Let $a, b, c$ be positive real numbers satisfying \[a+b+c = a^2 + b^2 + c^2.\]Let \[M = \max\left(\frac{2a^2}{b} + c, \frac{2b^2}{a} + c \right) \quad \text{ and } \quad N = \min(a^2 + b^2, c^2).\]Find the minimum possible value of $M/N$.
Source: 2024 Myanmar IMO Training
Tags: inequalities, ratio, inequalities proposed, algebra
Let $a, b, c$ be positive real numbers satisfying \[a+b+c = a^2 + b^2 + c^2.\]Let \[M = \max\left(\frac{2a^2}{b} + c, \frac{2b^2}{a} + c \right) \quad \text{ and } \quad N = \min(a^2 + b^2, c^2).\]Find the minimum possible value of $M/N$.
