1. Let ABC be an acute-angled triangle, let D, F 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.
    Discussion
  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