June 23 to 28
Tirana, Albania

ARTICLES | СТАТИИ

• RECENT ARTICLES | НАЈНОВИ СТАТИИ

SELECTED PROBLEMS | ОДБРАНИ ЗАДАЧИ

• Find all positive solutions of the system of equations $$\begin{cases}\dfrac{3 x}{x+1}+\dfrac{4 y}{y+1}+\dfrac{2 z}{z+1} &=1 \\ 8^{9} \cdot x^{3} \cdot y^{4} \cdot z^{2} &=1\end{cases}$$
• Prove that $$\left(a^{3}+b^{3}+c^{3}\right)\left(\frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}\right) \geq \frac{3}{2}\left(\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c}\right)$$ where $a, b, c$ are positive real numbers.
• Prove that $$(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+\frac{3(a-b)(b-c)(c-a)}{a b c} \geq 9$$ where $a, b, c$ are the lengths of the sides of a triangle.
• The circle inscribed a triangle $A B C$ meets the sides $B C$, $C A$, $A B$ at $D$, $E$, $F$, respectively. Let $H$ be the projection of $D$ on $E F$. Show that $\widehat{B H D}=\widehat{C H D}$.
• Let $p_{1}, p_{2}, \ldots, p_{n}$ and $q_{1}, q_{2}, \ldots, q_{m}$ be $m+n$ different prime numbers such that $p_{i} \geq n+1$ $(i=1,2, \ldots, n)$, $q_{j} \geq m+1$ $(j=1,2, \ldots, m)$. Put $$P=p_{1} p_{2} \ldots p_{n} \quad \text{and} \quad Q=q_{1} q_{2} \ldots q_{m}.$$ Prove that the equation $P^{s} x+Q^{t} y=1$, where $s$, $t$ are fixed positive integers, have infinitely many integral solutions $\left(x_{0}, y_{0}\right)$ such that $\left(P, x_{0}\right)=1$ and $\left(Q, y_{0}\right)=1$.
• Solve the equation $$\left(16 \cos ^{4} x+3\right)^{4}=2048 \cos x-768.$$
• Prove that $$\frac{x_{1}^{m}}{x_{2}^{n}+x_{3}^{n}+\ldots+x_{n}^{n}}+\ldots+\frac{x_{n}^{m}}{x_{1}^{n}+x_{2}^{n}+\ldots+x_{n-1}^{n}} \geq \frac{n}{n-1} \sqrt[n]{\left(\frac{2003}{n}\right)^{m-n}}$$ where $m \geq n>1$ are integers and $x_{i}$ $(i=1,2, \ldots, n)$ are positive numbers such that $$x_{1}^{n}+x_{2}^{n}+\ldots+x_{n}^{n}\geq 2003 .$$
• Show that for a triangle $A B C$ with three acute angles, $$\frac{\cos A \cos B}{\sin 2 C}+\frac{\cos B \cos C}{\sin 2 A}+\frac{\cos C \cos A}{\sin 2 B} \geq \frac{\sqrt{3}}{2}$$
• Construct in the base plane $(A B C)$ of a tetrahedron $S A B C$ the triangles $A_{1} B C$, $B_{1} C A$, $C_{1} A B$ on the same side as $\triangle A B C$ and $A_{2} B C$, $B_{2} C A$, $C_{2} A B$ on the opposite side as $A B C$ such that $$\Delta A_{1} B C=\Delta A_{2} B C=\Delta S B C,\,\Delta B_{1} C A=\Delta B_{2} C A=\triangle S C A,\,\Delta C_{1} A B=\Delta C_{2} A B=\Delta S A B.$$ Prove that $\dfrac{R_{1}}{R_{2}}=\dfrac{r}{r_{S}}$, where $R_{1}$, $R_{2}$ are the radii of the circumcircles of $A_{1} B_{1} C_{1}$, $A_{2} B_{2} C_{2}$ and $r$, $r_{S}$ are the radii of the spheres inscribed and escribed the tetrahedron $S A B C$ opposite to $S$, respectively.
• Let $$A=\frac{1}{14}+\frac{1}{29}+\ldots+\frac{1}{n^{2}+(n+1)^{2}+(n+2)^{2}}+\ldots+\frac{1}{1877}.$$ Prove that $0,15<A<0,25$.
• The median $B M$ and the angled-bisector $C D$ of a triangle $A B C$ intersect at $K$ such that $K B=K C$. Calculate the measures of the angles $\widehat{A B C}$, $\widehat{A C B}$, knowing that the measure of $\widehat{B A C}$ is $105^{\circ}$.
• Prove that the number $A=2^{n}+6^{n}+8^{n}+9^{n}$ ($n$ is a positive integer) is divisible by $5$ when and only when $n$ is not divisible by $4$.

PROBLEM OF THE WEEK | ЗАДАЧА НА НЕДЕЛАТА