x 2 y 2 z 2 6 xy yz zx 3 Prove that x 2 y 2 z 2 xy yz zx is a factor of x y n y z n z x n if n is not divisible by 3
Your proof is right but maybe it looks a bit of better by using a cyclic sum sum cyc x 2 xy frac 1 2 sum cyc 2x 2 2xy frac 1 2 sum cyc x 2 A working and fast proof color red 0 leq x y 2 x z 2 y z 2 2 color red left x 2 y 2 z 2 xy xz yz right
x 2 y 2 z 2 6 xy yz zx 3
x 2 y 2 z 2 6 xy yz zx 3
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If Xy yz xz 0 Then Find 1 x 2 Yz 1 y 2 Xz 1 z 2 Xy
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X 2 y 2 z 2 Greater Than Xy yz xz Proof Important For IIT JEE NDA
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So x y z 1 2 3 is a solution actually 3 6 solutions since any order of x y z is possible Bonus If there was not a simple solution that could be guessed how would you If x y z are positive real numbers prove that x y z 2 yz zx xy 2 leq 3 y 2 yz z 2 z 2 zx x 2 x 2 xy y 2 My best idea was to expand this and simplify Although that doesn t look very feasible
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x 2 y 2 z 2 6 xy yz zx 3 - So x y z 1 2 3 is a solution actually 3 6 solutions since any order of x y z is possible Bonus If there was not a simple solution that could be guessed how would you