BALKAN-SL 2007

Find the minimum and maximum value of the function \begin{align*} f(x,y)=ax^2+cy^2 \end{align*} Under the condition $ax^2-bxy+cy^2=d$ , where $a,b,c,d$ are positive real numbers such that $b^2 -4ac <0$

Find all values of $a \in \mathbb{R}$ for which the polynomial \begin{align*} f(x)=x^4-2x^3 + \left(5-6a^2 \right)x^2 + \left(2a^2-4 \right)x + \left(a^2 -2 \right)^2 \end{align*} has exactly three real roots.

Show that the sequence \begin{align*} a_n = \left \lfloor \left( \sqrt[3]{n-2} + \sqrt[3]{n+3} \right)^3 \right \rfloor \end{align*} contains infinitely many terms of the form $a_n^{a_n}$

Let $c>2$ and $a_0,a_1, \ldots$ be a sequence of real numbers such that \begin{align*} a_n = a_{n-1}^2 - a_{n-1} < \frac{1}{\sqrt{cn}} \end{align*} for any $n$ $\in$ $\mathbb{N}$ . Prove, $a_1=0$

Let $\omega$ be a circle with center $O$ and let $A$ be a point outside $\omega$ . The tangents from $A$ touch $\omega$ at points $B$ , and $C$ . Let $D$ be the point at which the line $AO$ intersects the circle such that $O$ is between $A$ and $D$ . Denote by $X$ the orthogonal projection of $B$ onto $CD$ , by $Y$ the midpoint of the segment $BX$ and by $Z$ the second point of intersection of the line $DY$ with $\omega$ . Prove that $ZA$ and $ZC$ are perpendicular to each other.

Points $M,N$ and $P$ on the sides $BC, CA$ and $AB$ of $\vartriangle ABC$ are such that $\vartriangle MNP$ is acute. Denote by $h$ and $H$ the lengths of the shortest altitude of $\vartriangle ABC$ and the longest altitude of $\vartriangle MNP$ . Prove that $h \le 2H$ .

Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$ , such that for any prime $p$ , $p > N$ , the $p$ -th term of the progression is also prime.

Let $p \geq 5$ be a prime and let \begin{align*} (p-1)^p +1 = \prod {i=1}^n q_i^{\beta_i} \end{align*} where $q_i$ are primes. Prove, \begin{align*} \sum{i=1}^n q_i \beta_i >p^2 \end{align*}