HMMT 二月 2013 · 冲刺赛 · 第 33 题

HMMT February 2013 — Guts Round — Problem 33

[ 25 ] Compute the value of 1 + 2 + 3 + . . . + 24 + 25 . If your answer is A and the correct answer ⌊ ( )⌋ ( ) ( ) 2 2 A C is C , then your score on this problem will be 25 min , . C A

解析

[ 25 ] Compute the value of 1 + 2 + 3 + . . . + 24 + 25 . If your answer is A and the correct answer ⌊ ( )⌋ ( ) ( ) 2 2 A C is C , then your score on this problem will be 25 min , . C A Answer: 66071772829247409 The sum is extremely unimodal, so we want to approximate it using its largest term. Taking logs of each term, we see that the max occurs when (26 − n ) log n peaks, and taking derivatives gives x + x log x = 26 From here it’s easy to see that the answer is around 10, and slightly less (it’s actually about 8 . 3, but in any case it’s hard to find powers of anything except 10). Thus the largest term will be something 16 16 like 10 , which is already an order of magnitude within the desired answer 6 . 6 × 10 . 26 − x To do better we’d really need to understand the behavior of the function x , but what approximately happens is that only the four or five largest terms in the sum are of any substantial size; thus it is 16 reasonable here to pick some constant from 4 to 20 to multiply our guess 10 ; any guess between 16 17