Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number

An $n\times n$ square table is divided into $n^2$ unit cells. Some unit segments of the obtained grid (i.e. the side of any unit cell) is colored black so that any unit cell of the given square has exactly one black side. Find a) the smallest b) the greatest possible number of black unit segments.

Let $\triangle ABC$ be an acute triangle. Let us denote the foot of the altitudes from the vertices $A, B$ and $C$ to the opposite sides by $D, E$ and $F,$ respectively, and the intersection point of the altitudes of the triangle $ABC$ by $H.$ Let $P$ be the intersection of the line $BE$ and the segment $DF.$ A straight line passing through $P$ and perpendicular to $BC$ intersects $AB$ at $Q.$ Let $N$ be the intersection of the segment $EQ$ with the perpendicular drawn from $A.$ Prove that $N$ is the midpoint of segment $AH.$

a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

Find all nonnegative integers $(x,y,z,u)$ with satisfy the following equation: $2^x + 3^y + 5^z = 7^u.$

In the beginning, there are $100$ cards on the table, and each card has a positive integer written on it. An odd number is written on exactly $43$ cards. Every minute, the following operation is performed: for all possible sets of $3$ cards on the table, the product of the numbers on these three cards is calculated, all the obtained results are summed, and this sum is written on a new card and placed on the table. A day later, it turns out that there is a card on the table, the number written on this card is divisible by $2^{2018}.$ Prove that one hour after the start of the process, there was a card on the table that the number written on that card is divisible by $2^{2018}.$