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INMO 2012

  1. Let ABCD be a quadrilateral inscribed in a circle. Suppose AB =\(\sqrt {2 + \sqrt {2} } \) and AB subtends 1350 at the center of the 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 \(<fn(x)> \) n∈N of functions as \(f_0(x )=1, f_1(x )=x \), \((f_n(x))^2 – 1 = f_{n-1} (x) f_{n+1} (x) \), for \(n \ge 1 \) . 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 and angle DEF = angle A and angle EFD = angle B. Show that ABC is equilateral.
  6. Let \(f :Z \mapsto Z \)be a function satisfying \(f(0) \neq 0 \) , \(f(1)=0 \) and
    1. \(f(xy) + f(x)f(y) = f(x) + f(y) \),
    2. \((f(x-y) – f(0) ) f(x) f(y) = 0 \) for all x , \(y in Z \) simultaneously.
      1. Find the set of all possible values of the function f.
      2. If \(f(10) \neq 0 \) and \(f(2) = 0 \), find the set of all integers n such that \(f(n) \neq 0 \) .
September 24, 2013

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