Question

CMR : nếu \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=2\)

Và 1 + b + c= abc

Thì \(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=2\)

Answer 1

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac.
(1/a + 1/b + 1/c)² = 1/a² + 1/b² + 1/c² + 2(1/ab + 1/bc + 1/ac) = 4
<=> 1/a² + 1/b² + 1/c² + 2(bcac + abac + abbc)/(a²b²c²) = 4
<=> 1/a² + 1/b² + 1/c² + 2abc(a + b + c)/(a²b²c²) = 4
<=> 1/a² + 1/b² + 1/c² + 2 = 4
(vì abc(a + b + c) = a² b² c²)
<=> 1/a² + 1/b² + 1/c² = 2