Consider a positive integer $n=\overline{a_1a_2...a_k},k\ge 2$.A trunk of $n$ is a number of the form $\overline{a_1a_2...a_t},1\le t\le k-1$.(For example,the number $23$ is a trunk of $2351$.) By $T(n)$ we denote the sum of all trunk of $n$ and let $S(n)=a_1+a_2+...+a_k$.Prove that $n=S(n)+9\cdot T(n)$.

Consider the set $A=\{1,2,...,120\}$ and $M$ a subset of $A$ such that $|M|=30$.Prove that there are $5$ different subsets of $M$,each of them having two elements,such that the absolute value of the difference of the elements of each subset is the same.

Solve in N $a^2 = 2^b3^c + 1$.

Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$, and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$. Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ $, show that the quadrilateral $ABCD$ is a square.

A lantern needs exactly $2$ charged batteries in order to work.We have available $n$ charged batteries and $n$ uncharged batteries,$n\ge 4$(all batteries look the same). A try consists in introducing two batteries in the lantern and verifying if the lantern works.Prove that we can find a pair of charged batteries in at most $n+2$ tries.

Let $ABCD$ be a cyclic quadrangle, let the diagonals $AC$ and $BD$ cross at $O$, and let $I$ and $J$ be the incentres of the triangles $ABC$ and $ABD$, respectively. The line $IJ$ crosses the segments $OA$ and $OB$ at $M$ and $N$, respectively. Prove that the triangle $OMN$ is isosceles.

Show that the edges of a connected simple (no loops and no multiple edges) finite graph can be oriented so that the number of edges leaving each vertex is even if and only if the total number of edges is even

Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime.

Given an integer $n \ge 2$ ,determine the numbers that written in the form $a_1$$a_2$$+$$a_2$$a_3$$+$$...$$a_{k-1}$$a_k$ , where $k$ is an integer greater than or equal to 2, and $a_1$ ,... $a_k$ are positive integers with sum $n$.