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  data-recalc-dims= 5" /> and q_1 data-recalc-dims=5" />. Prove that 30 divides p_1 - q_1.

3. Define a sequence langle f_n (x)rangle_{ninmathbb 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} ngeq 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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4 Comments
  • priyam ladha
    Posted at 23:41h, 21 April Reply

    when is next maths Olympiad is going to held in session 2012 to 2013 for class 11th please give me information by calling on 8107971000.
    thanking you!!!

    • Gonit Sora
      Posted at 13:33h, 24 April Reply

      The next Olympiad will be the RMO in the first Sunday of November 2012.

  • oindrila das
    Posted at 20:16h, 09 September Reply

    when is the next maths olympiad for the students of class11 this year

    • Gonit Sora
      Posted at 21:42h, 10 September Reply

      RMO in December 2012.

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