MathDB

5

Part of 2002 Irish Math Olympiad

Problems(2)

interesting

Source: Ireland 2002

7/5/2009
Let 0<a,b,c<1 0<a,b,c<1. Prove the inequality: \frac{a}{1\minus{}a}\plus{}\frac{b}{1\minus{}b}\plus{}\frac{c}{1\minus{}c} \ge \frac {3 \sqrt[3]{abc}}{1\minus{} \sqrt[3]{abc}}. Determine the cases of equality.
inequalities
equality case

Source: Ireland 2002

7/5/2009
Let ABC ABC be a triangle with integer side lengths, and let its incircle touch BC BC at D D and AC AC at E E. If |AD^2\minus{}BE^2| \le 2, show that AC\equal{}BC.
geometry