# Regional Math Olympiad 2013 (RMO 2013)

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