4 weeks ago
The Indian National Mathematical Olympiad 2012 was held on 5th February, 2012 at various centres all over the country. The North East had three centres at Guwahati, Shillong and Agartala. The questions are given below:
1. Let be a quadrilateral inscribed in a circle. Suppose and subtends degrees at center of circle . Find the maximum possible area of .
2. Let and be two sets of prime numbers, such that – and – . Suppose and . Prove that divides – .
3. Define a sequence of functions as
Prove that each is a polynomial with integer coefficients.
4. Let be a triangle. An interior point of is said to be good if we can find exactly rays emanating from intersecting the sides of the triangle such that the triangle is divided by these rays into smaller triangles of equal area. Determine the number of good points for a given triangle .
5. Let be an acute angled triangle. Let be points on such that is the median, is the internal bisector and is the altitude. Suppose that and Show that is equilateral.
6. Let be a function satisfying , and
for all , simultaneously.
Find the set of all possible values of the function .
If and , find the set of all integers such that .