Cool but may be tough(relatively) (BOMC-2)

Discussion on Bangladesh National Math Camp
sourav das
Posts: 461
Joined: Wed Dec 15, 2010 10:05 am
Location: Dhaka
Contact:

Cool but may be tough(relatively) (BOMC-2)

Unread post by sourav das » Thu Mar 29, 2012 9:49 pm

(Russian Math Olympiad 1999, Grade 11, #5) Four natural numbers have the property that
the square of the sum of any two of the numbers is divisible by the product of the other two.
Show that at least three of the four numbers are equal.
You spin my head right round right round,
When you go down, when you go down down......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

User avatar
*Mahi*
Posts: 1175
Joined: Wed Dec 29, 2010 12:46 pm
Location: 23.786228,90.354974
Contact:

Re: Cool but may be tough(relatively) (BOMC-2)

Unread post by *Mahi* » Thu Apr 05, 2012 12:56 am

Sorry for being late. I did not notice it earlier.
Steps:
1. Let us assume $\gcd (a,b,c,d)=1$. Then it can be shown that $\text{cyclic } \gcd(a,b)=1$
2. Then we have to show that $\text{cyclic} ab \mid 2(c+d)$
3.From that, inequality gives us $\text{cyclic }(a-1)(c+d) \leq 2b\;$, which can be manipulated to show that at least three of $a,b,c,d$ is $1$.
Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

User avatar
SANZEED
Posts: 550
Joined: Wed Dec 28, 2011 6:45 pm
Location: Mymensingh, Bangladesh

Re: Cool but may be tough(relatively) (BOMC-2)

Unread post by SANZEED » Thu Apr 05, 2012 12:59 am

I haven't solved it fully yet,but a useful hint:
If an integer $p$ divides $a$,then find the property of $p$ for this problem.
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$

User avatar
*Mahi*
Posts: 1175
Joined: Wed Dec 29, 2010 12:46 pm
Location: 23.786228,90.354974
Contact:

Re: Cool but may be tough(relatively) (BOMC-2)

Unread post by *Mahi* » Thu Apr 05, 2012 1:18 am

Share your full thought and progress... no need for "hints of progress".
Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

User avatar
zadid xcalibured
Posts: 217
Joined: Thu Oct 27, 2011 11:04 am
Location: mymensingh

Re: Cool but may be tough(relatively) (BOMC-2)

Unread post by zadid xcalibured » Thu Mar 07, 2013 6:16 pm

SANZEED wrote:I haven't solved it fully yet,but a useful hint:
If an integer $p$ divides $a$,then find the property of $p$ for this problem.
সত্যি এই ধরনের কমেন্ট দেখলে মেজাজ গরম হয়। :evil:

Post Reply