Let $p$ be an odd prime, and $n=\frac{2^{2p}-1}{3}$ in an integer. Prove $2^{n-1}\equiv 1\pmod{n}$.

Solve this modular equation: $$f(x)=4x^2+27x-9\equiv 0\pmod{15}$$

Compute $3^{2017}\pmod{1000}$.

Let integer $N=\left\lfloor{(\sqrt{29}+\sqrt{21})^{2020}}\right\rfloor$ where $\lfloor{x}\rfloor$ denotes the largest integer not exceeding $x$. Find the last two digits of $N$.

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

Find the largest integer $x$ such that for any positive integer $y$, the number $(7^y + 12y-1)$ is always a multiple of $x$.

Let $m$ and $n$ be two positive integers, find the minimal value of $\mid 12^m - 5^n\mid$.

Let $N$ be the product of four consecutive odd numbers. Show that $N\equiv 1\pmod{8}$.

Let the product of all odd positive integer not greater than $2019$ be $2019!!$. Find the last three digits of $2019!!$.

Let $S$ be the sum of squares of $10$ consecutive positive integers. Show $S$ cannot be a square.

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.

Show that there is at least one Friday $13^{th}$ in any year, including any leap year.

Show that there exists an infinite number of integers in the form of $(2^n+27)$ which are multiples of $7$.

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.

Determine all positive integer $n$ such that the following equation is solvable in integers: $$x^n + (2+x)^n + (2-x)^n = 0$$

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 $m$.