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]