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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, 4 months ago

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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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## 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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