Maths Olympiad Exam

Let P1(x) = ax^2 + bx + c, P2(x) = bx^2 + cx +a, P3(x) = cx^2 + ax + b be three quadratic polynomials where a, b, c are non-zero real numbers. Suppose there exists a real number
such that P1() = P2() = P3(). Prove that a = b = c.

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Maths Olympiad Other Question

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. Find the number of 4-digit numbers(in base 10) having non-zero digits and which are divisible by 4 but not by 8.