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Let nonnegative real numbers

a_{1},a_{2},a_{3},a_{4}

satisfy a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1. Prove that max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2. Where for all integers i, a_{i \plus{} 4} \equal{} a_{i} holds.

Problem Statement