INMO 2011

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.

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