Maths Olympiad Exam

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.

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Maths Olympiad Other Question

Suppose hx1, x2, . . . , xn, . . .i is a sequence of positive real numbers such that x1 x2
x3 x^n , and for all n
x1^1+x4^2+x9^3+..... +xn^2
n 1.
Show that for all k the following inequality is satisfied:
x1^1+x2^2+x3^3+ ...... +xk^k
n 3. (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?