HMMT 二月 2013 · 冲刺赛 · 第 33 题

英文解析

33. [ 25 ] Compute the value of 1 + 2 + 3 + . . . + 24 + 25 . If your answer is A and the correct answer
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    A Cis 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 usingits largest term. Taking logs of each term, we see that the max occurs when (26 − n ) log n peaks, andtaking derivatives givesx + 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, butin any case it’s hard to find powers of anything except 10). Thus the largest term will be something
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    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 approximatelyhappens is that only the four or five largest terms in the sum are of any substantial size; thus it isreasonable here to pick some constant from 4 to 20 to multiply our guess 10 ; any guess between 16
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