Regional Math Olympiad 2013 (RMO 2013)

  1. Find number of 8 digit numbers sum of whose digits is 4.
  2. Find the number of  4-tuples (a,b,c,d) of natural numbers with \(a \le b \le c \) and \(a! + b! + c! = 3^d \)
  3. 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.
  4. 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.
  5. 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 \(\angle A = 60^o \).
  6. 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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