Problem

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Tags: inequalities, floor function, function, inequalities proposed



Let $a_1,a_2,\ldots,a_n$ are real numbers, prove that; \[\sum_{i=1}^na_i^2-\sum_{i=1}^n a_i a_{i+1} \le \left\lfloor \frac{n}{2}\right\rfloor(M-m)^2.\] where $a_{n+1}=a_1,M=\max_{1\le i\le n} a_i,m=\min_{1\le i\le n} a_i$.