Determine all natural numbers $n$, such that it's possible to insert one digit at the right side of $n$ to obtain $13n$.

Points $P$ and $Q$ lie on the sides $AB$, $BC$ of the triangle $ABC$, such that $AC=CP =PQ=QB$ and $A \neq P$ and $C \neq Q$. If $\sphericalangle ACB = 104^{\circ}$, determine the measures of all angles of the triangle $ABC$.

Determine all triples $(x, y, z)$ of non-zero numbers such that \[ xy(x + y) = yz(y + z) = zx(z + x). \]

Let $ABCD$ be the rectangle. Points $E$, $F$ lies on the sides $BC$ and $CD$ respectively, such that $\sphericalangle EAF = 45^{\circ}$ and $BE = DF$. Prove that area of the triangle $AEF$ is equal to the sum of the areas of the triangles $ABE$ and $ADF$.

Let $a$, $b$, $c$ be the real numbers. It is true, that $a + b$, $b + c$ and $c + a$ are three consecutive integers, in a certain order, and the smallest of them is odd. Prove that the numbers $a$, $b$, $c$ are also consecutive integers in a certain order.

Let $ABCD$ be the parallelogram, such that angle at vertex $A$ is acute. Perpendicular bisector of the segment $AB$ intersects the segment $CD$ in the point $X$. Let $E$ be the intersection point of the diagonals of the parallelogram $ABCD$. Prove that $XE = \frac{1}{2}AD$.

There is the tournament for boys and girls. Every person played exactly one match with every other person, there were no draws. It turned out that every person had lost at least one game. Furthermore every boy lost different number of matches that every other boy. Prove that there is a girl, who won a match with at least one boy.

Let $ABC$ be such a triangle that $\sphericalangle BAC = 45^{\circ}$ and $ \sphericalangle ACB > 90^{\circ}.$ Show that $BC + (\sqrt{2} - 1)\cdot CA < AB.$