Maths Olympiad
Exam
Let a, b, c be three real numbers such that 1 ≥ a ≥ b ≥ c ≥ 0. Prove that if is a
root of the cubic equation x3 + ax2 + bx + c = 0 (real or complex),then || ≤ 1.
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Maths Olympiad Other Question
In a convex quadrilateral PQRS, PQ = RS, (√3+1)QR = SP and RSP−SPQ =
30◦. Prove that
PQR − QRS = 90◦.
For any natural number n, (n ≥ 3), let f(n) denote the number of non-congruent
integer-sided triangles with perimeter n (e.g., f(3) = 1, f(4) = 0, f(7) = 2). Show
that
(a) f(1999) > f(1996);
(b) f(2000) = f(1997).