An Interesting problem

For discussing Olympiad Level Number Theory problems
protik
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An Interesting problem

Unread post by protik » Tue Dec 21, 2010 9:48 pm

Find all positive integers n such that there are $$k\geq$$ 2 positive rational numbers a1, a2, ............... , ak
satisfying $$a_1+a_2+................+a_k$$=$$a_1a_2...............a_k$$ = n.

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Labib
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Re: An Interesting problem

Unread post by Labib » Tue Dec 21, 2010 11:21 pm

as much as I know $\pm 6\ and 0$ satisfy it!! haven't really tried it yet! :?
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Moon
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Re: An Interesting problem

Unread post by Moon » Wed Dec 22, 2010 12:19 am

I'd like to draw everyone's attention to the fact that the topics of many posts are not effective. Without using a proper topic, you actually break the rule number 1 of this forum.
viewtopic.php?f=9&t=6

For example, the title of this post is "An interesting problem". You can see below that there is already another topic with the same name! You could use something like find when sum is equal to product...I just want to request everyone to be careful about this thing. This is for our own good.
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

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Tahmid Hasan
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Re: An Interesting problem

Unread post by Tahmid Hasan » Wed Dec 22, 2010 5:32 pm

there are no solutions because the sum of $$n$$ integers $$\geq2$$ can not be equal to both their product and $$n$$.
If 1 of the conditions are removed,there might be a solution :?
বড় ভালবাসি তোমায়,মা

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Labib
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Re: An Interesting problem

Unread post by Labib » Wed Dec 22, 2010 10:54 pm

@ tahmid, If $k=3, a_1=1, a_2=2, a_3=3$ then $n=6=1\times 2\times 3=1+2+3$
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protik
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Joined: Wed Dec 08, 2010 7:21 am

Re: An Interesting problem

Unread post by protik » Thu Dec 23, 2010 3:15 pm

Moon wrote:I'd like to draw everyone's attention to the fact that the topics of many posts are not effective. Without using a proper topic, you actually break the rule number 1 of this forum.
viewtopic.php?f=9&t=6

For example, the title of this post is "An interesting problem". You can see below that there is already another topic with the same name! You could use something like find when sum is equal to product...I just want to request everyone to be careful about this thing. This is for our own good.
Moon bhaia, How to change the name? I can't see any options here?

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