5

Part of 2001 Irish Math Olympiad

Problems(2)

find the cases of equality

Source: Ireland 2001

Prove that for all real numbers

a,b

ab>0

we have: \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}10ab\plus{}b^2}{12} and find the cases of equality. Hence, or otherwise, prove that for all real numbers

a,b

\sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}ab\plus{}b^2}{3} and find the cases of equality.

inequalities proposedinequalities

function (maybe posted before)

Source: Ireland 2001

Determine all functions

f: \mathbb{N} \rightarrow \mathbb{N}

which satisfy: f(x\plus{}f(y))\equal{}f(x)\plus{}y for all

x,y \in \mathbb{N}

functionalgebra proposedalgebra