Maths Olympiad Exam

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?

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Maths Olympiad Other Question

Find all non-zero real numbers x, y, z which satisfy the system of equations:
(x^2 + xy + y^2)(y^2 + yz + z^2)(z^2 + zx + x^2) = xyz,
(x^4 + x^2y^2 + y^4)(y^4 + y^2z^2 + z^4)(z^4 + z^2x^2 + x^4) = x^3y^3z^3. 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.