This solution is from Chong Ching Tong, Chan Hei Leong, Chen Wei Jian and Ng Yan Shun from River Valley High School, Singapore.

If a + b + c = 4, ab + bc + ca = 6 and abc = 3, then

1/a + 1/b + 1/c = (bc + ac +ab) / abc
= 6/3
= 2

1/ab + 1/bc + 1/ca = c/abc + a/abc + b/abc
= ( c + a + b )/abc
= 4/3

Mondrian shape.

In the diagram the coloured squares have sides of length a, b and c. From the areas in the diagram we get

(a + b + c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca.

a 2 + b 2 + c 2 = (a + b + c) 2-2ab + bc + ca = 16 - 12 = 4.

Now think of a cube where each face is cut up in a similar way splitting the cube into 27 smaller cuboids. The volume of the cube is (a + b + c) 3 . We get the formula for the expansion of (a + b + c) 3 by adding the 27 volumes, as given in the table below.

Cube.

Top Layer a 3 2a 2b 2a 2c ab 2 ac 2 2abc
Middle Layer b 3 a 2b 2ab 2 2bc 2 bc 2 2abc
Bottom Layer c 3 a 2c 2ac 2 b 2c 2bc 2 2abc

Adding the volumes we get

(a+b+c) 3 = a 3 + b 3 + c 3+ 3a 2b+3a 2c+3ab 2+3ac 2 +3b 2c+3bc 2+6abc.