 ## 08 Mar Indian National Mathematical Olympiad 2018 Question Paper

The Indian National Mathematical Olympiad (INMO) is a high school mathematical competition held annually in India since 1989. It is the second tier in the Indian team selection procedure for the International Mathematical Olympiad and is conducted by the Homi Bhabha Centre for Science Education (HBCSE) under the aegis of the National Board of Higher Mathematics (NBHM). The exam has six questions which have to be solved in four hours. The 33rd INMO was held on 21st January.

1. Let ${ABC}$ be a non-equilateral triangle with integer sides. Let ${D}$ and ${E}$ be the respectively the midpoints of ${BC}$ and ${CA}$; let ${G}$ be the centroid of triangle ${ABC}$. Suppose ${D}$, ${C}$, ${E}$, ${G}$ are concyclic. Find the least possible perimeter of triangle ${ABC}$.

2. For any natural number ${n}$ consider a ${1\times n}$ rectangular board made up of ${n}$ unit squares. This is covered by three types of tiles: ${1\times 1}$ red tile, ${1\times 1}$ green tile and ${1\times 2}$ blue domino. Let ${t_n}$ denote the number of ways of covering ${1\times n}$ rectangular board by these three types of tiles. Prove that ${t_n}$ divides ${t_{2n+1}}$.

3. Let ${\Gamma_1}$ and ${\Gamma_2}$ be two circles with respective centres ${O_1}$ and ${O_2}$ intersecting in two distinct points ${A}$ and ${B}$ such that ${\angle O_1AO_2}$ is an obtuse angle. Let the circumcircle of triangle ${O_1AO_2}$ intersect ${\Gamma_1}$ and ${\Gamma_2}$ respectively in points ${C}$ and ${D}$. Let the line ${CB}$ intersect ${\Gamma_2}$ in ${E}$; let the line ${DB}$ intersect ${\Gamma_1}$ in ${F}$ . Prove that the points ${C}$, ${D}$, ${E}$, ${F}$ are concyclic.

4. Find all polynomials with real coefficients ${P(x)}$ such that ${P(x^2+x+1)}$ divides ${P(x^3-1)}$.

5. There are ${n \geq 3}$ girls in a class sitting around a circular table, each having some apples with her. Every time the teacher notices a girl having more apples than both of her neighbors combined, the teacher takes away one apple from that girl and gives one apple each to her neighbors. Prove that this process stops after a finite number of steps. (Assume that the teacher has an abundant supply of apples.)

6. Let ${\mathbb{N}}$ denote the set of all natural numbers and let ${f : \mathbb{N} \rightarrow \mathbb{N}}$ be a function such that: ${f (mn) = f (m)f (n)}$ for all ${m}$, ${n}$ in ${\mathbb{N}}$; and ${m + n}$ divides ${f (m) + f (n)}$ for all ${m}$, ${n}$ in ${\mathbb{N}}$. Prove that there exists an odd natural number ${k}$ such that ${f (n) = n^k}$ for all ${n}$ in ${\mathbb{N}}$.

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