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 $$DYX$$.