### USAJMO/USAMO 2017 P1

Posted:

**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 $a+b$.

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Posted: **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 $a+b$.

Posted: **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.

Posted: **Sat Apr 22, 2017 1:33 pm**

Quite easy as USAMO #1

Posted: **Sat Apr 22, 2017 3:19 pm**

Are you sure that helps? $a$ and $b$ need to be coprime.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.