RMO 2016

Let $ABC$ be a right-angled triangle with $\angle B=90^{\circ}$ . Let $I$ be the incenter of $ABC$ . Draw a line perpendicular to $AI$ at $I$ . Let it intersect the line $CB$ at $D$ . Prove that $CI$ is perpendicular to $AD$ and prove that $ID=\sqrt{b(b-a)}$ where $BC=a$ and $CA=b$ .

Find the number of all 6-digits numbers having exactly three odd and three even digits.

Let $ABC$ be a triangle with centroid $G$ . Let the circumcircle of triangle $AGB$ intersect the line $BC$ in $X$ different from $B$ ; and the circucircle of triangle $AGC$ intersect the line $BC$ in $Y$ different from $C$ . Prove that $G$ is the centroid of triangle $AXY$ .

Let $(a_1,a_2,\dots)$ be a strictly increasing sequence of positive integers in arithmetic progression. Prove that there is an infinite sub-sequence of the given sequence whose terms are in a geometric progression.

Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$ . Let $AD$ be the bisector of angle $A$ with $D$ on $BC$ . Let the circumcircle of triangle $ACD$ intersect $AB$ again at $E$ ; and let the circumcircle of triangle $ABD$ intersect $AC$ again at $F$ . Let $K$ be the reflection of $E$ in the line $BC$ . Prove that $FK = BC$ .

Show that the infinite arithmetic progression $\{1,4,7,10 \ldots\}$ has infinitely many 3 -term sub sequences in harmonic progression such that for any two such triples $\{a_1, a_2 , a_3 \}$ and $\{b_1, b_2 ,b_3\}$ in harmonic progression , one has $$\frac{a_1} {b_1} \ne \frac {a_2}{b_2}$$ .

Let $ABC$ be a triangle and $D$ be the mid-point of $BC$. Suppose the angle bisector of $\angle ADC$ is tangent to the circumcircle of triangle $ABD$ at $D$. Prove that $\angle A=90^{\circ}$.

Let $a,b,c$ be three distinct positive real numbers such that $abc=1$. Prove that $$\dfrac{a^3}{(a-b)(a-c)}+\dfrac{b^3}{(b-c)(b-a)}+\dfrac{c^3}{(c-a)(c-b)} \ge 3$$

Let $a,b,c,d,e,d,e,f$ be positive integers such that $\dfrac a b < \dfrac c d < \dfrac e f$. Suppose $af-be=-1$. Show that $d \geq b+f$.

There are $100$ countries participating in an olympiad. Suppose $n$ is a positive integers such that each of the $100$ countries is willing to communicate in exactly $n$ languages. If each set of $20$ countries can communicate in exactly one common language, and no language is common to all $100$ countries, what is the minimum possible value of $n$?

Let $ABC$ be a right-angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre if $ABC$. Extend $AI$ and $CI$; let them intersect $BC$ in $D$ and $AB$ in $E$ respectively. Draw a line perpendicular to $AI$ at $I$ to meet $AC$ in $J$, draw a line perpendicular to $CI$ at $I$ to meet $AC$ at $K$. Suppose $DJ=EK$. Prove that $BA=BC$.

(a). Given any natural number $N$, prove that there exists a strictly increasing sequence of $N$ positive integers in harmonic progression.

(b). Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.

Let $ABC$ be an isosceles triangle with $AB=AC.$ Let $\Gamma$ be its circumcircle and let $O$ be the centre of $\Gamma$ . let $CO$ meet $\Gamma$ in $D .$ Draw a line parallel to $AC$ thrugh $D.$ Let it intersect $AB$ at $E.$ Suppose $AE : EB=2:1$ .Prove that $ABC$ is an equilateral triangle.

The precent ages in years of two brothers $A$ and $B$ ,and their father $C$ are three distinct positive integers $a ,b$ and $c$ respectively .Suppose $\frac{b-1}{a-1}$ and $\frac{b+1}{a+1}$ are two consecutive integers , and $\frac{c-1}{b-1}$ and $\frac{c+1}{b+1}$ are two consecutive integers . If $a+b+c\le 150$ , determine $a,b$ and $c$ .

A box contains answer $4032$ scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of the scripts and keep the other outside. If there is one script with even score and the other with odd score, we put back the script with the odd score and keep the other script outside. After following this procedure a number of times, there are 3 scripts left among which there is at least one script each with odd and even scores. Find, with proof, the number of scripts with odd scores among the three left.

Let $ABC$ be a triangle , $AD$ an altitude and $AE$ a median . Assume $B,D,E,C$ lie in that order on the line $BC$ . Suppose the incentre of triangle $ABE$ lies on $AD$ and he incentre of triangle $ADC$ lies on $AE$ . Find ,with proof ,the angles of triangle $ABC$ .