11-th Czech-Slovak-Polish Match, 2011

Adding three AM-GM inequalities $$4a^{3}b+b^{3}c+2c^{3}a\ge 7a^{2}bc,$$ $$4b^{3}c+c^{3}a+2a^{3}b\ge 7b^{2}ca,$$ $$4c^{3}a+a^{3}b+2b^{3}c\ge 7c^{2}ab$$ we get $$a^{3}b+b^{3}c+c^{3}a\ge a^{2}bc+b^{2}ca+c^{2}ab(1)$$ The assumption $a^2<bc$ implies $-a^{3}b>-b^{2}ca$ and this together with (1) gives $$b^{3}c+c^{3}a>a^{2}bc+c^{2}ab$$ i.e. $$b^3+a{c}^2>ab(a+c)$$ □