要你填分母

新用户 · 发表于 2005-3-12 05:51:26

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!

yma16 · 发表于 2005-3-12 07:34:00

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

yma16 · 发表于 2005-3-12 17:13:45

number of fractions?

新用户 · 发表于 2005-3-12 18:06:16

我认为答案好像不止一组。能给出你的解题过程吗？

野菜花 · 发表于 2005-3-12 18:06:22

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？

野菜花 · 发表于 2005-3-12 18:43:06

1

=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

=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

=1-1/2+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

.......

yma16 · 发表于 2005-3-12 19:02:43

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.

野菜花 · 发表于 2005-3-12 19:44:39

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

=1-1/2+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?

fzy · 发表于 2005-3-12 20:20:22

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)
......

野菜花 · 发表于 2005-3-12 20:26:12

Great!

乱弹 · 发表于 2005-3-12 22:26:42

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

乱弹 · 发表于 2005-3-12 22:38:52

yma16 · 发表于 2005-3-13 01:03:15

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

乱弹 · 发表于 2005-3-13 02:52:10

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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要你填分母

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Can you prove that it is impossible for even

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回复：回复：Can you prove that it is impossible for even

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Yes by induction

Great! [@};-][@};-]

This problem has been discussed here. [:B]

野菜花® 出过一类似的题目 [>:D<]

乱弹, are you in charge of WXC's 脑筋急转?

Yes, I feel pretty sorry.

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要你填分母

回复：要你填分母

Can you prove that it is impossible for even

回复：回复：要你填分母

回复：Can you prove that it is impossible for even

More solutions

回复：回复：Can you prove that it is impossible for even

回复：回复：回复：Can you prove that it is impossible for e

Yes by induction

Great! [@};-][@};-]

This problem has been discussed here. [:B]

野 菜 花® 出过一类似的题目 [>:D<]

乱弹, are you in charge of WXC's 脑筋急转?

Yes, I feel pretty sorry.

野菜花® 出过一类似的题目 [>:D<]