If x, y, z are real numbers, Prove that y(z-x) < 4

[Math Tutor]

x, y, z are real numbers.
[tex]x^2+z^2=1[/tex] and [tex]y^2+2y(x+z)-6=0[/tex]
Prove that [tex]y(z-x)\le 4[/tex]

[Guest]

In order to give you a magic solution for this inequality you need to give me the most elementary proof
for the well-know problem with the broken stick (maybe it`s time to show that you know mathematics)

Problem: Pick two points uniformly at random on the stick, and break the stick at those points. What is the
probability that the three segments obtained in this way form a triangle?


YS.(when i say "the most elementary proof" i refer that even a child may use that way)

[Guest]

can't be proven ...
if you take [tex]y=3[/tex] than [tex]x+z=-0,5[/tex] and we know that [tex]x^2+z^2=1[/tex]
solving the system we get 2 roots one of them is [tex]z=\frac{\sqrt {7}-1 }{2}[/tex] and thus [tex]x=-\frac{\sqrt {7}}{2}[/tex]
and ...
[tex]3(\frac{\sqrt {7}-1 }{2}-(-\frac{\sqrt {7}}{2}))\le 4[/tex] is not true

i get to the bottom of this task as fallows ..
[tex]x=sin\varphi[/tex] and therefore [tex]z=cos\varphi[/tex]
after this its all about finding max and min of [tex]x+z[/tex] and [tex]z-x[/tex] (with derivatives) and a bit of manipulation of the second equality

[Guest]

You just entangled my plans with your explanations ... .

YS.

[Math Tutor]

Of course there is an elegant solution.
Try this:

[tex]z+x=u; \;z-x=v\Rightarrow u^2+v^2=2[/tex].

[Guest]

Of course: stick problem posted above has an elegant solution , as well. YS

[Math Tutor]

Can you solve the problem YS using the substitutions above?
I am curious.

[Guest]

I'm curious if all mathematicians in Bulgaria can solve my problem in the way I asked for. Of course I can solve it and
solved problems and issued theorems you`ll never dream of. (Working now on Riemann hypothesis)


YS.

[Math Tutor]

Have in mind that Bulgaria is every year in top 10 list of International Mathematical Olympiad and the first International Olympiad in Informatics was held in Bulgaria.

[shemet]

Have in mind that Bulgaria is every year in top 10 list of International Mathematical Olympiad and the first International Olympiad in Informatics was held in Bulgaria.

Are you crazy?!?!?!

[Guest]

Well, then it means that there is no problem to receive an answer for stick problem where it is required to provide with a solution that is at hand of any pupil of 7 years old or less. Maybe i`m a bit crazy but i`m not joking. Still waiting for that
elementary solution.

YS. (i don`t consider myself a mathematician but an amateur - having said all these
i dare to expect more from any mathematician)

[Guest]

One more thing: don`t google it cause you just waste your time {this was a helping hint}

[sajid121]

'm curious if all mathematicians in Bulgaria can solve my problem in the way I asked for. Of course I can solve it and
solved problems and issued theorems you`ll never dream of. (Working now on Riemann hypothesis)

[Guest]

then it means that there is no problem to receive an answer for stick problem where it is required to provide with a solution that is at hand of any pupil of 7 years old or less. Maybe i`m a bit crazy but i`m not joking. Still waiting for that
elementary solution.