IMO-2001-Problem-4

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sourav das
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IMO-2001-Problem-4

Unread post by sourav das » Wed Mar 28, 2012 6:39 pm

Let $n>1$ an odd integer and $c_1,c_2,c_3....c_n$ be integers. For each permutation $a=(a_1,a_2,...,a_n)$ of {$1,2,3,...n$} define $S(a)=\sum_{i=1}^{n}c_ia_i$. Prove that there exist permutations ($a$ not equal to $b$) of {$1,2,,,n$} such that $n$! divides $S(a)-S(b)$.
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itsnafi
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Re: IMO-2001-Problem-4

Unread post by itsnafi » Wed Mar 28, 2012 9:11 pm

Are $a_1,a_2,...,a_n$ also integers?
Besides,pls describe the problem in details in bengali.

sourav das
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Re: IMO-2001-Problem-4

Unread post by sourav das » Wed Mar 28, 2012 9:37 pm

As ($a_1,a_2,...a_n$) is permutation of {$1,2,3...n$} so, yes, they are integers.
Problem In Bengali (খাঁটি সোজা বাংলায় :) )

$n>1$ একটা বিজোড় সংখ্যা। মনে কর $c_1,c_2,c_3...c_n$ হল কিছু fix পূর্ণসংখ্যা । এখন মনে কর {$1,2,...n$} সংখ্যাগুলোর একটা বিন্যাস ($a_1,a_2,a_3,....a_n$) (উদাহরণঃ ($3,5,4,1,2,6,7....n$) হল এরকম একটা বিন্যাস)। এরকম একটা বিন্যাস কে একটা নাম দিয়ে উল্লেখ করা হলঃ $a=(a_1,a_2,a_3,....a_n)$. এখন প্রতিটা নির্দিষ্ট বিন্যাস যেমন $a$ এর জন্য $S(a)$ হল $c_1*a_1+c_2*a_2...+c_n*a_n$. প্রমাণ করতে হবে যে, এমন দুইটা ভিন্ন ভিন্ন বিন্যাস(অবশ্যই {$1,2,...n$} এর) $a,b$ পাওয়া যাবে যাতে $n$!, $S(a)-S(b)$ কে ভাগ করে।
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FahimFerdous
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Re: IMO-2001-Problem-4

Unread post by FahimFerdous » Wed Mar 28, 2012 9:41 pm

I solved it before and my solution was similar to the official one. Actually it's an obvious way. So, here's a hint (those who don't want to read, close your eyes, because I can't hide it from mobile):
try to work with residue class of (mod n!)

Edited
Moderator's note:Now you don't have to close your eyes :mrgreen:
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*Mahi*
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Re: IMO-2001-Problem-4

Unread post by *Mahi* » Wed Mar 28, 2012 11:08 pm

General Hint:
Whenever you see something like $x | a-b$ try converting it to $a \equiv b \text{ (mod x)}$. It kills the problem a lot, mind it, A LOT of times.
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Phlembac Adib Hasan
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Re: IMO-2001-Problem-4

Unread post by Phlembac Adib Hasan » Thu Mar 29, 2012 11:00 am

Nice problem. :D I have solved it just now.I also used residue class to build a contradiction.(But I didn't see the hints before solving it.Actually these come first to (almost) everyone's head seeing such problems. :) )
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