<s>[INST] Find the greatest integer in the expression below as a function of the given conditions Find the greatest integer less than $\sqrt{2^{100}+10^{10}}.$
Answer: $2^{50}$
I tried, but I can't finish:
$\sqrt{2^{100}+(2\cdot5)^{10}}=\sqrt{2^{10}(2^{90}+5^{10})}=2^5\sqrt{2^{90}+5^{10}}$
 [/INST] A little GM-AM can also be helpful:
\begin{eqnarray*} \sqrt{2^{100}+10^{10}}
& = & 2^{50} \sqrt{  1+\frac{10^{10}}{2^{100}}} \\
& \stackrel{GM-AM}{<} & 2^{50}\frac{1+1+\frac{10^{10}}{2^{100}}}2 \\
& = & 2^{50} + \frac{10^{10}}{2^{51}} \\
& < & 2^{50} + \frac{10^{10}}{2^{50}} \\
& = & 2^{50} + \left( \frac{10}{2^5}\right)^{10} \\
& < & 2^{50} + 1
\end{eqnarray*}
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