### triples (x,y,z) with particular properties

Posted:

**Wed May 15, 2013 1:52 pm UTC**Hello everyone, this problem is taken from the entrance test of an Italian university famous for being extremely hard, this is the first problem of the 13 pages long entrance test and I already have no idea on what to do with it (no, I'm not planning to even try this test!):

the original text for Italian speakers is:

"Si trovino le terne di numeri reali (x,y,z) con la proprieta' che la quarta potenza di ciascuno di essi e' uguale alla somma degli altri due."

That translates as:

"Find the triples of real numers (x,y,z) with the property that the fourth power of each of them is equal to the sum of the other two"

which is equivalent to the system of 3 equations:

x^4=y+z

y^4=x+z

z^4=y+x

but I cannot do anything better than pointing out the trivial solution for (0,0,0), any hint on how to solve this problem?

Thank in advance for any help!

the original text for Italian speakers is:

"Si trovino le terne di numeri reali (x,y,z) con la proprieta' che la quarta potenza di ciascuno di essi e' uguale alla somma degli altri due."

That translates as:

"Find the triples of real numers (x,y,z) with the property that the fourth power of each of them is equal to the sum of the other two"

which is equivalent to the system of 3 equations:

x^4=y+z

y^4=x+z

z^4=y+x

but I cannot do anything better than pointing out the trivial solution for (0,0,0), any hint on how to solve this problem?

Thank in advance for any help!