Olympiad problems

2015 India IMO Training Camp, 2

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2015 Taiwan TST Round 2, 3

Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2015 India IMO Training Camp, 2

Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2014 IMO Shortlist, N5

Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2015 Brazil Team Selection Test, 2

Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2015 Ukraine Team Selection Test, 3

Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]