<s>[INST] Find the sum of these variables. Five real numbers $a_1, a_2, a_3, a_4\;\text{and}\; a_5\;$ are such that
$$\sqrt{a_1- 1} + 2\sqrt{a_2- 4}+3\sqrt{a_3- 9}   
+4\sqrt{a_4- 16} + 5 \sqrt{a_4- 25} =\frac{a_1+a_2+a_3+a_4+a_5}{2}.$$ 
Find $a_1+a_2+a_3+a_4+a_5.$
Thanks for checking this out!
 [/INST] For real $\sqrt{a-b^2},$ we need $a-b^2\ge0$
and for $b>0,$ and as $\sqrt{a-b^2}\ge0$ by AM-GM inequality, 
$$\dfrac{(\sqrt{a-b^2})^2+(b)^2}2\ge b\sqrt{a-b^2}$$
the equality will occur if  $\sqrt{a-b^2}=b$
$$\implies\dfrac{(\sqrt{a_1-1})^2+1^2+\cdots+(\sqrt{a_5-5^2})^2+5^2}2=\dfrac{a_1+a_2+a_3+a_4+a_5}2 \ge \sqrt{a_1-1}+\cdots$$
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