Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 230

Find all the real

Find all the real a,b,c such that the equality

is valid for all the real x,y,z.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Find all the real

If the equation holds for all

it must hold in particular for

And for

Unfortunately, that’s not good enough.

I believe there are six solutions:

Last edited by JaneFairfax (2008-10-26 05:16:27)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Find all the real

Okay, try this.

First assume that

Now

means

must be either all non-negative or all non-positive, i.e. either

or

Putting

gives

Now think about it. The sum of three numbers having the same sign is ±1, and the difference between the largest and smallest of the three is 1. What does this mean? Well, it means that either

or

Considering all the possible orderings of

should yield all the six possible solutions.

Last edited by JaneFairfax (2008-10-26 06:01:24)