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Regional Math Olympiad 2013 (RMO 2013)

  1. In this post, there are questions from Regional Math Olympiad 2013. Try out the problems.
  3. Find the number of 8 digit numbers sum of whose digits are 4.
  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
  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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