A proof of inequality in Senior High School Let x, y, Z satisfy x + y + Z = 1 Verification: x ^ 2 / (y + 2Z) + y ^ 2 / (Z + 2x) + Z ^ 2 / (x + 2Y) > = 1 / 3

It is proved by Cauchy Inequality: [(y + 2Z) + (Z + 2x) + (x + 2Y)] [x ^ 2 / (y + 2Z) + y ^ 2 / (Z + 2x) + Z ^ 2 / (x + 2Y)] > = (x + y + z) ^ 2 = 1 and (y + 2Z) + (Z + 2x) + (x + 2Y) = 3 (x + y + Z) = 3, so x ^ 2 / (y + 2Z) + y ^ 2 / (Z + 2x) + Z ^ 2 / (x + 2Y) > = 1 / 3

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