珍珠湾ART

标题: 要你填分母 [打印本页]

作者: 新用户 时间: 2005-3-12 05:51
标题: 要你填分母

七个分子是一的数字相加要等于一, 要你填分母, 不能填一样的!

作者: yma16 时间: 2005-3-12 07:34
标题: 回复：要你填分母

2,3,12,18,72,108,216

作者: yma16 时间: 2005-3-12 17:13
标题: Can you prove that it is impossible for even

number of fractions?

作者: 新用户 时间: 2005-3-12 18:06
标题: 回复：回复：要你填分母

我认为答案好像不止一组。能给出你的解题过程吗？

作者: 野菜花 时间: 2005-3-12 18:06
标题: 回复：Can you prove that it is impossible for even

Your conjecture seems not right. The following is an example for 4 fractions.

1=
1+(-1/2+1/2)+(-1/3+1/3)+(-1/4+1/4)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+1/4
=1/2+1/6+1/12+1/4

用这个办法是否可以写出和为

1的任何多(not a product of two consecutive numbers)个分子为1而分母不同的分数？

作者: 野菜花 时间: 2005-3-12 18:43
标题: More solutions

=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7

=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)+1/7

=1/2+1/6+1/12+1/20+1/30+1/42+1/7

=1/2+1/3+1/6

=1/2+1/3+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10+1/10

=1/2+1/3+(1/6-1/7)+(1/7-1/8)+(1/8-1/9)+(1/9-1/10)+1/10

=1/2+1/3+1/42+1/56+1/72+1/90+1/10

=1-1/2+1/2-1/3+1/3-1/4+1/4

=1/2+1/6+1/12+1/4

=1/2+1/4+1/6+1/12-1/13+1/13-1/14+1/14-1/15+1/15

=1/2+1/4+1/6+1/156+1/182+1/210+1/15

.......

作者: yma16 时间: 2005-3-12 19:02
标题: 回复：回复：Can you prove that it is impossible for even

Thanks. Do you think "product of two consecutive numbers" 个分子为1而分母不同的分数 has no solution?

"用这个办法是否可以写出和为1的任何多(not a product of two consecutive numbers)个分子为1而分母不同的分数" - I believe so.

作者: 野菜花 时间: 2005-3-12 19:44
标题: 回复：回复：回复：Can you prove that it is impossible for e

I mean use that exact method we can at least conclude:"用这个办法是否可以写出和为1的任何多(not a product of two consecutive numbers)个分子为1而分母不同的分数" .

But I don't think "product of two consecutive numbers" 个分子为1而分母不同的分数 has no solution.

For example: 6=2*3www.ddhw.com

=1-1/2+1/2-1/3+1/3

=1/2+1/6+1/3

=1/2+1/3+1/6-1/7+1/7-1/8+1/8-1/9+1/9

=1/2+1/3+1/42+1/56+1/72+1/9

Can we prove there is always solutions for any n>2?

作者: fzy 时间: 2005-3-12 20:20
标题: Yes by induction

1 = 1/2 + 1/3 + 1/6
1 = 1/2 + 1/3 + 1/6 - 1/7 + 1/7
= 1/2 + 1/3 + 1/7 + 1/42
1 = 1/2 + 1/3 + 1/7 + 1/42 - 1/43 + 1/43
= 1/2 + 1/3 + 1/7 + 1/43 + 1/(42*43)
......

作者: 野菜花 时间: 2005-3-12 20:26
标题: Great! [@};-][@};-]

Great!

作者: 乱弹 时间: 2005-3-12 22:26
标题: This problem has been discussed here. [:B]

Not long time ago. The difference is, that time the problem wanted 10 fractions, now 7.

作者: 乱弹 时间: 2005-3-12 22:38
标题: 野菜花® 出过一类似的题目 [>:D<]

作者: yma16 时间: 2005-3-13 01:03
标题: 乱弹, are you in charge of WXC's 脑筋急转?

That board has blocked 野菜花, which is very unfair. Thank you for the posts.

作者: 乱弹 时间: 2005-3-13 02:52
标题: Yes, I feel pretty sorry.

If she wants to come back, I would tell the site admin to solve the problem. However, she is happy to be here only.

You can find my previous posts here in which we discussed this issue.

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