MathDB

5

Part of 2008 Irish Math Olympiad

Problems(2)

Obtuse Triangle Ratio Condition

Source: Irish Mathematical Olympiad - Paper 1 - 2008 -Question 5

5/10/2008
A triangle ABC ABC has an obtuse angle at B B. The perpindicular at B B to AB AB meets AC AC at D D, and |CD| \equal{} |AB|. Prove that |AD|^2 \equal{} |AB|.|BC| if and only if \angle CBD \equal{} 30^\circ.
ratiogeometryparallelogramgeometry proposed
2 Inequalities When xyz>1

Source: Irish Mathematical Olympiad - Paper 2 - 2008 -Question 5

5/11/2008
Suppose that x,y x, y and z z are positive real numbers such that xyz≥1 xyz \ge 1. (a) Prove that 27 \le (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2, with equality if and only if x \equal{} y \equal{} z \equal{} 1. (b) Prove that (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2 \le 3(x \plus{} y \plus{} z)^2, with equality if and only if x \equal{} y \equal{} z \equal{} 1.
inequalitiesinequalities proposed