Maths Olympiad Exam

Find all pairs (x, y) of real numbers such that
16x^2 + y + 16x + y^2 = 1.

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Maths Olympiad Other Question

Let ABC be a triangle and let BB1, CC1 be respectively the bisectors of angle B, C with B1 on AC and C1 on AB. Let E, F be the feet of perpendiculars drawn from A onto BB1, CC1 respectively. Suppose D is the point at which the incircle of ABC touches AB. Prove that AD = EF Let ABCDEF be a convex hexagon in which the diagonals AD, BE, CF are concurrent at O. Suppose the area of traingle OAF is the geometric mean of those of OAB and OEF; and the area of triangle OBC is the geometric mean of those of OAB and OCD. Prove that the area of triangle OED is the geometric mean of those of OCD and OEF.