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Found problems: 1

2011 Romania National Olympiad, 1
Tags: inequalities , formulas , algebra , factorization , sum of cubes
Let be a natural number $ n $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $$ a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} ,\quad\forall m\in\{ 1,2,\ldots ,n \} . $$ Prove that $ a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} . $