RMO 2007

  1. Let ABC be an acute-angled triangle; AD be the bisector of angle BAC with D on BC; and BE be the altitude from B on AC.
    Show that \(\angle CED > 45^\circ \) . [weightage 17/100]
  2. Let a, b, c be three natural numbers such that a < b < c and gcd (c – a, c – b) = 1. Suppose there exists an integer d such that a + d, b + d, c + d form the sides of a right-angled triangle. Prove that there exist integers, l,m such that \(c + d = l^{2} + m^{2} \). [Weightage 17/100]
  3. Find all pairs (a, b) of real numbers such that whenever \(\alpha \) is a root of \(x^{2} + ax + b = 0, \alpha^{2} – 2 \) is also a root of the equation.[Weightage 17/100]
  4. How many 6-digit numbers are there such that-:
    1. The digits of each number are all from the set {1,2,3,4,5}
    2. b)any digit that appears in the number appears at least twice ?
      (Example: 225252 is valid while 222133 is not) [weightage 17/100]
  5. A trapezium ABCD, in which AB is parallel to CD, is inscribed in a circle with centre O. Suppose the diagonals AC and BD of the trapezium intersect at M, and OM = 2.
    1. If \(\angle AMB \) is \(60^\circ \) , find, with proof, the difference between the lengths of the parallel sides.
    2. If \(\angle AMD \) is \(60^\circ \) , find, with proof, the difference between the lengths of the parallel sides.
      [Weightage 17/100]
  6. Prove that:
    1. \(5<\sqrt {5}+\sqrt [3]{5}+\sqrt [4]{5} \)
    2. \(8>\sqrt {8}+\sqrt [3]{8}+\sqrt [4]{8} \)
    3. \(n>\sqrt {n}+\sqrt [3]{n}+\sqrt [4]{n} \) for all integers \(\ngeq 9 \).[Weightage 16/100]

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