Question 1
Let $a,b,c\gt 0$ satisfy $a+b+c=4\sqrt[3]{abc}$. Prove that
$$2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.$$
Proposed by Titu Andreescu. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.
$$2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.$$
Proposed by Titu Andreescu. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.