An easy system with radicals

[ins-]

Rak so dna, nnosipov thank you both for the time and effort put on this problem as well as interesting ideas and materials.

[ins-]

I found something interesting. In a book was given a different way to solve this system and an answer in the form: $(\frac{a}{b^2+c^2-a^2}, \frac{b}{c^2+a^2-b^2}, \frac{c}{a^2+b^2-c^2})$. If the answer given from Rak so dna is correct there are dependencies between $a$, $b$ and $c$.

[Rak so dna]

ins- this solution is incorrect as the degree $x(a,b,c), y(a,b,c), z(a,b,c)$ must be 1. You have $(-1)$

[ins-]

Rak so dna, proably you can see the critical part here:
https://services.artofproblemsolving.co ... J0LmpwZw==
I think the idea, even with a wrong implementation, is interesting.

[ins-]

You are correct. Their mistake is in solving the last linear system.

[ins-]

Rak so dna

Excuse me for reviving an old topic, but I found something interesting about this system. It appears also here

http://booksshare.net/index.php?id1=4&c ... &book=1960

(Chapter 13, problem 10, page 90 and page 508 from the djvu file)

with an interesting geometric interpretation (in the solution) and answers with some mistakes.

[Rak so dna]

ins- I think that standard systems are best solved by standard methods. And try these methods when others did not work.

[ins-]

Rak so dna You are right. Geometric, trigonometric etc. ways are useful when pure algebra is hard to be used, to have a different point of view on a problem and sometimes for creation of new problems.