Math contest proof problem

Question

Could someone help me with this?

Let $x,y,z,w$ be positive real numbers such that $x + y + z = w.$ Show that $${(w−x)(w−y)(w−z) \over (w + x)(w + y)(w + z)} \le \frac 18$$

Note that because the equation is homogeneous you can rescale if you want so, for example, $w=1$ — Mark Bennet, Aug 27 '13 at 09:55

score 5 · Answer 1 · answered Aug 27 '13 at 14:21

5

Use the substitution $a = y+z, b = z+x, c = x+y$, then the inequality is

$$ \frac{ abc} {(a+b)(b+c)(c+a) } \leq \frac{1}{8}. $$

Cross multiply and apply $AM-GM$ directly to the terms, to get

$$ (a+b)(b+c)(c+a) \geq 2 \sqrt{ab} \times 2 \sqrt{bc} \times 2 \sqrt{ca} = 8abc$$

answered Aug 27 '13 at 14:21

Calvin Lin

68,864

That must be it! Thanks so much for the help, it was a much more simpler/elegant approach than some of the cumbersome messes that could be used. – user87611 just now edit delete – Aug 27 '13 at 22:45

score 1 · Answer 2 · answered Aug 27 '13 at 10:19

1

With a bit of algebra, you can show this is equivalent to: $$x^2 (y+z)+y^2 (x+z)+z^2 (x+y)\le2 \left(x^3+y^3+z^3\right)$$ Then note that if $x=y=z$, equality holds. Now show that if, say, $z>y>x$, then the left hand side must be smaller.

answered Aug 27 '13 at 10:19

Nathaniel Bubis

33,018

2

$x^3+y^3 \ge x^2y+y^2x$,you get answer directly. – chenbai Aug 27 '13 at 11:27

Math contest proof problem

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