When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

Define the sequence of integers $\langle a_n\rangle$ as; \[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\] Prove that $a_{2012}-2010$ is divisible by $2011.$

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]

If $a$, $b$, $c$ are rational, show that \[ \frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}+\frac{1}{(a-b)^2} \] is the square of a rational.

$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$

Let $P,Q$ be two monic polynomials with complex coefficients such that $P(P(x))=Q(Q(x))$ for all $x$. Prove that $P=Q$. [i]Marius Cavachi[/i]

In a class, the total numbers of boys and girls are in the ratio $4 : 3$. On one day it was found that $8$ boys and $14$ girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?

[b](a)[/b] Sketch the diagram of the function $f$ if \[f(x)=4x(1-|x|) , \quad |x| \leq 1.\] [b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$ [b](c)[/b] Let $g$ be a function such that \[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\] Is the function $g$ continuous in the point $x=0 \ ?$ [b](d)[/b] Sketch the diagram of $g.$

For values of $ x$ less than $ 1$ but greater than $ \minus{}4$, the expression \[ \frac{x^2 \minus{} 2x \plus{} 2}{2x \minus{} 2} \] has: $ \textbf{(A)}\ \text{no maximum or minimum value}\qquad \\ \textbf{(B)}\ \text{a minimum value of }{\plus{}1}\qquad \\ \textbf{(C)}\ \text{a maximum value of }{\plus{}1}\qquad \\ \textbf{(D)}\ \text{a minimum value of }{\minus{}1}\qquad \\ \textbf{(E)}\ \text{a maximum value of }{\minus{}1}$

For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$. Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$. [i]Proposed by Georgia[/i]

Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$

Some numbers are written along the ring. If inequality $(a-d)(b-c) < 0$ is held for the four arbitrary numbers in sequence $a,b,c,d$, you have to change the numbers $b$ and $c$ places. Prove that you will have to do this operation finite number of times.

Solve in real numbers the equation $$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$

For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.

Let be a finite field $ K. $ Say that two polynoms $ f,g $ from $ K[X] $ are [i]neighbours,[/i] if the have the same degree and they differ by exactly one coefficient. [b]a)[/b] Show that all the neighbours of $ 1+X^2 $ from $ \mathbb{Z}_3[X] $ are reducible in $ \mathbb{Z}_3[X] . $ [b]b)[/b] If $ |K|\ge 4, $ show that any polynomial of degree $ |K|-1 $ from $ K[X] $ has a neighbour from $ K[X] $ that is reducible in $ K[X] , $ and also has a neighbour that doesn´t have any root in $ K. $

Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.

The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

Aidée draws ten squares of different sizes. The diagonal of the first square measures $1$ cm, the diagonal of the second measures $2$ cm, the diagonal of the third measures $3$ cm, and so on until the diagonal of the tenth square measures $10$ cm. How much are the areas of the ten squares?

Given an infinite sequence of numbers $a_1, a_2, a_3,...$ . For each positive integer $k$ there exists a positive integer $t = t(k)$ such that $a_k = a_{k+t} = a_{k+2t} =...$. Is this sequence necessarily periodic? That is, does a positive integer $T$ exist such that $a_k = a_{k+T}$ for each positive integer k?

Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Denote $$s=\sum_{k=1}^na_k\text{ and }s'=\sum_{k=1}^na_k^{1-\frac1k}.$$ (a) Let $\lambda>1$ be a real number. Show that $s'<\lambda s+\frac\lambda{\lambda-1}$. (b) Deduce that $\sqrt{s'}<\sqrt s+1$.

Find all postitive integers n such that $$\left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2$$ where $\lfloor x \rfloor$ represents the largest integer less than the real number $x$.

Let $\{a_i\}_{i=0}^\infty$ be a sequence of real numbers such that \[\sum_{n=1}^\infty\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\cdots\] for all $|x|<1$. Find $a_{1000}$. [i]Proposed by David Altizio[/i]

Does there exist an operation $*$ on the set of all integers such that the following conditions hold simultaneously: $(1)$ for all integers $x,y,z$, $(x*y)*z=x*(y*z)$; $(2)$ for all integers $x$ and $y$, $x*x*y=y*x*x=y$?

Find the function $f(x)$ such that : \[f(x)=\cos x+\int_0^{2\pi} f(y)\sin (x-y)\ dy\]