 This post contains RMO 2018 solutions, problems, and discussions.

1. Let $ABC$ be a triangle with integer sides in which $AB < AC$. Let the tangent to the circumcircle of triangle $ABC$ at $A$ intersect the line $BC$ at $D$. Suppose $AD$ is also an integer. Prove that gcd$(AB,AC) >1$.
2. Let $n$ be a natural number. Find all real numbers $x$ satisfying the equation    $\displaystyle\sum_{k=1}^{n} \frac{kx^k}{1+x^{2k}}$$=\frac{n(n+1)}{4}$
3. For a rational number $r$, its period is the length of the smallest repeating block in its decimal expansion. For example, the number $r=0.123123123..$ has period $3$. If $S$ denotes the set of all rational numbers $r$ of the form $r=0.\overline{abcdefgh}$ having period $8$, find the sum of all elements of $S$.
4. Let $E$ denotes the set of $25$ points $(m,n)$ in the$xy$-plane, where $m$,$n$ are natural numbers, $1 \leq m \leq 5$, $1 \leq m \leq 5$. Suppose the points of $E$ are arbitrarily coloured using two colours, red and blue. Show that there always  exist  four points in the set $E$ of the form $(a,b)$, $(a+k,b)$, $(a+k,b+k)$, $(a,b+k)$ for some positive integer $k$ such that at least three of these four points have the same colour. ( That is there always  exist  four points in the set $E$ which form the vertices of a square with sides parallel to axes and having at least three points of the same colour.)
5. Find all natural numbers $n$ such that $1+[\sqrt{2n}]$ divides $2n$. (For any real number $x$, $[x]$ denotes the largest integer not exceeding x.)
6. Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $I$ be the incentre of triangle $ABC$, and let $D$,$E$,$F$ be the points at which its incircle touches the side $BC$,$CA$,$AB$ respectively. Let $BI$, $CI$ meet the line $EF$ at $Y$,$X$, respectively. Further assume that both $X$ and $Y$ are outside the triangle $ABC$. Prove that
1. $B$,$C$,$Y$,$X$ are concyclic; and
2. $I$ is also the incentre of triangle $DYX$.