# Problem 5b

Solve for $$x,y,z$$
$\frac { x }{ y } +\frac { y }{ z } +\frac { z }{ x } =\frac { y }{ x } +\frac { z }{ y } +\frac { x }{ z } =x+y+z=3$
This a part of my set NMTC 2nd Level (Junior) held in 2014.

Note by Siddharth G
6 years, 6 months ago

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$\dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{x}\ge 3$ by AM-GM, with equality case if and only if $x=y=z$

Since $x+y+z=3$, then we know the only real solution is $(x,y,z)=\boxed{(1,1,1)}$

- 6 years, 6 months ago

Good observation.

Note that the application of AM-GM assumes that the variables are positive.
Are there solutions over all reals?
What about over all complex numbers?

Staff - 6 years, 6 months ago

$minimum\quad value\quad of\quad \frac { x }{ y } \quad +\quad \frac { y }{ z } +\frac { z }{ x } =3\quad \\ from\quad this\quad how\quad can\quad we\quad conclude\quad that\quad x=y=z????$

- 6 years, 6 months ago

Hint: Make common denominator for$\frac{ x}{y} + \frac{y}{z} + \frac{z}{x} - \frac{ y}{x} - \frac{ z}{y} - \frac{ x}{z}$.
Factorize the numerator.

Staff - 6 years, 6 months ago

$\frac{ x}{y} + \frac{y}{z} + \frac{z}{x} - \frac{ y}{x} - \frac{ z}{y} - \frac{ x}{z} = \frac{z-y}x+\frac{x-z}y+\frac{y-x}z=\frac{(z-y)zy+(x-z)xz+(y-x)yx}{xyz}=\frac{(y-z)(x-z)(y-x)}{xyz}=0$

Therefore $x,y,z\ne0$ and ($y=z$ or $x=z$ or $y=x$).

- 6 years, 6 months ago

Great work.

This is a short step away from finishing the solution.

Staff - 6 years, 6 months ago

When $y=z$, $\frac xy+1+\frac yx=\frac yx+1+\frac xy=x+2y=3$

$\frac xy+\frac yx=2$, $\frac xy=1$, $x=y$.

Therefore, $x=y=z$.

From $x+y+z=3$, we have $x=y=z=1$ as the only real solution.

- 6 years, 6 months ago

x=1,y=1,z=1

- 6 years, 6 months ago

Write a comment or ask a question... (1,1,1)

- 6 years, 6 months ago

How do you know that there are no other solutions?

Staff - 6 years, 6 months ago

x=y=z=1 therefore the answer is 3 ,basic algebra question

- 6 years, 6 months ago