Prove that if $ n $ is a non-negative integer, then number $$ 2^{n+2} + 3^{2n+1}$$ is divisible by $7$.

Let $f(x)=x^2+4x+2$. Let $r$ be the difference between the largest and smallest real solutions of the equation $f(f(f(f(x))))=0$. Then $r=a^{\frac{p}{q}}$ for some positive integers $a$, $p$, $q$ so $a$ is square-free and $p,q$ are relatively prime positive integers. Compute $a+p+q$.

For any positive integer $a$, define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$. Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$.

Prove that $9$-digit number, that contains all the decimal digits except zero and does not ends with $5$ can not be exact square.

What is the hundreds digit of $2011^{2011}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9 $

Positive integers $a, b$ and $c$ are all less than $2020$. We know that $a$ divides $b + c$, $b$ divides $a + c$ and $c$ divides $a + b$. How many such ordered triples $(a, b, c)$ are there? Note: In an ordered triple, the order of the numbers matters, so the ordered triple $(0, 1, 2)$ is not the same as the ordered triple $(2, 0, 1)$.

If $x_i$ is an integer greater than 1, let $f(x_i)$ be the greatest prime factor of $x_i,x_{i+1} =x_i-f(x_i)$ ($i\ge 0$ and i is an integer). (1) Prove that for any integer $x_0$ greater than 1, there exists a natural number$k(x_0)$, such that $x_{k(x_0)+1}=0$ Grade 10: (2) Let $V_{(x_0)}$ be the number of different numbers in $f(x_0),f(x_1),\cdots,f(x_{k(x_0)})$. Find the largest number in $V(2),V(3),\cdots,V(781)$ and give reasons. Note: Bai Lu Zhou Academy was founded in 1241 and has a history of 781 years. Grade 11: (2) Let $V_{(x_0)}$ be the number of different numbers in $f(x_0),f(x_1),\cdots,f(x_{k(x_0)})$. Find the largest number in $V(2),V(3),\cdots,V(2022)$ and give reasons.

Determine all positive integers $n$ for which $2^{n+1}-n^2$ is a prime number.

Prove that for any integer $k > 2$, there exist infinitely many positive integers $n$ such that the least common multiple of $n$, $n + 1$,$...$, $n + k - 1$ is greater than the least common multiple of $n + 1$,$n + 2$,$...$, $n + k$.

Determine all prime numbers $p$ such that $p^2 - 6$ and $p^2 + 6$ are both prime numbers.

Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Find the number of five-digit positive integers, $n$, that satisfy the following conditions: (a) the number $n$ is divisible by $5$, (b) the first and last digits of $n$ are equal, and (c) the sum of the digits of $n$ is divisible by $5$.

Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has 19. Prove that for any odd integer $n \geq 1$, there is a way to number each of the 33 beads with an integer from the sequence \[ \{ n, n+1, n+2, \dots, n+32 \} \] so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace'' is viewed as a circle in which each bead is adjacent to two other beads.)

Determine all $(x,y) \in \mathbb{N}^2$ which satisfy $x^{2y} + (x+1)^{2y} = (x+2)^{2y}.$

Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.

Let $ a,n$ be positive integers such that $ a\ge(n\minus{}1)!$. Prove that there exist $ n$ [i]distinct[/i] prime numbers $ p_1,\ldots,p_n$ so that $ p_i|a\plus{}i$, for all $ i\equal{}\overline{1,\ldots,n}$.

Find all triples $(a, b, c)$ of positive integers such that $a^2 + b^2 = n\cdot lcm(a, b) + n^2$ $b^2 + c^2 = n \cdot lcm(b, c) + n^2$ $c^2 + a^2 = n \cdot lcm(c, a) + n^2$ for some positive integer $n$.

Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

Prove that there exists a subset $S$ of positive integers such that we can represent each positive integer as difference of two elements of $S$ in exactly one way.

Solve in the integers the diophantine equation $$x^4-6x^2+1 = 7 \cdot 2^y.$$

Find all pairs of integers $m, n \geq 2$ such that $$n\mid 1+m^{3^n}+m^{2\cdot 3^n}.$$

Let $a_1<a_2< \cdots$ be a strictly increasing sequence of positive integers. Suppose there exist $N$ such that for all $n>N$, $$a_{n+1}\mid a_1+a_2+\cdots+a_n$$ Prove that there exist $M$ such that $a_{m+1}=2a_m$ for all $m>M$. [i]Proposed by Ivan Chan Kai Chin[/i]

Let $P$ be the probability that the product of $2020$ real numbers chosen independently and uniformly at random from the interval $[-1, 2]$ is positive. The value of $2P - 1$ can be written in the form $\left(\frac{m}{n} \right)^b$, where $m$, $n$ and $b$ are positive integers such that $m$ and $n$ are relatively prime and $b$ is as large as possible. Compute $m + n + b$.

Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$