2005 Indonesia MO Problems/Problem 6

Problem

Find all triples $(x,y,z)$ of integers which satisfy

$x(y + z) = y^2 + z^2 - 2$

$y(z + x) = z^2 + x^2 - 2$

$z(x + y) = x^2 + y^2 - 2$ .

Solution

By using the Distributive Property on all three equations, we get $\begin{align*} xy + xz &= y^2 + z^2 - 2 \\ yz + yx &= x^2 + z^2 - 2 \\ zx + zy &= x^2 + y^2 - 2. \end{align*}$ Add all three equations, rearrange, and factor to get $\begin{align*} 2xy + 2yz + 2xz &= 2x^2 + 2y^2 + 2z^2 - 6 \\ 6 &= x^2 - 2xy + y^2 + y^2 - 2yz + z^2 + z^2 - 2xz + x^2 \\ 6 &= (x-y)^2 + (y-z)^2 + (z-x)^2. \end{align*}$ Notice that since all the variables are integers, the squares must also be integers. The only combination of three integer squares that sum to six is $1, 1, 4$ .

Also note that the system is symmetric, so any permutation of a solution will work. Therefore, we can, WLOG, let $x \ge y \ge z$ .

Therefore, $(x-y)^2 = 1$ , $(y-z)^2 = 1$ , and $(x-z)^2 = 4$ , so $x-y = 1$ , $y-z = 1$ , and $x-z = 2$ . By rearrangement, $y = x-1$ and $z = x-2$ .

Substitute and solve to get $\begin{align*} x(x-1) + x(x-2) &= 2x^2 - 6x + 5 - 2 \\ 2x^2 - 3x &= 2x^2 - 6x + 3 \\ 3x &= 3 \\ x &= 1 \end{align*}$ Therefore, one solution is $(1,0,-1)$ , and plugging that value back in satisfies the system. Since any permutation is also a solution, the solutions are $\boxed{(1,0,-1), (1,-1,0), (0,1,-1), (0,-1,1), (-1,0,1), (-1,1,0)}$ , and plugging all the values back into the original system satisfies the system.

2005 Indonesia MO (Problems)
Preceded by Problem 5	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8	Followed by Problem 7
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2005 Indonesia MO Problems/Problem 6

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