Question 15
Let $n \ge 1$ be an integer. We have two sequences, each of $n$ positive real numbers, $a_1, a_2, ..., a_n$ and $b_1, b_2, ..., b_n$, such that $a_1 + a_2 + ... + a_n = 1$ and $b_1 + b_2 + ... + b_n = 1$. Find the smallest possible value that the sum
$$
\frac{a_1^2}{a_1+b_1} + \frac{a_2^2}{a_2+b_2} + \dots + \frac{a_n^2}{a_n+b_n}
$$
can take.
$$
\frac{a_1^2}{a_1+b_1} + \frac{a_2^2}{a_2+b_2} + \dots + \frac{a_n^2}{a_n+b_n}
$$
can take.