HMMT 2025 U36

Ethan initially writes some numbers on a blackboard, each of which is either a 3 or a 5. He then repeatedly picks two numbers and replaces them with their sum, difference, product, or quotient (if the divisor is nonzero). Let $f(n)$ denote the minimum number of numbers Ethan must initially write for him to be able to eventually write the number $n$ . For example, $f(2025) \leq 6$ because Ethan could start with 3, 3, 3, 3, 5, and 5 on the board, then repeatedly multiply two numbers at a time to eventually get 2025.

Submit a comma- separated ordered 8- tuple of integers corresponding to the values of $f(164)$ , $f(187)$ , $f(191)$ , $f(224)$ , $f(255)$ , $f(286)$ , $f(374)$ , and $f(479)$ , in that order, or an X for any value you wish to leave blank. For instance, if you think $f(164) = 9$ and $f(224) = 8$ , you should submit "9, X, X, 8, X,

X, X, X". You will earn $\left|0.6^{W} \cdot \frac{(C + 1)^{2}}{4}\right|$ points, where $C$ is the number of correct answers you submit and $W$ is the number of incorrect (non- blank) answers.