Maths Olympiad Exam

(i) Consider two positive integers a and b which are such that aabb is divisible by 2000.
What is the least possible value of the product ab?
(ii) Consider two positive integers a and b which are such that abba is divisible by 2000.
What is the least possible value of the product ab?

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Maths Olympiad Other Question
The internal bisector of angle A in a triangle ABC with AC > AB, meets the circumcircle
􀀀 of the triangle in D. Join D to the centre O of the circle 􀀀 and suppose DO meets AC
in E, possibly when extended. Given that BE is perpendicular to AD, show that AO is
parallel to BD. Find all real values of a for which the equation
x^4 - 2ax^2 + x + a^2 - a = 0 has all its roots real.