Let $ f_m: \mathbb R \to \mathbb R$, $ f_m(x)=(m^2+m+1)x^2-2(m^2+1)x+m^2-m+1,$ where $ m \in \mathbb R$. 1) Find the fixed common point of all this parabolas. 2) Find $ m$ such that the distance from that fixed point to $ Oy$ is minimal.

Find $ f(x): (0,+\infty) \to \mathbb R$ such that \[ f(x)\cdot f(y) + f(\frac{2008}{x})\cdot f(\frac{2008}{y})=2f(x\cdot y)\] and $ f(2008)=1$ for $ \forall x \in (0,+\infty)$.

From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$. The point $ M \in (AE$ is such that $ M$ external to $ ABC$, $ \angle AMB = 20 ^\circ$ and $ \angle AMC = 30 ^ \circ$. What is the measure of the angle $ \angle MAB$?

Let $ n$ be a positive integer. Find all $ x_1,x_2,\ldots,x_n$ that satisfy the relation: \[ \sqrt{x_1-1}+2\cdot \sqrt{x_2-4}+3\cdot \sqrt{x_3-9}+\cdots+n\cdot\sqrt{x_n-n^2}=\frac{1}{2}(x_1+x_2+x_3+\cdots+x_n).\]

Determine the polynomial P(X) satisfying simoultaneously the conditions: a) The remainder obtained when dividing P(X) to the polynomial X^3 −2 is equal to the fourth power of quotient. b) P(−2) + P(2) = −34.

find x and y in R $\begin{array}{l} (\frac{1}{{\sqrt[3]{x}}} + \frac{1}{{\sqrt[3]{y}}})(\frac{1}{{\sqrt[3]{x}}} + 1)(\frac{1}{{\sqrt[3]{y}}} + 1) = 18 \\ \frac{1}{x} + \frac{1}{y} = 9 \\ \end{array}$

Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.

Let $ (a_{n})_{n\ge 1} $ be a sequence such that: $ a_{1}=1; a_{n+1}=\frac{n}{a_{n}+1}.$ Find $ [a_{2008}] $

Consider the equation $ x^4 - 4x^3 + 4x^2 + ax + b = 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a + b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 + x_2 = 2x_1x_2$.

Find the exact value of $ E=\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.

In the usual coordinate system $ xOy$ a line $ d$ intersect the circles $ C_1:$ $ (x+1)^2+y^2=1$ and $ C_2:$ $ (x-2)^2+y^2=4$ in the points $ A,B,C$ and $ D$ (in this order). It is known that $ A\left(-\frac32,\frac{\sqrt3}2\right)$ and $ \angle{BOC}=60^{\circ}$. All the $ Oy$ coordinates of these $ 4$ points are positive. Find the slope of $ d$.

Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p=\displaystyle\frac{\binom p0}{2\cdot 4}-\frac{\binom p1}{3\cdot5}+\frac{\binom p2}{4\cdot6}-\ldots+(-1)^p\cdot\frac{\binom pp}{(p+2)(p+4)}$. Find $ \lim_{n\to\infty}(a_0+a_1+\ldots+a_n)$.

Find the least positive integer $ n$ so that the polynomial $ P(X)=\sqrt3\cdot X^{n+1}-X^n-1$ has at least one root of modulus $ 1$.

Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n=\frac1{\sqrt{n^2+8n-1}}+\frac1{\sqrt{n^2+16n-1}}+\frac1{\sqrt{n^2+24n-1}}+\ldots+\frac1{\sqrt{9n^2-1}}$.

Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC = 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} = 3$. Find the locus of the point $ A$ so that $ F\in BC$.

Evaluate $ \displaystyle I = \int_0^{\frac\pi4}\left(\sin^62x + \cos^62x\right)\cdot \ln(1 + \tan x)\text{d}x$.