RMO 2018

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 $\gcd(AB,AC)>1$ .

Let $n$ be a natural number. Find all real numbers $x$ satisfying the equation $$\sum^n_{k=1}\frac{kx^k}{1+x^{2k}}=\frac{n(n+1)}4.$$

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 of the form $r=\overline{abcdefgh}$ having period $8$ , find the sum of all elements in $S$ .

Let $E$ denote the set of $25$ points $(m,n)$ in the $\text{xy}$ -plane, where $m,n$ are natural numbers, $1\leq m\leq5,1\leq n\leq5$ . 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 the axes and having at least three points of the same colour.)

Find all natural numbers $n$ such that $1+[\sqrt{2n}]~$ divides $2n$ .

( For any real number $x$ , $[x]$ denotes the largest integer not exceeding $x$ . )

Let $ABC$ be an acute angled triangle and let $D$ be an interior point of the segment $BC$ . Let the circumcircle of $ACD$ intersect $AB$ at $E$ ( $E$ between $A$ and $B$ ) and let circumcircle of $ABD$ intersect $AC$ at $F$ ( $F$ between $A$ and $C$ ). Let $O$ be the circumcenter of $AEF$ . Prove that $OD$ bisects $\angle EDF$ .

Find the set of all real values of $a$ for which the real polynomial equation $P(x)=x^2-2ax+b=0$ has real roots, given that $P(0)\cdot P(1)\cdot P(2)\neq 0$ and $P(0),P(1),P(2)$ form a geometric progression.

Suppose $100$ points in the plane are coloured using two colours, red and white such that each red point is the centre of circle passing through at least three white points. What is the least possible number of white points?

In a cyclic quadrilateral $ABCD$ with circumcenter $O$ , the diagonals $AC$ and $BD$ intersect at $X$ . Let the circumcircles of triangles $AXD$ and $BXC$ intersect at $Y$ . Let the circumcircles of triangles $AXB$ and $CXD$ intersect at $Z$ . If $O$ lies inside $ABCD$ and if the points $O,X,Y,Z$ are all distinct, prove that $O,X,Y,Z$ lie on a circle.

Define a sequence $\{a_n\}_{n\geq 1}$ of real numbers by $$a_1=2, a_{n+1} = \frac{a_n^2+1}{2}, \text{ for } n\geq 1.$$ Prove that $$\sum_{j=1}^{N} \frac{1}{a_j + 1} < 1$$ for every natural number $N$ .