HMMT 二月 2026 · 冲刺赛 · 第 33 题

HMMT February 2026 — Guts Round — Problem 33

[20] From each of the first 6 sets of problems in this Guts round, Mark selects one of the four (correct) answers from that set. Compute the minimum possible range of these 6 values. 1 / 3 Submit a real number E . If the correct answer is A , you will receive max(0 , ⌊ 20 . 99 − 18 | E − A | ⌋ ) points. (The range of n values x ≤ x ≤ · · · ≤ x is given by x − x .) 1 2 n n 1 4

解析

[20] From each of the first 6 sets of problems in this Guts round, Mark selects one of the four (correct) answers from that set. Compute the minimum possible range of these 6 values. 1 / 3 Submit a real number E . If the correct answer is A , you will receive max(0 , ⌊ 20 . 99 − 18 | E − A | ⌋ ) points. (The range of n values x ≤ x ≤ · · · ≤ x is given by x − x .) 1 2 n n 1 Proposed by: Derek Liu 2251 Answer: ≈ 15 . 6319444444444 144 √ Solution: One set of answers that achieves this range is (4 , 11 , 1 , 16 , 53 / 144 , 6 2) from problems 3 , 7 , 10 , 15 , 20 , and 23 , respectively. The range is entirely determined by sets 4 and 5 . The correct answers from those sets are in the following table. Set 4 Set 5 π 2024 First problem 18 · 7 1 − ≈ 0 . 215 4 165 Second problem 136 = 82 . 5 2 Third problem 16 6723 √ 53 Fourth problem 3 30 ≈ 16 . 432 ≈ 0 . 368 144 53 2251 The minimal positive difference between any set 4 answer and any set 5 answer is 16 − = . 144 144 4