For a positive integer $a$, $a'$ is the integer obtained by the following method: the decimal writing of $a'$ is the inverse of the decimal writing of $a$ (the decimal writing of $a'$ can begin by zeros, but not the one of $a$); for instance if $a=2370$, $a'=0732$, that is $732$. Let $a_{1}$ be a positive integer, and $(a_{n})_{n \geq 1}$ the sequence defined by $a_{1}$ and the following formula for $n \geq 1$: \[a_{n+1}=a_{n}+a'_{n}. \] Can $a_{7}$ be prime?

Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$. Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]

Let $A,B,C,D$ be four distinct points on a circle such that the lines $(AC)$ and $(BD)$ intersect at $E$, the lines $(AD)$ and $(BC)$ intersect at $F$ and such that $(AB)$ and $(CD)$ are not parallel. Prove that $C,D,E,F$ are on the same circle if, and only if, $(EF)\bot(AB)$.

Do there exist $5$ points in the space, such that for all $n\in\{1,2,\ldots,10\}$ there exist two of them at distance between them $n$?

Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

Click for solution I will write the solutions with more details. Using the above subtitution we obtain that $g(x+g(y))=g(x)+y$ (1). For $x=0$ the last relation becomes $g(g(y))=g(0)+y$ From this we find that $g$ is surjective ,so also f is surjective so there exist u such that $f(u)=0$ .Setting in the first relation where x,y the u we obtain $f(f(u))=2f(u)$ so $f(0)=0$ so $g(g(y))=y$ (2) .Now put in (1) where y the g(y) to have $g(x+y)=g(x)+g(y)$ (3) due to (2) and (3) holds for all $x,y\in Z$ due to the surjectivity of g. So $g(x)=cx$ where $c=g(1)$ so $f(x)=x+xc'$ . But subtituting this in the first relation gives c'=-1 or c'=1 so the fuctions are $f(x)=2x$ and $f(x)=0$

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.