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# RMO Resource Center

RMO is the second step in Math Olympiad in India. Past papers, sequential hints and training resources.

# PAST PAPERS

Past Papers of RMO (Regional Math Olympiad India)

## Regional Math Olympiad (RMO) 2016 Telengana Region

1. Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre of $\triangle ABC$. Suppose $AI$ is extended to meet $BC$ at $F$ . The perpendicular on $AI$ at $I$ is extended to meet $AC$ at $E$ . Prove that $IE = IF$. Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre of $\triangle ABC$. Suppose $AI$ is extended to meet $BC$ at $F$ . The perpendicular on $AI$ at $I$ is extended to meet $AC$ at $E$ . Prove that $IE = IF$.
2. Let $a,b,c$ be positive real numbers such that $\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1$.Prove that $abc\leq\frac{1}{8}$.
3. For any natural number $n$, expressed in base $10$, let $S(n)E$ denote the sum of all digits of $n$. Find all positive integers $n$ such that $n^3$ = $8Sn^3$+$6Sn(n+1)$.
4. Find all $6$ digit natural numbers, which consist of only the digits $1,2$ and $3$, in which $3$ occurs exactly twice and the number is divisible by $9$.
5. 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 $FK = BC$.
6. Show that the infinite arithmetic progression {$1,4,7,10 \cdots$} 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$$\frac{a_1} {b_1} \neq \frac {a_2}{b_2}$$

### Regional Math Olympiad (RMO) 2016 Bengal Region

1. 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 $ABC$ 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  $\angle A=90^{\circ}$.
2. Let $a,b,c$ be three distinct positive real numbers such that $abc=1$. Prove that $$\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-c)(b-a)}+\frac{c^3}{(c-a)(c-b)} \geq 3$$
3. Let $a,b,c,d,e,d,e,f$ be positive integers such that $\frac a b <$; $\frac c d <$; $\frac e f$. Suppose $af-be=-1$. Show that $d \geq b+f$.
4. 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 $n$?
5. 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  $BA=BC$.
6. (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.

### Regional Math Olympiad (RMO) 2016 Maharashtra Region

1. Find distinct positive integers $n_1<n_2<\cdots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \cdots \times n_7$ is divisible by $2016$. Find distinct positive integers $n_1<n_2<\cdots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \cdots \times n_7$ is divisible by $2016$.
2. At an international event there are $100$ countries participating, each with its own flag. There are $10$ distinct flagpoles at the stadium, labelled $1,2,...,10$ in a row. In how many ways can all the $100$ flags be hoisted on these $10$ flagpoles, such that for each $i$ from $1$ to $10$, the flagpole $i$ has at least $i$ flags? (Note that the vertical order of the flagpoles on each flag is important)
3. Find all integers $k$ such that all roots of the following polynomial are also integers:$$f(x)=x^3-(k-3)x^2-11x+(4k-8)$$.
4. Let $\triangle ABC$ be scalene, with $BC$ as the largest side. Let $D$ be the foot of the perpendicular from $A$ on side $BC$. Let points (K,L) be chosen on the lines $AB$ and $AC$ respectively, such that $D$ is the midpoint of segment $KL$. Prove that the points $B,K,C,L$ are concyclic if and only if $\angle BAC=90^{\circ}$.
5. Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that$$(x^3+2y)(y^3+2z)(z^3+2x) \geq 27.$$
6. $ABC$ is an equilateral triangle with side length $11$ units. Consider the points $P_1,P_2, \cdots, P_10$ dividing segment $BC$ into $11$ parts of unit length. Similarly, define $Q_1, Q_2, \cdots, Q_10$ for the side $CA$ and $R_1,R_2,\cdots, R_10$ for the side $AB$. Find the number of triples ($i,j,k$) with $i,j,k$ in {$1,2,\cdots,10$} such that the centroids of $\triangle ABC$ and $P_iQ_jR_k$ coincide.

### Regional Math Olympiad (RMO) 2016 Mumbai Region

1. 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 $CA=b$.
2. Let $a,b,c$ be positive real numbers such that$$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$Prove that $abc \leq \frac{1}{8}$.
3. For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all natural numbers $n$ such that $n=2S(n)^2$.
4. Find the number of all 6-digits numbers having exactly three odd and three even digits.