$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.

Prove that the polynomial $P(X) = (X^2-12X +11)^4+23$ can not be written as the product of three non-constant polynomials with integer coefficients. (Le Anh Vinh)

Let $i$ be a root of the equation $x^2+1=0$ and let $\omega$ be a root of the equation $x^2+x+1=0$ . Construct a polynomial $$f(x)=a_0+a_1x+\cdots+a_nx^n$$ where $a_0,a_1,\cdots,a_n$ are all integers such that $f(i+\omega)=0$ .

Let $a$ and $b$ be positive integers and $K=\sqrt{\frac{a^2+b^2}2}$, $A=\frac{a+b}2$. If $\frac KA$ is a positive integer, prove that $a=b$.

Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that: a/(1-x)+b/(1-y)=1 Prove that: ∛ay+∛bx≤1.

Determine the remainder when $(x^4-1)(x^2-1)$ is divided by $1+x+x^2$.

For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$

Solve the equation $$(x^2-x-1)^2-x^3=5$$

For $a,b,c$ positive real numbers such that $ab+bc+ca=3$, prove: $ \frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

How many distinct arrangements are possible for wearing five different rings in the five fingers of the right hand? (We can wear multiple rings in one finger) [list=1] [*] $\frac{10!}{5!}$ [*] $5^5$ [*] $\frac{9!}{4!}$ [*] None of these [/list]

$5.$ Let $P(x)$ be a non - zero polynomial with integer coefficients. If $P(n)$ is divisible by $n$ for each integer polynomial $n.$ What is the value of $P(0) ?$

Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds: $a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$ $b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$

For constants $a,\ b,\ c,\ d$ consider a process such that the point $(p,\ q)$ is mapped onto the point $(ap+bq,\ cp+dq)$. Note : $(a,\ b,\ c,\ d)\neq (1,\ 0,\ 0,\ 1)$. Let $k$ be non-zero constant. All points of the parabola $C: y=x^2-x+k$ are mapped onto $C$ by the process. (1) Find $a,\ b,\ c,\ d$. (2) Let $A'$ be the image of the point $A$ by the process. Find all values of $k$ and the coordinates of $A$ such that the tangent line of $C$ at $A$ and the tangent line of $C$ at $A'$ formed by the process are perpendicular at the origin.

Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as: $u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$ a) With $a=0$, prove that there exist a finite limit of $(u_n)$ and find that limit b) With $a \ge 0$, prove that there exist a finite limit of $(u_n)$

An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

Mummy-trolley huts are located on a straight line at points with coordinates $x_1, x_2,...., x_n$. In this village are going to build $3$ stores $A, B$ and $C$, of which will be brought every day to all Moomin-trolls chocolates, bread and water. For the delivery of chocolate, the store takes the distance from the store to the hut, raised to the square; for bread delivery , take the distance from the store to the hut; for water delivery take distance $1$, if the distance is greater than $1$ km, but do not take anything otherwise. a) Where to build each of the stores so that the total cost of all Moomin-trolls for delivery wasthe smallest? b) Where to place the TV tower, if the fee for each Moomin-troll is the maximum distance from the TV tower to the farthest hut from it? c) How will the answer change if the Moomin-troll huts are not located in a straight line, and on the plane? [hide=original wording] На прямiй розташованi хатинки Мумi-тролей в точках з координатами x1, x2, . . . , xn. В цьому селi бираються побудувати 3 магазина A, B та C, з яких будуть кожен день привозити всiм Мумi-тролям шоколадки, хлiб та воду. За доставку шоколадки мага- зин бере вiдстань вiд магазину до хатинки, пiднесену до квадрату; за доставку хлiба – вiдстань вiд магазину до хатинки; за доставку води беруть 1, якщо вiдстань бiльша 1 км, та нiчого не беруть в супротивному випадку. 1. Де побудувати кожний з магазинiв, щоб загальнi витрати всiх Мумi-тролей на доставку були найменшими? 2. Де розташувати телевежу, якщо плата для кожного Мумi-троля – максимальна вiдстань вiд телевежi до самої вiддаленої вiд неї хатинки? 3. Як змiниться вiдповiдь, якщо хатинки Мумi-тролей розташованi не на прямiй, а на площинi?[/hide]

$a)$ Prove that $2x^3-3x^2+1\geq 0,~(\forall)x\geq0.$ $b)$ Let $x,y,z\geq 0$ such that $\frac{2}{1+x^3}+\frac{2}{1+y^3}+\frac{2}{1+z^3}=3.$ Prove that $\frac{1-x}{1-x+x^2}+\frac{1-y}{1-y+y^2}+\frac{1-z}{1-z+z^2}\geq 0.$

Let $p > 1$ and let $a, b, c, d$ be positive numbers such that $$(a + b + c + d) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)= 16p^2.$$ Find all values of the ratio $ R =\frac{\max \{a, b, c, d\}}{\min \{a, b, c, d\}}$ (depending on the parameter $p$)

On a circular table sit $\displaystyle {n> 2}$ students. First, each student has just one candy. At each step, each student chooses one of the following actions: (A) Gives a candy to the student sitting on his left or to the student sitting on his right. (B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right. At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps. Find the number of legitimate distributions. (Two distributions are different if there is a student who has a different number of candy in each of these distributions.) （Forgive my poor English）

Let $x, y, z$ be nonnegative real numbers with $xy + yz + zx = 1$. Show that: $$\frac{4}{x + y + z} \le (x + y)(\sqrt3 z + 1).$$

Justify if continuity can be affirmed, denied or cannot be decided in the point$ x = 0$ of a real function $f(x)$ of real variable, in each of the three (independent) cases . a) It is known only that for all natural $n$: $f\left( \frac{1}{2n}\right)= 1$ and $f\left( \frac{1}{2n+1}\right)= -1$. b) It is known that for all nonnegative real $x$ is $f(x) = x^2$ and for negative real $x$ is $f(x) = 0$. c) It is only known that for all natural $n$ it is $f\left( \frac{1}{n}\right)= 1$.

Paca-Vaca decided to note every day a single quadratic polynomial of the form $x^2+ax+b$, where $a$ and $b$ are positive integers, less or equal than $100$. He follows the rule that the polynomial he writes must not have any common roots with the polynomials previously noted. What is the maximum amount of days Paca-Vaca can follow this plan?

Evaluate the following expression: $2013^2 + 2011^2 + … + 5^2 + 3^2 -2012^2 -2010^2-…-4^2-2^2$