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:

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

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:

$\displaystyle x^2 + y^2 + z^2 = 2(yz + 1)$
And
$\displaystyle x + y + z = 4018.$
Write the first equation as $\displaystyle x^2 + (y-z)^2 = 2$. 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 $\displaystyle x^2 + (y-z)^2 = 2$. 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 $\displaystyle x^2 + (y-z)^2 = 2$. 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 $\displaystyle x^2+y^2+z^2 = 2(yz+1) = 2yz+2$ then $\displaystyle x^2+y^2-2yz+z^2 = 2$. If you're aiming to do olympiad questions then you ought to be able to factorise $\displaystyle y^2-2yz+z^2$.

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