HMMT 二月 2026 · COMB 赛 · 第 6 题

HMMT February 2026 — COMB Round — Problem 6

Derek currently owes π units of a currency called Money of Indiscrete Type, or MIT for short. Every day, the following happens: • He flips a fair coin to decide how much of his debt to pay. If he flips heads, he decreases his debt by 1 MIT. If he flips tails, he decreases his debt by 2 MITs. If Derek’s debt ever becomes nonpositive, Derek becomes debt-free. • Afterwards, his remaining debt doubles. Compute the probability that Derek ever becomes debt-free. (MITs are continuous, so the debt is never rounded.)

解析

Derek currently owes π units of a currency called Money of Indiscrete Type, or MIT for short. Every day, the following happens: • He flips a fair coin to decide how much of his debt to pay. If he flips heads, he decreases his debt by 1 MIT. If he flips tails, he decreases his debt by 2 MITs. If Derek’s debt ever becomes nonpositive, Derek becomes debt-free. • Afterwards, his remaining debt doubles. Compute the probability that Derek ever becomes debt-free. (MITs are continuous, so the debt is never rounded.) Proposed by: Derek Liu 4 − π π Answer: = 2 − 2 2 Solution: Instead of the debt doubling, we can imagine that every day, Derek’s payment halves. − i − i Label the first day as 0 ; then, on day i , he randomly decides between paying off 2 and 2 · 2 of his remaining debt (which never increases). It is clear that this formulation of the problem is equivalent. − i Suppose that on day i , Derek pays (1 + b ) · 2 of his debt, so each b is either 0 or 1 with equal i i probability. Then, he pays off his full debt if and only if ∞ ∞ ∑ ∑ − i − i (1 + b )2 ≥ π ⇐⇒ b 2 ≥ π − 2 . i i i =0 i =0 ©2026 HMMT ∑ ∞ − i Note that b 2 is a uniformly distributed real number from 0 to 2 , so the answer is i i =0 2 − ( π − 2) 4 − π = . 2 2