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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