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Solve for positive integers x, y & z :

$x + y - z = 4$

$x^2 - y^2 + z^2 = 4$

$xyz = 6$

Note by Dev Sharma 3 years, 10 months ago

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2^{34}

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\frac{2}{3}

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Usual Method:

There're no integral solutions. As $(x,y,z)$ will only be the 9 permutations of $(1,1,6)$ and $(1,2,3)$. None of them satisfy.

Brute Force Method:

I also did some nonsense calculations to find the values:

$x+y = 4+z, xy=\dfrac{6}{z}$

Thus I obtained $x-y = \sqrt{\dfrac{z^3+8z^2+16z-24}{z}}$

$(x+y)(x-y) = 4 - z^2 \Rightarrow (x+y)^2(x-y)^2 = (4 - z^2)^2$

$\Rightarrow (4+z)^2 \left(\dfrac{z^3+8z^2+16z-24}{z} \right)= (4 - z^2)^2$

And then visit this.

No integral solutions obtained by Brute Force as well.

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Thanks sir.

But how do you find x - y

$(x-y)^2 = (x+y)^2 - 4xy$

@Dev Sharma : Is this any of NMTC's problems? Nowadays, NMTC is trending among the young students of India!

@Satyajit Mohanty – sir, it RMO problem. What is full form of NMTC?

@Dev Sharma – National Mathematics Talent Contests

@Satyajit Mohanty – Sir, are you professor?

@Dev Sharma – As you can see, I'm only 18 years old. So, I'm a student, not a professor :D And please don't call me Sir. Sounds weird :/

@Nihar Mahajan @Swapnil Das @Calvin Lin @Niranjan Khanderia @Satyajit Mohanty

I made it to a bi quadratic in z but no integral sol.

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$</code> ... <code>$</code>...<code>."> Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

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boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

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Remember to wrap math in $</span> ... <span>$ or $</span> ... <span>$ to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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TopNewestUsual Method:There're no integral solutions. As $(x,y,z)$ will only be the 9 permutations of $(1,1,6)$ and $(1,2,3)$. None of them satisfy.

Brute Force Method:I also did some nonsense calculations to find the values:

$x+y = 4+z, xy=\dfrac{6}{z}$

Thus I obtained $x-y = \sqrt{\dfrac{z^3+8z^2+16z-24}{z}}$

$(x+y)(x-y) = 4 - z^2 \Rightarrow (x+y)^2(x-y)^2 = (4 - z^2)^2$

$\Rightarrow (4+z)^2 \left(\dfrac{z^3+8z^2+16z-24}{z} \right)= (4 - z^2)^2$

And then visit this.

No integral solutions obtained by Brute Force as well.

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Thanks sir.

But how do you find x - y

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$(x-y)^2 = (x+y)^2 - 4xy$

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@Dev Sharma : Is this any of NMTC's problems? Nowadays, NMTC is trending among the young students of India!

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National Mathematics Talent Contests

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@Nihar Mahajan @Swapnil Das @Calvin Lin @Niranjan Khanderia @Satyajit Mohanty

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I made it to a bi quadratic in z but no integral sol.

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