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Easy inequality with square roots

Source: ARO Regional stage 2023 9.9

February 16, 2023
algebraRussiainequalities proposedInequality

Problem Statement

Find the largest real mm, such that for all positive real a,b,ca, b, c with sum 11, the inequality abab+c+bcbc+a+caca+b≥m\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m is satisfied.
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