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Minimum value of a ratio with given constraints

Source: 2024 Myanmar IMO Training

July 4, 2024
inequalitiesratioinequalities proposedalgebra

Problem Statement

Let a,b,ca, b, c be positive real numbers satisfying a+b+c=a2+b2+c2.a+b+c = a^2 + b^2 + c^2. Let M = \max\left(\frac{2a^2}{b} + c, \frac{2b^2}{a} + c \right)   \text{ and }   N = \min(a^2 + b^2, c^2). Find the minimum possible value of M/NM/N.
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