# RMO 2008

1. Let ABC be an acute-angled triangle , let D,E be the mid-points of BC, AB respectively. Let the perpendicular from F to AC and the perpendicular at B to BC meet in N. Prove that ND is equal to circum-radius of ABC.
2. Prove that there exists two infinite sequences $${\left \langle a_n \right \rangle}_{n \ge 1}$$ and $${\left \langle b_n \right \rangle}_{n \ge 1}$$ of positive integers such that the following conditions holds simultaneously:
1. $$1<a_1<a_2<a_3<…..$$;
2. $$a_n < b_n < a_n^{2}$$, for all $$n \ge 1$$;
3. $$a_{n}-1$$ divides $$b_{n}-1$$, for all $$n \ge 1$$;
4. $$a_{n}^2-1$$ divides $$b_{n}^2-1$$, for all $$n \ge 1$$.
3. Suppose a and b are real numbers such that the roots of the cubic equation $$ax^3-x^2+bx+1=0$$ are all positive real numbers. Prove that i) $$0<3ab<1$$ and ii) $$b \ge \sqrt{3}$$.
4. Find the number of all 6-digits natural number such that the sum of their digits is 10 and each of the digits 0,1,2,3 occurs at least once in them.
5. Three non-zero numbers a,b,c are said to be in harmonic progression if $$\frac{1}{a}+\frac{1}{c}=\frac{2}{b}$$. Find all three term harmonic progressions a,b,c of strictly increasing positive integers in which a=20 and b divides c.
6. Find the number of integer-sided isosceles obtuse-angled triangles with perimeter 2008.
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