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BMO Question: Two Equations, Three Variables By Kit George xyzൌ4018 x 2 y 2 z 2 ൌ2yz2 Find all possible integer values of x, y and z.
SOLUTION
x 2 y 2 z 2 ൌ2yz2 y 2 z 2 -2yzൌ2-x 2 ሺy-zሻ 2 ൌ2-x 2 y-zൌ√ሺ2-x 2 ሻ ∴ √ሺ2-x 2 ሻ is a real number ∴ ሺ2-x 2 ሻ is positive ∴ x 2 2 ∴ -√2x√2 x must be an integer ∴ xൌ-1, 0 or 1 ∴ y-zൌ√1 or √2 ൌ1, -1 or √2 y and z are integers ∴ y-zൌ1 ∴ xൌ-1 or 1 xyzൌ4018 yzൌ4018-x
∴ yzൌ4019, if xൌ-1, or 4017, if xൌ1 y-zൌ-1 ∴ If xൌ-1, yൌ2010, zൌ2009 or yൌ2009, zൌ2010
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