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

Prove that there exists two infinite sequences and of positive integers such that the following conditions holds simultaneously:

;

, for all ;

divides , for all ;

divides , for all .

Suppose a and b are real numbers such that the roots of the cubic equation are all positive real numbers. Prove that i) and ii) .

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.

Three non-zero numbers a,b,c are said to be in harmonic progression if . Find all three term harmonic progressions a,b,c of strictly increasing positive integers in which a=20 and b divides c.

Find the number of integer-sided isosceles obtuse-angled triangles with perimeter 2008. Discussion

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