$\sum\limits_{cyc}\frac{a_1a_2}{1-(a_1-a_2)^2}\le\frac12$

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整数$n\ge2$,正数$a_1,a_2,\cdots,a_n$满足$\sum\limits_{i=1}^na_i=1$,求证\[\sum\limits_{cyc}\frac{a_1a_2}{1-(a_1-a_2)^2}\le\frac12\]