June 23 to 28
Tirana, Albania

ARTICLES | СТАТИИ

• RECENT ARTICLES | НАЈНОВИ СТАТИИ

SELECTED PROBLEMS | ОДБРАНИ ЗАДАЧИ

• Solve the system of equations $$\begin{cases} x+y+z &=-1 \\ x y z &=1 \\ \dfrac{x}{y^{2}}+\dfrac{y}{z^{2}}+\dfrac{z}{x^{2}} &=\dfrac{y^{2}}{x}+\dfrac{z^{2}}{y}+\dfrac{x^{2}}{z}\end{cases}$$
• Find the least value of the expression $$\frac{a^{3}}{1+b}+\frac{b^{3}}{1+a},$$ where $a$ and $b$ are positive numbers satisfying the condition $a b=1$.
• Let $A B C D$ a convex quadrilateral such that the diagonals $C A$ and $D B$ are the angled-bisectors of $\widehat{B C D}$ and $\widehat{A D C}$. Let $E$ be the point of intersection of $C A$ and $D B$. Prove that $$E C \cdot E D=E A \cdot E B+E A \cdot E D+E B \cdot E C$$ when and only when $\widehat{A E D}=45^{\circ}$.
• Let be given a semi-circle with diameter $A B$ and a fixed point $C$ on the segment $A B$ ($C$ distinct from $A$, $B$). $M$ is a point on the semi-circle. The line passing through $M$, perpendicular to $M C$, cuts the tangents of the semi-circle at $A$ and $B$ respectively at $E$ and $F$. Find the least value of the area of triangle $C E F$ when $M$ moves on the semi-circle.
• Prove that $$(a+b)(b+c)(c+a) \geq 2(1+a+b+c)$$ where $a, b, c$ are positive numbers satisfying the condition $a b c=1$.
• Solve the following equation with parameter $m$ $$x^{3}+5 x^{2}+(5 m+1) x+m^{2}=\left(x^{2}-x+1\right)^{2}$$
• Prove the inequalities $$\frac{2 x^{n}}{1+x^{n+1}} \leq\left(\frac{1+x}{2}\right)^{n-1} \leq \frac{x^{n}-1}{n(x-1)}$$ for positive number $x \neq 1$ and positive integer $n$.
• Prove that for arbitrary acute triangle $A B C$ $$\tan A+\tan B+\tan C+6(\sin A+\sin B+\sin C) \geq 12 \sqrt{3}.$$
• Let $S A B C$ be a tetrahedron with $S A=B C$, $S B=C A$, $S C=A B$. A plane passing through the incenter of triangle $A B C$ cuts the rays $S A$, $S B$, $S C$ respectively at $M$, $N$, $P$. Prove that $$S M+S N+S P \geq S A+S B+S C.$$
• Compare the fractions $$A=\frac{2003^{2003}+1}{2003^{2004}+1}, \quad B=\frac{2003^{2002}+1}{2003^{2003}+1}$$ Can you do it by distinct methods?
• Let $A B C$ be a triangle. Construct the segment $B D$ so that $\widehat{A B D}=60^{\circ}$, $B D=B A$ and the ray $B A$ lies between the rays $B C$, $B D$. Construct the segment $B E$ so that $\widehat{C B E}=60^{\circ}$, $B E=B C$ and the ray $B C$ lies between the rays $B A$, $B E$. Let $M$ be the midpoint of $D E$, $P$ be the point of intersection of the perpendicular bisectors of the segments $B A$ and $B D$. Calculate the angles of triangle $C M P$.
• Let $x_{1}$, $x_{2}$ be the solutions of the equation $x^{2}-4 x+1=0$. Prove that for every positive integer $n$, $x_{1}^{n}+x_{2}^{n}$ can be written as the sum of the squares of three consecutive integers.

PROBLEM OF THE WEEK | ЗАДАЧА НА НЕДЕЛАТА