Problem

Source: Chinese TST 2009 5th P3

Tags: inequalities, inequalities proposed



Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} + a_{2} + a_{3} + a_{4} = 1.$ Prove that $ max\{\sum_{1}^4{\sqrt {a_{i}^2 + a_{i}a_{i - 1} + a_{i - 1}^2 + a_{i - 1}a_{i - 2}}},\sum_{1}^4{\sqrt {a_{i}^2 + a_{i}a_{i + 1} + a_{i + 1}^2 + a_{i + 1}a_{i + 2}}}\}\ge 2.$ Where for all integers $ i, a_{i + 4} = a_{i}$ holds.