Maths Olympiad Exam

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.

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Maths Olympiad Other Question

How many 6-tuples (a1, a2, a3, a4, a5, a6) are there such that each of a1, a2, a3, a4, a5, a6 is from the set {1, 2, 3, 4} and the six expressions a2 j - ajaj +1 + a2 j+1
for j = 1, 2, 3, 4, 5, 6(where a7 is to be taken as a1) are all equal to one another? 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