Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

INMO 2011 | Problems

  1. This post contains problem from Indian National Mathematics Olympiad, INMO 2011. Try them out and share your solution in the comments.
  2.  
  3. Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC is equilateral.
  4. Call a natural number n faithful, if there exist natural numbers a < b < c such that a divides b, b divides c and n = a + b + c. (i) Show that all but a finite number of natural numbers are faithful. (ii) Find the sum of all natural numbers which are not faithful.
  5. Consider two polynomials P(x)=a_nx^n+a_{n-1}x^{n-1}+......+a_1x+a_0=0 and Q(x)=b_nx^n+b_{n-1}x^{n-1}+......+b_1x+b_0=0 with integer coefficients such that a_n-b_n is a prime, a_{n-1}=b_{n-1} and a_nb_0-a_0b_n \neq 0 . Suppose there exists a rational number r such that P(r) = Q(r) = 0. Prove that r is an integer.
  6. Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
  7. Let ABCD be a quadrilateral inscribed in a circle \Gamma . Let E, F, G, H be the midpoints of the arcs AB, BC, CD, DA of the circle \Gamma . Suppose AC.BD = EG .FH. Prove that AC, BD, EG, FH are concurrent.
  8. Find all functions f : R \mapsto R such that f(x+y)f(x-y)=(f(x)+f(y))^2-4x^2f(y) ,  for all x, y ∈ R, where R denotes the set of all real numbers.

Some Useful Links:

INMO 2012 Problems

INMO 2018 Problem 6

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com