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Indian National Mathematical Olympiad 2012


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 ABCD be a quadrilateral inscribed in a circle. Suppose AB=\sqrt{2+\sqrt{2}} and AB subtends 135 degrees at center of circle . Find the maximum possible area of ABCD.

2. Let p_1<p_2<p_3<p_4 and q_1<q_2<q_3<q_4 be two sets of prime numbers, such that p_4 - p_1 = 8 and q_4 - q_1= 8. Suppose p_1 > 5 and q_1>5. Prove that 30 divides p_1 - q_1.

3. Define a sequence \langle f_n (x)\rangle_{n\in\mathbb N_0} of functions as
f_0(x)=1, f_1(x)=x, \ (f_n(x))^2-1=f_{n-1}(x)f_{n+1}(x), \ \text{for}\ n\geq 1.
Prove that each f_n(x) is a polynomial with integer coefficients.

4. Let ABC be a triangle. An interior point P of ABC is said to be good if we can find exactly 27 rays emanating from P intersecting the sides of the triangle ABC such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given triangle ABC.

5. Let ABC be an acute angled triangle. Let D,E,F be points on BC, CA, AB such that AD is the median, BE is the internal bisector and CF is the altitude. Suppose that \angle FDE=\angle C, \angle DEF=\angle A and \angle EFD=\angle B. Show that ABC is equilateral.

6. Let f : \mathbb{Z} \to \mathbb{Z} be a function satisfying f(0) \ne 0, f(1) = 0 and

(i) f(xy) + f(x)f(y) = f(x) + f(y)

(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0

for all x,y \in \mathbb{Z}, simultaneously.

(a) Find the set of all possible values of the function f.

(b) If f(10) \ne 0 and f(2) = 0, find the set of all integers n such that f(n) \ne 0.


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