. Regional Mathematics Olympiad-2014 Time: 3 hours December 07, 2014
Instructions: Calculators (in any form) and protractors are not allowed.
Rulers ands compasses are allowed.
Answer all the questions.
All questions carry equal marks. Maximum marks: 102
In an acute-angled , is the largest angle. The perpendicular bisectors of BC and BA intersect AC at X and Y respectively. Prove that circumcentre of is incenter of
. Let be positive real numbers. Prove that
Find all pairs of (x,y) of positive integers such that divides .
. For any positive integer let denote the largest prime not exceeding n. Let denote the next prime larger than . (For example, and .) If is a prime number, prove that the value of the sum
. Let be a triangle with . Let be a point on line beyond such that . Let be the mid-point of and let be a point on the side such that . Prove that
. Each square of an grid is arbitrarily filled with either by or by . Let and denote the product of all numbers in the row and the column respectively, . Prove that
Note: In Question No.6, is an odd number.
This is RMO 2014 Coastal Andhra and Rayalaseema region. I had attempted first 4 questions. And 30 members will be selected from our region. And I want to know whether my answers are correct or not. So please try solve and keep the solutions. And, Thanks in Advance.