There are $n \ge 2$ houses on the northern side of a street. Going from the west to the east, the houses are numbered from 1 to $n$. The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the currnet number plates are swapped exactly once during the day. How many different sequences of number plates are possible at the end of the day?

Let $n\ge 4$ be an integer. You have two $n\times n$ boards. Each board contains the numbers $1$ to $n^2$ inclusive, one number per square, arbitrarily arranged on each board. A move consists of exchanging two rows or two columns on the first board (no moves can be made on the second board). Show that it is possible to make a sequence of moves such that for all $1 \le i \le n$ and $1 \le j \le n$, the number that is in the $i-th$ row and $j-th$ column of the first board is different from the number that is in the $i-th$ row and $j-th$ column of the second board.

Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions: [list] [*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square. [*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$. [/list] Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$. And determine the value of $\sum_{k=N}^{N+2021}a_k$.

Denote by $\lfloor x\rfloor$ the greatest positive integer less than or equal to $x$. Let $m\ge2$ be an integer, and let $s$ be a real number between $0$ and $1$. Define an infinite sequence of real numbers $a_1, a_2, a_3,\ldots$ by setting $a_1 = s$ and $ak = ma_{k-1}-(m-1)\lfloor a_{k-1}\rfloor$ for all $k\ge2$. For example, if $m = 3$ and $s = \tfrac58$, then we get $a_1 = \tfrac58$, $a_2 = \tfrac{15}8$, $a_3 = \tfrac{29}8$, $a_4 = \tfrac{39}8$, and so on. Call the sequence $a_1, a_2, a_3,\ldots$ $\textbf{orderly}$ if we can find rational numbers $b, c$ such that $\lfloor a_n\rfloor = \lfloor bn + c\rfloor$ for all $n\ge1$. With the example above where $m = 3$ and $s = \tfrac58$, we get an orderly sequence since $\lfloor a_n\rfloor = \left\lfloor\tfrac{3n}2-\tfrac32\right\rfloor$ for all $n$. Show that if $s$ is an irrational number and $m\ge2$ is any integer, then the sequence $a_1, a_2, a_3,\ldots$ is $\textbf{not}$ an orderly sequence.

Let $a_0,a_1,a_2,\dots$ be a periodic sequence of real numbers(that is, there is a fixed positive integer $k$ such that $a_n=a_{n+k}$ for every integer $n\geq 0$). The following equality is true, for all $n\geq 0$: $a_{n+2}=\frac{1}{n+2} (a_n - \frac{n+1}{a_{n+1}})$ if $a_0=2020$, determine the value of $a_1$.

Let $n$ be a positive integer and let $V$ be a $(2n-1)$-dimensional vector space over the two-element field. Prove that for arbitrary vectors $v_1,\dots,v_{4n-1} \in V,$ there exists a sequence $1\leq i_1<\dots<i_{2n}\leq 4n-1$ of indices such that $v_{i_1}+\dots+v_{i_{2n}}=0.$

Given a strictly increasing infinite sequence $\{a_n\}$ of positive real numbers such that for any $n\in N$: $$a_{n+2}=(a_{n+1}-a_{n})^{\sqrt{n}}+n^{-\sqrt{n}}$$ Prove that for any $C>0$ there exist a positive integer $m(C)$ (depended on $C$) such that $a_{m(C)}>C$.

Consider the sequence of real numbers $(a_n)_{n\ge1}$ such that $$\lim_{n\to\infty}\frac1{n^r}\sum_{k=1}^n\frac{a_k}k=l\in\mathbb R,r\in\mathbb N^*$$ Show that: $$\lim_{n\to\infty}\left(\dfrac{\displaystyle\sum_{p=n+1}^{2n}\sum_{k=1}^p\sum_{i=1}^k\frac{a_i}{p\cdot i}}{n^{r+1}}\right)=l\left(\frac{2^{r+1}}{r(r+1)}-\frac{2^r}{(r+1)^2}\right)$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]

Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$ \[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \] Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.

Find all positive integers $ n$ such that acute-angled $ \triangle ABC$ with $ \angle BAC<\frac{\pi}{4}$ could be divided into $ n$ quadrilateral. Every quadrilateral is inscribed in circle and radiuses of circles are in geometric progression. [hide] be carefull ! :lol: [/hide]

Find, with proof, all pairs of positive integers $(n,d)$ with the following property: for every integer $S$, there exists a unique non-decreasing sequence of $n$ integers $a_1,a_2,...,a_n$ such that $a_1 + a_2 + ... + a_n = S$ and $a_n-a_1=d.$

Let $n$ be a positive integer, and let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers. Alex the Kat writes down the $n^2$ numbers of the form $\min(a_i, a_j)$, and Kelvin the Frog writes down the $n^2$ numbers of the form $\max(b_i, b_j)$. Let $x_n$ be the largest possible size of the set $\{a_1, \ldots, a_n, b_1, \ldots, b_n\}$, such that Alex the Kat and Kelvin the Frog write down the same collection of numbers. Determine the number of distinct integers in the sequence $x_1, x_2, \ldots, x_{10,000}$.

Let $f(n) = n + \lfloor \sqrt{n} \rfloor$. Prove that for every positive integer $m$, the integer sequence $m, f(m), f(f(m)), \dots$ contains at least one square of an integer.

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties: (i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$; (ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite. Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic. [i]Proposed by Ang Jie Jun, Singapore[/i]

For a positive integer $n$, let $p(n)$ denote the number of prime divisors of $n$, counting multiplicity (i.e. $p(12)=3$). A sequence $a_n$ is defined such that $a_0 = 2$ and for $n > 0$, $a_n = 8^{p(a_{n-1})} + 2$. Compute $$\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$