IMO Shortlist 1966 problem 26


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2. travnja 2012.
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Prove the inequality

a.) \left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left(a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) ,

where k\geq 1 is a natural number and a_{1}, a_{2}, ..., a_{k} are arbitrary real numbers.

b.) Using the inequality (1), show that if the real numbers a_{1}, a_{2}, ..., a_{n} satisfy the inequality

a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left(a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },

then all of these numbers a_{1}, a_{2}, \ldots, a_{n} are non-negative.
Izvor: Međunarodna matematička olimpijada, shortlist 1966