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- This post contains RMO 2007 problems. Try to solve these problems.
- 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 . [weightage 17/100] - 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 . [Weightage 17/100]

- Find all pairs (a, b) of real numbers such that whenever is a root of is also a root of the equation.[Weightage 17/100]

- How many 6-digit numbers are there such that-:
- The digits of each number are all from the set {1,2,3,4,5}
- b)any digit that appears in the number appears at least twice ?

(Example: 225252 is valid while 222133 is not) [weightage 17/100]

- 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.
- If is , find, with proof, the difference between the lengths of the parallel sides.
- If is , find, with proof, the difference between the lengths of the parallel sides.

[Weightage 17/100]

- Prove that:
- for all integers .[Weightage 16/100]

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