2000 Pan African MO Problems/Problem 4

Problem

Let $a$, $b$ and $c$ be real numbers such that $a^2+b^2=c^2$, solve the system: \[z^2=x^2+y^2\] \[(z+c)^2=(x+a)^2+(y+b)^2\] in real numbers $x, y$ and $z$.

Solution

Expanding the last equation and simplifying results in \begin{align*} z^2 + 2zc + c^2 &= x^2 + 2xa + a^2 + y^2 + 2yb + b^2 \\ 2zc &= 2xa + 2yb \\ zc &= xa + yb. \end{align*} Isolating $z$ means that $z = \frac{xa+yb}{c}$. Substituting $z$ in the second equation results in \begin{align*} \frac{(xa)^2 + 2xayb + (yb)^2}{c^2} &= x^2 + y^2 \\ (xa)^2 + 2xayb + (yb)^2 &= (xc)^2 + (yc)^2 \\ 2xayb &= (xb)^2 + (ya)^2 \\ 0 &= (xb-ya)^2  \end{align*} By the Zero Product Property, $xb = ya$. If $a = 0$, then either $b = 0$ or $x = 0$. Thus, ordered pair $(a,b,c,x,y,z)$ can be $(0,0,0,x,y,z)$ or $(0,b,b,0,y,y)$. Otherwise, $x = \frac{ya}{b}$, so another ordered pair can be $(a,b,c,\frac{ya}{b},y,\frac{yc}{b})$. Setting $\frac{y}{b} = n$ means that the ordered pair can be rewritten as $(a,b,c,na,nb,nc)$.


Because the case $(0,b,b,0,y,y)$ is part of $(a,b,c,na,nb,nc)$, the ordered pairs that are solutions are $\boxed{(0,0,0,x,y,z), (a,b,c,na,nb,nc)}$.

See Also

2000 Pan African MO (Problems)
Preceded by
Problem 3
1 2 3 4 5 6 Followed by
Problem 5
All Pan African MO Problems and Solutions
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