let $a>b>c>d$ be positive integers and suppose
$ac+bd=(b+d+ac)(b+da+c)$.
prove that $ab+cd$ is composite.[easiest IMO6 of 21th century from my point of view ]
IMO20016
 Tahmid Hasan
 Posts: 665
 Joined: Thu Dec 09, 2010 5:34 pm
 Location: Khulna,Bangladesh.
IMO20016
বড় ভালবাসি তোমায়,মা

 Posts: 461
 Joined: Wed Dec 15, 2010 10:05 am
 Location: Dhaka
 Contact:
Re: IMO20016
First of all, feeling easy and hard is relative. And i think until you see the solution, the intuitions are really unique. So in my point of view, every IMO problem is quite unique in contest and has a beautiful solution.
The real trick (I'm consenting that I've seen this part from an article)
With the help of this hint, i have solved the problem. So, everyone should work alone after this hint...
The real trick (I'm consenting that I've seen this part from an article)
You spin my head right round right round,
When you go down, when you go down down......($from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......($from$ "$THE$ $UGLY$ $TRUTH$" )
 FahimFerdous
 Posts: 176
 Joined: Thu Dec 09, 2010 12:50 am
 Location: Mymensingh, Bangladesh
Re: IMO20016
I totally agree with Sourav. I also used the hint that Sourav gave and solved the problem. I got the hint from 'How To Solve It'. But it's true that the official solution is much more beautiful and cruxy (a new adjective maybe) than this one and needs unique ideas.
Your hot head might dominate your good heart!
 Tahmid Hasan
 Posts: 665
 Joined: Thu Dec 09, 2010 5:34 pm
 Location: Khulna,Bangladesh.
Re: IMO20016
oops,i forgo post post a reply . actually i used the same lemma as well.
another lemma i used was if $\frac {a^2+b^2c^2d^2}{cos x}$ is a natural number for $0^{\circ}<x<180^{\circ}$ then $cos x$ is rational.
@Sourav vaiya,could you please share a link to the article if possible?
another lemma i used was if $\frac {a^2+b^2c^2d^2}{cos x}$ is a natural number for $0^{\circ}<x<180^{\circ}$ then $cos x$ is rational.
@Sourav vaiya,could you please share a link to the article if possible?
বড় ভালবাসি তোমায়,মা
 FahimFerdous
 Posts: 176
 Joined: Thu Dec 09, 2010 12:50 am
 Location: Mymensingh, Bangladesh
Re: IMO20016
Your hot head might dominate your good heart!