Monday, June 1, 2015

China IMO problem: Simplify $(ay+bx)^3-(ax+by)^3+(a^3-b^3)(x^3-y^3)$

Simplify $(ay+bx)^3-(ax+by)^3+(a^3-b^3)(x^3-y^3)$.

We know this is a trick problem because it couldn't be required us to expand all the three terms and then collect like terms to simplify it.

So, if we don't expand the expression, how are we going to simplify it?

If we choose the expand the first two expressions [MATH]\color{black}\bbox[5px,orange]{(ay+bx)^3-(ax+by)^3}\color{black}{+(a^3-b^3)(x^3-y^3)}[/MATH]:

We will get another $8$ single terms and it is highly likely that we need to further expand the last term, the product of two factors to simplify for more.

So, we will try to keep the first two terms but expand the last term, [MATH]\color{black}{(ay+bx)^3-(ax+by)^3+}\color{yellow}\bbox[5px,blue]{(a^3-b^3)(x^3-y^3)}[/MATH] :

[MATH]\color{yellow}\bbox[5px,blue]{(a^3-b^3)(x^3-y^3)}[/MATH]

$=a^3x^3-a^3y^3-b^3x^3+b^3y^3$

$=\,\,\,\,\,\,\,\,\underbrace{(a^3x^3+b^3y^3)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underbrace{(a^3y^3+b^3x^3)}$

$=((ax+by)^3-3axby(ax+by))-((ay+bx)^3-3aybx(ay+bx))$

$=(ax+by)^3-(ay+bx)^3-3abxy((ax+by)-(ay+bx))$

$=(ax+by)^3-(ay+bx)^3-3abxy(x(a-b)-y(a-b))$

$=(ax+by)^3-(ay+bx)^3-3abxy(x-y)(a-b)$

$=(ax+by)^3-(ay+bx)^3+3abxy(y-x)(a-b)$

Therefore, we get:

$(ay+bx)^3-(ax+by)^3+(a^3-b^3)(x^3-y^3)=3abxy(y-x)(a-b)$ and we're done.