USAJMO/USAMO 2017 P1

For discussing Olympiad Level Number Theory problems
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Zawadx
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USAJMO/USAMO 2017 P1

Unread post by Zawadx » Sat Apr 22, 2017 8:24 am

Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.

aritra barua
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Re: USAJMO/USAMO 2017 P1

Unread post by aritra barua » Sat Apr 22, 2017 12:55 pm

When $a,b$ € $N$ and it follows that $a^x$=$b^y$,there exists $t$ € $N$ such that $a$=$t^k$,$b$=$t^q$.This lemma can be quite handy in this problem.

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Atonu Roy Chowdhury
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Re: USAJMO/USAMO 2017 P1

Unread post by Atonu Roy Chowdhury » Sat Apr 22, 2017 1:33 pm

Quite easy as USAMO #1
We'll show $(a,b)=(2n+1,2n-1)$ works. It is easy to see that $gcd(2n+1, 2n-1) =1 $.
$(2n+1)^{2n-1} + (2n-1)^{2n+1} \equiv (2n+1) + (2n-1) \equiv 0 (mod 4n) $ because $(2n \pm 1)^2 \equiv 4n^2 \pm 4n +1 \equiv 1 (mod 4n) $
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Thanic Nur Samin
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Re: USAJMO/USAMO 2017 P1

Unread post by Thanic Nur Samin » Sat Apr 22, 2017 3:19 pm

aritra barua wrote:When $a,b$ € $N$ and it follows that $a^x$=$b^y$,there exists $t$ € $N$ such that $a$=$t^k$,$b$=$t^q$.This lemma can be quite handy in this problem.
Are you sure that helps? $a$ and $b$ need to be coprime.
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