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Mayra Berry 2022-06-27 Answered
If
x = c y + b z y = a z + c x z = b x + a y
where x , y , z are not all zero, prove that a 2 + b 2 + c 2 + 2 a b c = 1
Further if at least one of a , b , c is a proper fraction, prove that:
(i) a 2 + b 2 + c 2 < 3
(ii) a b c > 1
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Answers (1)

Kaydence Washington
Answered 2022-06-28 Author has 32 answers
Assuming a is a proper fraction, then
a 2 < 1
Considering quadratic in c,
c 2 + 2 a b c + a 2 + b 2 1 = 0
Admitting real value of c, Δ 0 4 ( a 2 1 ) ( b 2 1 ) 0 b 2 1 0
Hence,
b 2 1
Similarly,
c 2 1
Hence,
a 2 + b 2 + c 2 < 3
Also
a b c = 1 a 2 b 2 c 2 2 > 1
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