Rmo was conducted on 9th October in Gujarat.
1)Let ABC be a right angled triangle with B=90°.Let I be tye incentre of ABC.Let AI extended intersect BC at F.Draw a line perpendicular to AI at I.Let it intersect AC at E.Prove that IE=IF.
2)Let a,b,c be positive reals such that
Prove that abc≤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
4)How many 6 digit natural numbers containing only the digits 1,2,3 are there in which 3 occurs exactly twice and the number Is divisible by 9?
5)Let ABC be a right angled triangle with B=90°.Let AD be the bisector of A with D on BC.Let the circumcircle of triangle ABD intersect AC again in E,and let the circumcircle of triangle ABD intersect AC again in 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......) Has infinitely many 3-term subsequences on harmonic progression such that for any two such triples (a,b,c) and (p,q,r) in harmonic progression ,one has a/p≠b/q.