ZT from WXC: 美国数学天才选拔赛

constant · 发表于 2006-1-17 19:50:32

美国数学天才选拔赛 (USA Math Talent Search)www.ddhw.com

2. (难度：+.5) Anna writes a sequence of integers starting with the number 12. Each subsequent integer she writes is chosen randomly with equal chance from among the positive divisors of the previous integer (including the possibility of the integer itself). She keeps writing integers until she writes the integer 1 for the first time, and then she stops. One such sequence is

12, 6, 6, 3, 3, 3, 1.www.ddhw.com

What is the expected value of the number of terms in Anna’s sequence?

5. (难度：++.5) Lisa and Bart are playing a game. A round table has n lights evenly spaced around its circumference. Some of the lights are on and some of them off; the initial configuration is random. Lisa wins if she can get all of the lights turned on; Bart wins if he can prevent this from happening.www.ddhw.com

On each turn, Lisa chooses the positions at which to flip the lights, but before the lights are flipped, Bart, knowing Lisa’s choices, can rotate the table to any position that he chooses (or he can leave the table as is). Then the lights in the positions that Lisa chose are flipped: those that are off are turned on and those that are on are turned off.

Lisa can take as many turns as she needs to win, or she can give up if it becomes clear to her that Bart can prevent her from winning.

(a) Show that if n = 7 and initially at least one light is on and at least one light is off, then Bart can always prevent Lisa from winning.

(b) Show that if n = 8, then Lisa can always win in at most 8 turns.

king19 · 发表于 2006-1-19 10:51:58

可否中文大哥

husonghu · 发表于 2006-1-19 18:16:03

我想，把这译成中文再打印出来是一项比较费事的工作.

heavenarch · 发表于 2006-1-19 18:33:08

E(2)=E(3)=2,E(4)=5/2,E(6)=8/3, E(12)=9.1/3

本贴由[husonghu]最后编辑于：2006-1-19 18:13:22

constant · 发表于 2006-1-20 05:44:47

E(2)=3

dfdfdf · 发表于 2006-1-20 08:51:12

how comes?

heavenarch · 发表于 2006-1-20 14:37:57

but during the caculation it is convienent to take it discount. One just need to add 1 to my E(12) to get the exact answer. Thanks for the correction.

constant · 发表于 2006-1-20 21:02:02

前一题很简单，后一题和我以前贴的题有点类似。
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ZT from WXC: 美国数学天才选拔赛

回复：ZT from WXC: 美国数学天才选拔赛

心感歉意。[:>][:>]

E(2)=E(3)=2,E(4)=5/2,E(6)=8/3, E(12)=9.1/3

E(2)=3

how comes?

you are right if the first number is counted.

前一题很简单，后一题和我以前贴的题有点类似。