5.\[\sum_{cyc}a^2b^2=\frac{(\sum_{cyc}a)^2\sum_{cyc}a^2b^2}{(\sum_{cyc}a^2)^2}=\frac{\sum_{sym}a^4b^2+2\sum_{cyc}a^3b^3+2\sum_{sym}a ^3b^2c+3a^2b^2c^2}{(\sum_{cyc}a^2)^2}\]
\[\le\frac{\sum_{sym}a^5b+\sum_{cyc}a^4bc+2\sum_{cyc}a^3b^3+2\sum_{sym}a^3b^2c}{(\sum_{cyc}a^2)^2}=\frac{(\sum_{cyc}a^2)^2\sum_{cy c}ab}{(\sum_{cyc}a^2)^2}=\sum_{cyc}ab\]
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