RMO 2014

Let $a_1,a_2 \cdots a_{2n}$ be an arithmetic progression of positive real numbers with common difference $d$ . Let $(i)$ $\sum_{i=1}^{n}a_{2i-1}^2 =x$ $(ii)$ $\sum _{i=1}^{n}a_{2i}^2=y$ $(iii)$ $a_n+a_{n+1}=z$ Express $d$ in terms of $x,y,z,n$

Suppose for some positive integers $r$ and $s$ , $2^r$ is obtained by permuting the digits of $2^s$ in decimal expansion. Prove that $r=s$ .

In an acute-angled triangle $ABC, \angle ABC$ is the largest angle. The perpendicular bisectors of $BC$ and $BA$ intersect AC at $X$ and $Y$ respectively. Prove that circumcentre of triangle $ABC$ is incentre of triangle $BXY$ .

Let $x, y, z$ be positive real numbers. Prove that $\frac{y^2 + z^2}{x}+\frac{z^2 + x^2}{y}+\frac{x^2 + y^2}{z}\ge 2(x + y + z)$ .

Find all pairs of $(x, y)$ of positive integers such that $2x + 7y$ divides $7x + 2y$ .

Let $ABC$ be a triangle with $AB > AC$ . Let $P$ be a point on the line $AB$ beyond $A$ such that $AP +P C = AB$ . Let $M$ be the mid-point of $BC$ and let $Q$ be the point on the side $AB$ such that $CQ \perp AM$ . Prove that $BQ = 2AP.$

Suppose $n$ is odd and each square of an $n \times n$ grid is arbitrarily filled with either by $1$ or by $-1$ . Let $r_j$ and $c_k$ denote the product of all numbers in $j$ -th row and $k$ -th column respectively, $1 \le j, k \le n$ . Prove that $$\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0$$

Let $ABC$ be a triangle with $\angle ABC$ as the largest angle. Let $R$ be its circumcenter. Let the circumcircle of triangle $ARB$ cut $AC$ again at $X$ . Prove that $RX$ is perpendicular to $BC$ .

Let $ABC$ be a triangle and let $X$ be on $BC$ such that $AX=AB$ . let $AX$ meet circumcircle $\omega$ of triangle $ABC$ again at $D$ . prove that circumcentre of triangle $BDX$ lies on $\omega$ .

For any natural number, let $S(n)$ denote sum of digits of $n$ . Find the number of $3$ digit numbers for which $S(S(n)) = 2$ .

let $ABCD$ be a isosceles trapezium having an incircle with $AB$ parallel to $CD$ . let $CE$ be the perpendicular from $C$ on $AB$ prove that $CE^2 = AB. CD$

let $ABC$ be a right angled triangle with inradius $1$ find the minimum area of triangle $ABC$

let $ABC$ be a triangle and $I$ be its incentre. let the incircle of $ABC$ touch $BC$ at $D$ . let incircle of triangle $ABD$ touch $AB$ at $E$ and incircle of triangle $ACD$ touch $AC$ at $F$ . prove that $B,E,I,F$ are concyclic.

In the adjacent figure, can the numbers $1,2,3, 4,..., 18$ be placed, one on each line segment, such that the sum of the numbers on the three line segments meeting at each point is divisible by $3$ ?

Attachments:

Three positive real numbers $a,b,c$ are such that $a^2+5b^2+4c^2-4ab-4bc=0$ . Can $a,b,c$ be the lengths of te sides of a triangle? Justify your answer.

The roots of the equation $$x^3-3ax^2+bx+18c=0$$ form a non-constant arithmetic progression and the roots of the equation $$x^3+bx^2+x-c^3=0$$ form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$ .

Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$ .

A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at a point $(x,y)$ , she takes a step based on the following rules: (a) if $x+y$ is even she moves to either $(x+1,y)$ or $(x+1,y+1)$ ; (b) if $x+y$ is odd she moves to either $(x,y+1)$ or $(x+1,y+1)$ . How many distinct paths can she take to go from $(0,0)$ to $(8,8)$ given that she took exactly three steps to the right $((x,y)$ to $(x+1,y))$ ?

Let $a,b,c$ be positive real numbers such that $$\cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1.$$ Prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$ . When does equality hold?

Let $D,E,F$ be the points of contact of the incircle of an acute-angled triangle $ABC$ with $BC,CA,AB$ respectively. Let $I_1,I_2,I_3$ be the incentres of the triangles $AFE, BDF, CED$ , respectively. Prove that the lines $I_1D, I_2E, I_3F$ are concurrent.