In terms of $n\ge2$, find the largest constant $c$ such that for all nonnegative $a_1,a_2,\ldots,a_n$ satisfying $a_1+a_2+\cdots+a_n=n$, the following inequality holds: \[\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.\] [i]Calvin Deng.[/i]

Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings: (1) $\sum_{v\in V} f(v)=|E|$; (2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$. Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.

Let ${\bf R}$ denote the set of all real numbers. Find all functions $f$ from ${\bf R}$ to ${\bf R}$ satisfying: (i) there are only finitely many $s$ in ${\bf R}$ such that $f(s)=0$, and (ii) $f(x^4+y)=x^3f(x)+f(f(y))$ for all $x,y$ in ${\bf R}$.

In the following, a [i]word[/i] will mean a finite sequence of letters "$a$" and "$b$". The [i]length[/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word. A word $w$ of length $\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\ell}$ in this order is called a [i]palindrome[/i] if and only if $x_j=x_{\ell+1-j}$ holds for every $j$ such that $1\leq j\leq\ell$. For instance, $baaab$ is a palindrome; so is the empty word. For two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$. Let $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.

Let $k$ and $l$ be two given positive integers and $a_{ij}(1 \leq i \leq k, 1 \leq j \leq l)$ be $kl$ positive integers. Show that if $q \geq p > 0$, then \[(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.\]

Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?

In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off). Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.

Show that $ \frac{1}{2^2} +\frac{1}{3^2} +...+\frac{1}{n^2} <\frac{2}{3}$ for all $n \ge 2 $.

The set of positive nonzero real numbers are partitioned into three mutually disjoint non-empty subsets $(A\cup B\cup C)$. a) show that there exists a triangle of side-lengths $a,b,c$, such that $a\in A, b\in B, c\in C$. b) does it always happen that there exists a right triangle with the above property ?

Let be some rational numbers with the property that their sum, as well as the product of any two of them is integer. Prove that all these are integers.

Let $k$ be a fixed positive integer. The sequence $\{a_{n}\}_{n\ge1}$ is defined by \[a_{1}=k+1, a_{n+1}=a_{n}^{2}-ka_{n}+k.\] Show that if $m \neq n$, then the numbers $a_{m}$ and $a_{n}$ are relatively prime.

The positive sequence $\{a_n\}$ satisfies:$a_1=1+\sqrt 2$ and $(a_n-a_{n-1})(a_n+a_{n-1}-2\sqrt n)=2(n\geq 2).$ (1)Find the general formula of $\{a_n\}$; (2)Find the set of all the positive integers $n$ so that $\lfloor a_n\rfloor=2022$.

Let $\{x_{n}\}_{n\ge0}$ and $\{y_{n}\}_{n\ge0}$ be two sequences defined recursively as follows \[x_{0}=1, \; x_{1}=4, \; x_{n+2}=3 x_{n+1}-x_{n},\] \[y_{0}=1, \; y_{1}=2, \; y_{n+2}=3 y_{n+1}-y_{n}.\] [list=a][*] Prove that ${x_{n}}^{2}-5{y_{n}}^{2}+4=0$ for all non-negative integers. [*] Suppose that $a$, $b$ are two positive integers such that $a^{2}-5b^{2}+4=0$. Prove that there exists a non-negative integer $k$ such that $a=x_{k}$ and $b=y_{k}$.[/list]

Let $P_1$, $P_2$, $\ldots$, $P_{1993} = P_0$ be distinct points in the $xy$-plane with the following properties: (i) both coordinates of $P_i$ are integers, for $i = 1, 2, \ldots, 1993$; (ii) there is no point other than $P_i$ and $P_{i+1}$ on the line segment joining $P_i$ with $P_{i+1}$ whose coordinates are both integers, for $i = 0, 1, \ldots, 1992$. Prove that for some $i$, $0 \leq i \leq 1992$, there exists a point $Q$ with coordinates $(q_x, q_y)$ on the line segment joining $P_i$ with $P_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.

During a break, $n$ children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of $n$ for which eventually, perhaps after many rounds, all children will have at least one candy each.

Suppose $\alpha_i > 0, \beta_i > 0$ for $1 \leq i \leq n, n > 1$ and that \[ \sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi. \] Prove that \[ \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i). \]

Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]

The sequence $\{x_{n}\}$ is defined by \[x_{0}\in [0, 1], \; x_{n+1}=1-\vert 1-2 x_{n}\vert.\] Prove that the sequence is periodic if and only if $x_{0}$ is irrational.

There is a broken computer such that only three primitive data $c$, $1$ and $-1$ are reserved. Only allowed operation may take $u$ and $v$ and output $u \cdot v + v.$ At the beginning, $u,v \in \{c, 1, -1\}.$ After then, it can also take the value of the previous step (only one step back) besides $\{c, 1, -1\}$. Prove that for any polynomial $P_{n}(x) = a_0 \cdot x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ with integer coefficients, the value of $P_n(c)$ can be computed using this computer after only finite operation.

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that \[\frac{a^{2}+b^{2}}{ab+1}\] is the square of an integer.

Let $n > 3$ be a natural number, and let $x_1$, $x_2$, ..., $x_n$ be $n$ positive real numbers whose product is $1$. Prove the inequality \[ \frac {1}{1 + x_1 + x_1\cdot x_2} + \frac {1}{1 + x_2 + x_2\cdot x_3} + ... + \frac {1}{1 + x_n + x_n\cdot x_1} > 1. \]

Let $n$ be a positive integer. Show that \begin{align*}&\quad\,\,\frac{1}{\binom{n}{1}}+\frac{1}{2\binom{n}{2}}+\frac{1}{3\binom{n}{3}}+\cdots+\frac{1}{n\binom{n}{n}}\\&=\frac{1}{2^{n-1}}+\frac{1}{2\cdot2^{n-2}}+\frac{1}{3\cdot2^{n-3}}+\cdots+\frac{1}{n\cdot2^0}.\end{align*}

Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$. Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$

The sequences $(a_n), (b_n),$ and $(c_n)$ are defined by $a_0 = 1, b_0 = 0, c_0 = 0,$ and \[a_n = a_{n-1} + \frac{c_{n-1}}{n}, b_n = b_{n-1} +\frac{a_{n-1}}{n}, c_n = c_{n-1} +\frac{b_{n-1}}{n}\] for all $n \geq1$. Prove that \[\left|a_n -\frac{n + 1}{3}\right|<\frac{2}{\sqrt{3n}}\] for all $n \geq 1$.

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.