# 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.
Discussion