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
