Question 165
Given that $a_1, a_2, \dots, a_{10}$ are positive real numbers, determine the smallest possible value of
$$
\sum_{i=1}^{10} \left\lfloor \frac{7a_i}{a_i + a_{i+1}} \right\rfloor
$$
where we define $a_{11} = a_1$.
$$
\sum_{i=1}^{10} \left\lfloor \frac{7a_i}{a_i + a_{i+1}} \right\rfloor
$$
where we define $a_{11} = a_1$.