HMMT 二月 2012 · 冲刺赛 · 第 33 题
HMMT February 2012 — Guts Round — Problem 33
题目详情
英文原题
[23] Compute the decimal expansion of√π. Your score will be min(23 , k), where kis the number of consecutive correct digits immediately following the decimal point in your answer . For each of the remaining three problems, it is difficult to obtain an exact answer. Instead, give an interval [L, U], where LandUare positive real numbers written in decimal . If [L, U]contains the answer to the problem, you will receive credit based off of how close L andUare. Otherwise, you will receive no credit.
解析
英文解析
- [ 23 ] Compute the decimal expansion of π . Your score will be min(23 , k ), where k is the number ofconsecutive correct digits immediately following the decimal point in your answer.
Answer: 1.77245385090551602729816... For this problem, it is useful to know the following square
√ √
root algorithm that allows for digit-by-digit extraction of x and gives one decimal place of x foreach two decimal places of x . We will illustrate how to extract the second digit after the decimal point
√ √
of π , knowing that π = 3 . 1415 · · · and π = 1 . 7 · · · .
Let d be the next decimal digit. Then d should be the largest digit such that (1 . 7 + 0 . 01 d ) < π , which 2
2 2
in this case we will treat as (1 . 7 + 0 . 01 d ) < 3 . 1415. Expanding this, we get 2 . 89 + 0 . 034 d + 0 . 0001 d <