Maths Olympiad Exam

Let ABCD be a quadrilateral inscribed in a circle. Let E, F, G, H be the midpoints of the arcs AB, BC, CD, DA of the circle. Suppose AC· BD = EG · FH. Prove that AC, BD, EG, FH are concurrent.

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Maths Olympiad Other Question

Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show
that one can select four among these five such that they are the vertices of a trapezium. Find all functions f : R + R such that
f(x + y)f(x - y) = f(x) + f(y)
2 - 4x^2f(y), (1)
for all x, y - R, where R denotes the set of all real numbers.