Maths Olympiad Exam

Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that
BD = CE = AF and BDF = CED = AFE. Prove that ABC is equilateral

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Maths Olympiad Other Question

Let ABC be an acute-angled triangle with altitude AK. Let H be its ortho-centre and O
be its circum-centre. Suppose KOH is an acute-angled triangle and P its circum-centre.
Let Q be the reflection of P in the line HO. Show that Q lies on the line joining the
mid-points of AB and AC. 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.