Question

a^3 + b^3 + c^3 = 3abc and a, b, c are positive numbers, then prove that a = b = c.​

Answer 1

Answer:

Step-by-step explanation:

WKT

$a^3+b^3+c^3=3abc-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\\\\a^3+b^3+c^3=3abc-\frac{1}{2}(a+b+c)(2a^2+2b^2+2c^2-2ab-2bc-2ca)\\\\a^3+b^3+c^3=3abc-\frac{1}{2}(a+b+c)((a-b)^2+(b-c)^2+(c-a)^2)\\\\given\\\\a^3+b^3+c^3=3abc\\\\=> (a+b+c)((a-b)^2+(b-c)^2+(c-a)^2)=0\\\\either\\a+b+c=0\\\\or\\\\(a-b)^2+(b-c)^2+(c-a)^2=0\\\\=>a=b\;and\;b=c\;and\;c=a\\\\=>a=b=c\\\\Hence \;Proved$

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