Time: 3 hours
Calculators (in any form) and protractors are not allowed.
Rulers and compasses are allowed.
Answer all the questions.
All questions carry equal marks. Maximum marks: 102.
1 Let ABC be an acute-angled triangle and suppose is the largest angle of the triangle. Let R be it's circumcentre. Let the circumcircle of triangle ARB cut AC again in X. Prove that RX is perpendicular to BC.
2 Find all real numbers and such that
3 Prove that there does not exist any positive integer , such that is a multiple of .
4 Find all positive real number triplets which satisfy
5 Let be a triangle. Let be on the segment , such that . Let meet the circumcircle of triangle again at . Show that the circumcentre of lies on .
6 For any natural number , let denote the sum of digits of . Find the number of all 3-digit numbers such that .
RMO was held in various centres in India on 7th December. You can take it as a test. Please let me know your score. 17 students will be selected from my state, Rajasthan. Pls Like and Reshare and Enjoy!