4a

Part of 2023 Abelkonkurransen Finale

Problems(1)

Sharp inequality [cyclic sum (a^2+b^2-c^2)/(ab) > 2] in triangle

Source: 2023 Abelkonkurransen Finale, Problem 4a

a,b,c

are the side-lengths of a triangle, show that \begin{align*} \frac{a^2+b^2-c^2}{ab} + \frac{b^2+c^2-a^2}{bc} + \frac{c^2+a^2-b^2}{ca} > 2. \end{align*} Also show that the inequality does not necessarily hold if you replace

2

(on the right-hand side) by a bigger by a bigger number.

inequalitiesgeometry