c$ satisfy $a+b+c=8$. Find the minimal value of $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+9}$.

Show that there exists a perfect sqaure whose leading $2024$ digits are all $1$.

Design an algorithm that finds the number of ways in which you can traverse $N$ meters by doing jumps of $1$, $2$, $3$, $4$, or $5$ meter lengths.

On a $M\times N$ board, some cells are occupied. Find the size of the largest square of unoccupied cells.