Regional Mathematical Olympiad 2014 (Mumbai Region)
1 Three positive real numbers are such that . Can be the lengths of sides of a triangle? Justify your answer.
2 The roots of the equation
form a non-constant arithmetic progression and the roots of the equation
form a non-constant geometric progression. Given that are real numbers, find all positive integral values and .
3 Let be an acute-angled triangle in which is the largest angle. Let be its circumcentre. The perpendicular bisectors of and meet at and respectively. The internal bisectors of and meet and at and respectively. Prove that is perpendicular to if is parallel to .
4 A person moves in the plane moving along points with integer co-ordinates and only. When she is at point , she takes a step based on the following rules:
(a) if is even she moves either to or ;
(b) if is odd she moves either to or .
How many distinct paths can she take to go from to given that she took exactly three steps to right to ?
5 Let be positive numbers such that
Prove that . When does the equality hold?
6 Let be the points of contact of the incircle of an acute-angled triangle with respectively. Let be the incenters of the triangles respectively. Prove that the lines are concurrent.
Post your innovative solutions below!! Enjoy!!!!!!!!