Regional Math Olympiad 2013 (RMO 2013)

1. In this post, there are questions from Regional Math Olympiad 2013. Try out the problems.
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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 $latex a \le b \le c $ and $latex a! + b! + c! = 3^d $
   Discussion
5. In an acute-angled triangle ABC with AB < AC the circle $latex \Gamma $ touches AB at B and passes through C intersecting AC again at D. Prove that the orthocenter of triangle ABD lies on $latex \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 $latex A_i B_i C_i $ with \( i \ge 0\) as follows. $latex A_0 B_0 C_0 = ABC $ and for $latex i \ge 0 A_{i+1} B_{i+1} C_{i+1} $ are the reflections of the orthocenter of triangle $latex A_i B_i C_i $ in the sides $latex B_i C_i , C_i A_i , A_i B_i $  respectively. Assume that $latex \angle A_n = \angle A_m $ for some distinct natural numbers m, n. Prove that $latex \angle A = 60^o $.
8. Let $latex n \ge 4 $ be a natural number. Let $latex A_1 , A_ 2 .... A_n $ be a regular polygon and X = { 1, 2, ..., n }. A subset $latex { i_1 , i_2 , ... i_k } $, $latex 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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