HMMT 二月 2012 · 冲刺赛 · 第 33 题

HMMT February 2012 — Guts Round — Problem 33

[ 23 ] Compute the decimal expansion of π . Your score will be min(23 , k ), where k is 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 L and U are 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 and U are. Otherwise, you will receive no credit.

解析

[ 23 ] Compute the decimal expansion of π . Your score will be min(23 , k ), where k is the number of consecutive 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 for each 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 · · · . 2 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 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 <