iran mathematical olympiad

Discussion on International Mathematical Olympiad (IMO)
mahathir
Posts: 24
Joined: Tue Feb 15, 2011 11:01 pm

iran mathematical olympiad

Unread post by mahathir » Sat Sep 24, 2011 12:05 am

if $a,b,c,d$ are natural numbers and $ab=cd$,then,prove that,$a+b+c+d$ cannot be a prime.

User avatar
Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm
Location: Dhaka,Bangladesh

Re: iran mathematical olympiad

Unread post by Masum » Sun Sep 25, 2011 7:22 pm

Try to prove the general one.
If $ab=cd$, prove that $a^n+b^n+c^n+d^n$ is not prime.
One one thing is neutral in the universe, that is $0$.

User avatar
nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

Re: iran mathematical olympiad

Unread post by nayel » Sun Sep 25, 2011 9:37 pm

The following method works for many problems of similar sort.
Let $g=(a,c), a=ga', c=gc'$. Then $a'b=c'd$ so $a'\mid c'd$. But $(a',c')=1$, hence $a'\mid d$. Let $d=ha'$. Then $a'b=c'ha'$ so $b=hc'$. Thus $a^n+b^n+c^n+d^n=(ga')^n+(hc')^n+(gc')^n+(ha')^n=(g^n+h^n)(a'^n+c'^n)$.
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

Post Reply