How Cheenta works to ensure student success?
Explore the Back-Story

Indian National Math Olympiad, INMO 2018 Problems

This post contains problems from Indian National Mathematics Olympiad, INMO 2018. Try them and share your solution in the comments.

INMO 2018, Problem 1

Let ABC be a non-equilateral triangle with integer sides. Let D and E be respectively the mid-points BC and C A, let G be the centroid of triangle ABC. Suppose D, C, E, G are concyclic. Find the least possible perimeter of triangle ABC.

INMO 2018, Problem 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. (For example, we can have 5 types of tiling when n = 2; red-red; red-green; green-red; green-green; and  blue.) 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}.

INMO 2018, Problem 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_1}A{O_2} is an obtuse angle. Let the circumcircle of triangle {O_1}A{O_2} intersect \Gamma_1 T_2 respectively in points C{(\not= A)} and D{(\not= A)}. Let the line C B intersect \Gamma_2 in E; let the line D B intersect \Gamma_1 in F . Prove that the points C, D, E, F are concyclic.

INMO 2018, Problem 4

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

INMO 2018, Problem 5

There are n\ge 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 neighnors 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 numberof steps. (Assume that the teacher has an abundant supply of apples.)

INMO 2018, Problem 6

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

Also Visit: Math Olympiad Program

This post contains problems from Indian National Mathematics Olympiad, INMO 2018. Try them and share your solution in the comments.

INMO 2018, Problem 1

Let ABC be a non-equilateral triangle with integer sides. Let D and E be respectively the mid-points BC and C A, let G be the centroid of triangle ABC. Suppose D, C, E, G are concyclic. Find the least possible perimeter of triangle ABC.

INMO 2018, Problem 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. (For example, we can have 5 types of tiling when n = 2; red-red; red-green; green-red; green-green; and  blue.) 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}.

INMO 2018, Problem 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_1}A{O_2} is an obtuse angle. Let the circumcircle of triangle {O_1}A{O_2} intersect \Gamma_1 T_2 respectively in points C{(\not= A)} and D{(\not= A)}. Let the line C B intersect \Gamma_2 in E; let the line D B intersect \Gamma_1 in F . Prove that the points C, D, E, F are concyclic.

INMO 2018, Problem 4

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

INMO 2018, Problem 5

There are n\ge 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 neighnors 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 numberof steps. (Assume that the teacher has an abundant supply of apples.)

INMO 2018, Problem 6

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

Also Visit: Math Olympiad Program

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
Menu
Trial
Whatsapp
Math Olympiad Program
magic-wandrockethighlight