Let $\{ a_1, a_2, \cdots, a_{2n+1}\}$ be a set of integers such that after removing any element, the remaining ones can always be equally divided into two groups with equal sum. Show that all these $a_i$, $(1 \le i \le 2n+1)$ are equal.

Show that the number $(2n^{3k}+4n^{k}+10)$ cannot be a product of consecutive integers for any positive integers $n$ and $k$.

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 integers $l > m > n$ be the side lengths of a triangle satisfying $\left\{\frac{3^l}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$ where function $\{x\}$ returns the decimal part of real number $x$. Find the least possible value of this triangle's perimeter.

Let $n$ be an integer which is divisible by neither $2$ nor $5$. Show that $n$ must be divisible by a number whose digits are all $1$.

Let $N=4568^{7777}$, $a$ be the sum of digits in $N$, $b$ be the sum of digits in $a$, and $c$ be the sum of digits in $b$. Find $c$.

Show that for any integer $x$, the number $\left(\frac{x^5}{5}+\frac{x^3}{3}+\frac{7x}{15}\right)$ is an integer.

Let $m$ and $n$ be positive integers, $m$ be odd, and $(m, 2^{n} - 1)=1$. Show that $\displaystyle\sum_{k=1}^{m}k^n$ is a multiple of $m$.

Let $n$ be an odd positive integer, and $$N=6^n + \binom{n}{1}\cdot 6^{n-1}+\cdots + \binom{n}{n-1}\cdot 6-1$$

Find the remainder when $N$ is being divided by $8$.

Let the binary representation of positive integer $n$ be $b_tb_{t-1}\cdots b_1b_0$. Show that $$\binom{n}{2^j} \equiv b_j \pmod{2}$$

where $j$ is a non-negative integer. Note that $\binom{n}{m} = 0$ if $m > n$.

Find the last $4$ digits of $2018^{2019^{2020}}$.

$\textbf{Guess the Card}$

At a work picnic, Todd invites his coworkers, Ava and Bruce, to play a game. Ava and Bruce will each draw a random card from a standard $52$-card deck and place it on their own forehead. So they can see the other's card, but not his or her own. Meanwhile, they cannot communicate in any way. Then they will each write down a guess of his or her own card's color, i.e. red or black. If at least one of them guesses correctly, Todd will pay them $\$50$ each. If both guesses are incorrect, they shall each pay Todd $\$50$. If Ava and Bruce are given a chance to discuss a strategy before the game starts, can they guarantee to win?

After this game, Todd invites two more colleagues, Charlie and Doug, to join a new game. These four players will each draw a card and place it on their own foreheads so only others can see. What is different this time is that instead of color, they should guess the suite, i.e. spade, heart, club, and diamond. If at least one of them makes a correct guess, Todd will pay each of them $\$50$. Otherwise, they should each pay Todd $\$50$. Can these four co-workers guarantee to win if they are given a chance to discuss a strategy before the game starts?

Given a pointer pointing to the header of a linked list, how to detect whether the linked list has a loop?

An additional question: how can math knowledge help here?

Determine whether a given integer can be expressed as the sum of two perfect squares.