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 \(DYX\).

RMO 2018 Solutions

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Problem 2

Problem 4

Problem 4 Part 2

Problem 5

Problem 6