## An Interesting problem

For discussing Olympiad Level Number Theory problems
protik
Posts: 35
Joined: Wed Dec 08, 2010 7:21 am

### An Interesting problem

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.

Labib
Posts: 411
Joined: Thu Dec 09, 2010 10:58 pm

### Re: An Interesting problem

as much as I know $\pm 6\ and 0$ satisfy it!! haven't really tried it yet!
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"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes

Moon
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
Contact:

### Re: An Interesting problem

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

learn how to write equations, and don't forget to read Forum Guide and Rules.

Tahmid Hasan
Posts: 665
Joined: Thu Dec 09, 2010 5:34 pm

### Re: An Interesting problem

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

Labib
Posts: 411
Joined: Thu Dec 09, 2010 10:58 pm

### Re: An Interesting problem

@ tahmid, If $k=3, a_1=1, a_2=2, a_3=3$ then $n=6=1\times 2\times 3=1+2+3$
Please Install $L^AT_EX$ fonts in your PC for better looking equations,
Learn how to write equations, and don't forget to read Forum Guide and Rules.

"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes

protik
Posts: 35
Joined: Wed Dec 08, 2010 7:21 am

### Re: An Interesting problem

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?