Question 83
Let $N$ be the set of positive integers. A function $f: N \to N$ satisfies $f(34) = 2024$, and for any positive integers $a, b, c$, there exists a triangle with side lengths $a+f(b)$, $b+f(c)$, $c+f(a)$. Find the smallest possible value of $f(100) + f(101) + \dots + f(199)$ under these conditions. Note that three points lying on the same line do not form a triangle.