2

Part of 2007 Irish Math Olympiad

Problems(2)

right-angled triangle

Source:

Prove that the triangle ABC is right-angled if it holds:

\sin^2 A+\sin^2 B+\sin^2 C = 2

nice inequality

Source: Ireland 2007

a,b,

c

are positive real numbers. Prove that: \frac{a\plus{}b\plus{}c}{3} \le \sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{3}} \le \frac {\frac{ab}{c}\plus{}\frac{bc}{a}\plus{}\frac{ca}{b}}{3}. For each of the inequalities, find the conditions on

a,b,

c

such that equality holds.

inequalitiesinequalities proposed