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.

#Algebra

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

$\dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{x}\ge 3$ 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 $(x,y,z)=\boxed{(1,1,1)}$

Daniel Liu - 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?

Calvin Lin Staff - 6 years, 6 months ago

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

Prashant Ramnani - 6 years, 5 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.

Calvin Lin 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 $y=x$ ).

Kenny Lau - 6 years, 6 months ago

Great work.

This is a short step away from finishing the solution.

Calvin Lin Staff - 6 years, 6 months ago

@Calvin Lin – 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.

Kenny Lau - 6 years, 5 months ago

x=1,y=1,z=1

deepak bisht - 6 years, 6 months ago

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

John Wilcox - 6 years, 6 months ago

How do you know that there are no other solutions?

Calvin Lin Staff - 6 years, 6 months ago

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

Praguna Manvi - 6 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$