Maths Olympiad Exam

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).

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Maths Olympiad Other Question

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. Let ABC be a triangle in which no angle is 90◦. For any point P in the plane
of the triangle, let A1,B1,C1 denote the reflections of P in the sides BC,CA,AB
respectively. Prove the following statements:
(a) If P is the incentre or an excentre of ABC, then P is the circumcentre of A1B1C1;
(b) If P is the circumcentre of ABC, then P is the orthocentre of A1B1C1;
(c) If P is the orthocentre of ABC, then P is either the incentre or an excentre of
A1B1C1.