Titu's Lemma
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Titu's lemma states that:
![\[\frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } \geq \frac{ (a_1 + a_2 + \cdots+ a_n ) ^2 } { b_1 + b_2 + \cdots+ b_n }.\]](http://latex.artofproblemsolving.com/2/6/4/264fff665becc96f5f9e05a178f2a31ee3f2603f.png)
It is a direct consequence of Cauchy-Schwarz theorem
