Question 147
Let $a, b, c$ be real numbers with $1 \lt a \lt b \lt c$ that satisfy the equations
$$
\begin{gathered}
\log_{a} b + \log_{b} c + \log_{c} a = 6.5 \\
\log_{b} a + \log_{c} b + \log_{a} c = 5
\end{gathered}
$$
Then $\max \left\{ \log_{a} b, \log_{b} c, \log_{c} a \right\}$ can be written in the form $\sqrt{x} + \sqrt{y}$, where $x$ and $y$ are positive integers. What is $x + y$?
$$
\begin{gathered}
\log_{a} b + \log_{b} c + \log_{c} a = 6.5 \\
\log_{b} a + \log_{c} b + \log_{a} c = 5
\end{gathered}
$$
Then $\max \left\{ \log_{a} b, \log_{b} c, \log_{c} a \right\}$ can be written in the form $\sqrt{x} + \sqrt{y}$, where $x$ and $y$ are positive integers. What is $x + y$?