RMO 2007 Problems

This post contains RMO 2007 problems. Try to solve these problems

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 $latex \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 $latex 60^\circ $ , find, with proof, the difference between the lengths of the parallel sides.
   2. If $ \angle AMD $ is $latex 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 $latex \ngeq 9 $.[Weightage 16/100]