# Thread: Simultaneous Equations (BMO Past Question)

1. ## Simultaneous Equations (BMO Past Question)

So yeah, I'm looking at British Mathematical Olympiad questions, and this one has me utterly stumped. Any ideas?

Find all integers x, y and z such that:

$x^2 + y^2 + z^2 = 2(yz + 1)$
And
$x + y + z = 4018.$

2. Originally Posted by SuperCalculus
So yeah, I'm looking at British Mathematical Olympiad questions, and this one has me utterly stumped. Any ideas?

Find all integers x, y and z such that:

$x^2 + y^2 + z^2 = 2(yz + 1)$
And
$x + y + z = 4018.$
Write the first equation as $x^2 + (y-z)^2 = 2$. Then clearly both x and y–z must be 1 or –1. Is that enough of a hint?

3. "Then clearly both x and y–z must be 1 or –1."

How the hell did you get that? O_o

4. Originally Posted by satx
"Then clearly both x and y–z must be 1 or –1."

How the hell did you get that? O_o
What other integers could make 2 when squared and added?

5. Oh, my bad. Didn't see the "integers" part.

6. Originally Posted by Opalg
Write the first equation as $x^2 + (y-z)^2 = 2$. Then clearly both x and y–z must be 1 or –1. Is that enough of a hint?
Yeah, I reckon I've got it providing that is true, but can you tell me how you got to that equation? (I understand how it can only be 1 or -1)

7. Originally Posted by SuperCalculus
Originally Posted by Opalg
Write the first equation as $x^2 + (y-z)^2 = 2$. Then clearly both x and y–z must be 1 or –1. Is that enough of a hint?
Yeah, I reckon I've got it providing that is true, but can you tell me how you got to that equation? (I understand how it can only be 1 or -1)
If $x^2+y^2+z^2 = 2(yz+1) = 2yz+2$ then $x^2+y^2-2yz+z^2 = 2$. If you're aiming to do olympiad questions then you ought to be able to factorise $y^2-2yz+z^2$.

8. Originally Posted by Opalg
If $x^2+y^2+z^2 = 2(yz+1) = 2yz+2$ then $x^2+y^2-2yz+z^2 = 2$. If you're aiming to do olympiad questions then you ought to be able to factorise $y^2-2yz+z^2$.
Ugh, I knew it was something obvious. I was thinking in all these complicated terms, and all I had to do was basic rearranging >.>