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December 1, 2013

Regional Math Olympiad 2013 (RMO 2013)

  1. In this post, there are questions from Regional Math Olympiad 2013. Try out the problems.
  2.  
  3. Find the number of 8 digit numbers sum of whose digits are 4.
    Discussion
  4. Find the number of  4-tuples (a,b,c,d) of natural numbers with a \le b \le c and a! + b! + c! = 3^d
    Discussion
  5. In an acute-angled triangle ABC with AB < AC the circle \Gamma touches AB at B and passes through C intersecting AC again at D. Prove that the orthocenter of triangle ABD lies on \Gamma if and only if it lies on the perpendicular bisector of BC.
  6. A polynomial is called a Fermat Polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose f(x) is a Fermat Polynomial such that f(0) = 1000. Show that f(x) + 2x is not a Fermat Polynomial.
  7. Let ABC be a triangle which is not right angled. Define a sequence of triangles A_i B_i C_i with \( i \ge 0\) as follows. A_0 B_0 C_0 = ABC and for i \ge 0 A_{i+1} B_{i+1} C_{i+1} are the reflections of the orthocenter of triangle A_i B_i C_i in the sides B_i C_i , C_i A_i , A_i B_i   respectively. Assume that \angle A_n = \angle A_m for some distinct natural numbers m, n. Prove that \angle A = 60^o .
  8. Let n \ge 4 be a natural number. Let A_1 , A_ 2 .... A_n be a regular polygon and X = { 1, 2, ..., n }. A subset { i_1 , i_2 , ... i_k } , k \ge 1 , \( i_1 < i_2 < ... < i_k \) is called a good subset if the angles of the polygon angles \( A_{i_1} ... A_{i_k}\) when arranged in an increasing order is an arithmetic progression.  If n is prime then show that a PROPER good subset of X contains exactly 4 elements.
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RMO2013

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