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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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