Chứng minh rằng: (2a + 2b - c/3)^3 + (2b + 2c - a/3)^2 + (2c + 2a - b/3) = a^2 + b^2 + c^2

Để chứng minh rằng (2a+2b-c/3)^3+(2b+2c-a/3)^2 +(2c+2a-b/3)=a^2+b^2+c^2, ta sẽ chứng minh từng phần bên trái và bên phải của phương trình.

Bên trái của phương trình: (2a+2b-c/3)^3+(2b+2c-a/3)^2 +(2c+2a-b/3)

= (8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27) + (4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9) + (4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9)

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b - 4ac^2 + 12ab^2 + 8b^3 - 4bc^2 - 4a^2c - 4b^2c + c^3/27 + 4b^2 + 4c^2 + 4bc - 4ac - 4ab + a^2/9 + 4c^2 + 4a^2 + 4ac - 4bc - 4ab + b^2/9

= 8a^3 + 12a^2b -