Find all real numbers $x,y,zin [0,1]^3$ such that $(x^2+y^2)sqrt{1-z^2}ge z$…
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3
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Such that:
$$(x^2+y^2)sqrt{1-z^2}ge z$$
and
$$(z^2+y^2)sqrt{1-x^2}ge x$$
and
$$(x^2+z^2)sqrt{1-y^2}ge y$$
Since $x,y,z$ $in ]0,1[^3$
then , there are some real numbers $a,b,c$ such that
$cos a=x, cos b=y , cos c=z$
After some manipulations , we find that :
$$frac{1}{1+tan^2 a}+frac{1}{1+tan^2 b}ge frac{1}{tan c}$$
.... same for other inequalities
I don't know what i must do now
Source : Test N°1 for IMO 2020 in Morocoo
inequality contest-math
add a comment |
up vote
3
down vote
favorite
Such that:
$$(x^2+y^2)sqrt{1-z^2}ge z$$
and
$$(z^2+y^2)sqrt{1-x^2}ge x$$
and
$$(x^2+z^2)sqrt{1-y^2}ge y$$
Since $x,y,z$ $in ]0,1[^3$
then , there are some real numbers $a,b,c$ such that
$cos a=x, cos b=y , cos c=z$
After some manipulations , we find that :
$$frac{1}{1+tan^2 a}+frac{1}{1+tan^2 b}ge frac{1}{tan c}$$
.... same for other inequalities
I don't know what i must do now
Source : Test N°1 for IMO 2020 in Morocoo
inequality contest-math
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Such that:
$$(x^2+y^2)sqrt{1-z^2}ge z$$
and
$$(z^2+y^2)sqrt{1-x^2}ge x$$
and
$$(x^2+z^2)sqrt{1-y^2}ge y$$
Since $x,y,z$ $in ]0,1[^3$
then , there are some real numbers $a,b,c$ such that
$cos a=x, cos b=y , cos c=z$
After some manipulations , we find that :
$$frac{1}{1+tan^2 a}+frac{1}{1+tan^2 b}ge frac{1}{tan c}$$
.... same for other inequalities
I don't know what i must do now
Source : Test N°1 for IMO 2020 in Morocoo
inequality contest-math
Such that:
$$(x^2+y^2)sqrt{1-z^2}ge z$$
and
$$(z^2+y^2)sqrt{1-x^2}ge x$$
and
$$(x^2+z^2)sqrt{1-y^2}ge y$$
Since $x,y,z$ $in ]0,1[^3$
then , there are some real numbers $a,b,c$ such that
$cos a=x, cos b=y , cos c=z$
After some manipulations , we find that :
$$frac{1}{1+tan^2 a}+frac{1}{1+tan^2 b}ge frac{1}{tan c}$$
.... same for other inequalities
I don't know what i must do now
Source : Test N°1 for IMO 2020 in Morocoo
inequality contest-math
inequality contest-math
edited Nov 23 at 20:54
Michael Rozenberg
95k1588183
95k1588183
asked Nov 23 at 19:20
user600785
2610
2610
add a comment |
add a comment |
3 Answers
3
active
oldest
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up vote
3
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accepted
The inequality is symetric.
so we can suppose that $xge y ge z$
the second inequality becomes
$$2x^2sqrt{1-x^2}ge x$$
$$2xsqrt{1-x^2}ge 1$$
$$4x^2-4x^4-1ge 0$$
$$-(2x^2-1)^2ge 0$$
$$2x^2-1=0$$
$$x=frac{1}{sqrt{2}}$$
By the same way , after remplacing $x$ by its value
we will find that $x=y=z=frac{1}{sqrt{2}}$
Also there is the case $x=0$, which gives $x=y=z=0$. Nice! +1.
– Michael Rozenberg
Nov 23 at 21:27
but I wrotte $]0;1[$ , sooo we can't have $0$ as a solution.
– user600785
Nov 23 at 21:56
But by the given it gives a solution. You can't change the given.It's not fair.
– Michael Rozenberg
Nov 23 at 22:07
add a comment |
up vote
2
down vote
You were very close, but you made a mistake. You should have
$$frac{1}{1+tan^2 a}+frac{1}{1+tan^2b}geq frac{1}{tan c}$$
instead. Here is a similar approach.
First, we assume that $x,y,z>0$. So, we can define $p,q,rgeq 0$ to be $frac{sqrt{1-x^2}}{x}$, $frac{sqrt{1-y^2}}{y}$, and $frac{sqrt{1-z^2}}{z}$, respectively. The three inequalities become
$$frac{1}{1+p^2}+frac{1}{1+q^2}geq frac1rwedge frac{1}{1+q^2}+frac{1}{1+r^2}geq frac1p wedge frac{1}{1+r^2}+frac{1}{1+p^2}geq frac1q.$$
Adding all of these and then dividing the result by $2$ yield
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}geq frac{1}{2p}+frac{1}{2q}+frac{1}{2r}.tag{1}$$
However, by AM-GM, $1+p^2ge 2p$, $1+q^2ge 2q$, and $1+r^2ge 2r$. That is,
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}leq frac{1}{2p}+frac{1}{2q}+frac{1}{2r}.tag{2}$$
From (1) and (2), we must have
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}= frac{1}{2p}+frac{1}{2q}+frac{1}{2r},$$
which implies $1+p^2=2p$, $1+q^2=2q$, $1+r^2=2r$, so $p=q=r=1$ and $x=y=z=frac1{sqrt{2}}$.
Now, WLOG, if $x=0$, then $y^2sqrt{1-z^2}geq z$ and $z^2sqrt{1-y^2}geq y$. Multiplying the two inequalities gives
$$y^2z^2sqrt{1-y^2}sqrt{1-z^2}geq yz.$$
If $yzneq 0$, then dividing by $yz$, we have
$$yzsqrt{1-y^2}sqrt{1-z^2}geq 1.$$
But by AM-GM, $ysqrt{1-y^2}=sqrt{y^2(1-y^2)}leq frac{y^2+(1-y^2)}{2}=frac12$ and similarly, $zsqrt{1-z^2}leq frac12$. So,
$$yzsqrt{1-y^2}sqrt{1-z^2}leq frac12cdotfrac12=frac14<1.$$
This is a contradiction, so $yz=0$, so $y=0$ or $z=0$, so $x=y=z=0$. Therefore, if any of the variables is $0$, all of them are $0$. There are then two solutions $$x=y=z=0 wedge x=y=z=1/sqrt{2}.$$
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up vote
2
down vote
The hint:
$$(x^2+y^2)sqrt{1-z^2}geq z$$ it's
$$(x^2+y^2)^2(1-z^2)geq z^2$$ or
$$(x^2+y^2)^2geq((x^2+y^2)^2+1)z^2,$$ which gives
$$z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$$
By the same way $$y^2leqfrac{x^2+z^2}{2}$$ and $$x^2leqfrac{y^2+z^2}{2},$$ which gives
$$x=y=z$$ and all inequalities the are equalities.
why it gives $z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$
– user600785
Nov 24 at 10:16
@user600785 I used AM-GM: $(x^2+y^2)^2+1geq2sqrt{(x^2+y^2)^2cdot1}=2(x^2+y^2).$ See also my post, I added something.
– Michael Rozenberg
Nov 24 at 12:20
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
The inequality is symetric.
so we can suppose that $xge y ge z$
the second inequality becomes
$$2x^2sqrt{1-x^2}ge x$$
$$2xsqrt{1-x^2}ge 1$$
$$4x^2-4x^4-1ge 0$$
$$-(2x^2-1)^2ge 0$$
$$2x^2-1=0$$
$$x=frac{1}{sqrt{2}}$$
By the same way , after remplacing $x$ by its value
we will find that $x=y=z=frac{1}{sqrt{2}}$
Also there is the case $x=0$, which gives $x=y=z=0$. Nice! +1.
– Michael Rozenberg
Nov 23 at 21:27
but I wrotte $]0;1[$ , sooo we can't have $0$ as a solution.
– user600785
Nov 23 at 21:56
But by the given it gives a solution. You can't change the given.It's not fair.
– Michael Rozenberg
Nov 23 at 22:07
add a comment |
up vote
3
down vote
accepted
The inequality is symetric.
so we can suppose that $xge y ge z$
the second inequality becomes
$$2x^2sqrt{1-x^2}ge x$$
$$2xsqrt{1-x^2}ge 1$$
$$4x^2-4x^4-1ge 0$$
$$-(2x^2-1)^2ge 0$$
$$2x^2-1=0$$
$$x=frac{1}{sqrt{2}}$$
By the same way , after remplacing $x$ by its value
we will find that $x=y=z=frac{1}{sqrt{2}}$
Also there is the case $x=0$, which gives $x=y=z=0$. Nice! +1.
– Michael Rozenberg
Nov 23 at 21:27
but I wrotte $]0;1[$ , sooo we can't have $0$ as a solution.
– user600785
Nov 23 at 21:56
But by the given it gives a solution. You can't change the given.It's not fair.
– Michael Rozenberg
Nov 23 at 22:07
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
The inequality is symetric.
so we can suppose that $xge y ge z$
the second inequality becomes
$$2x^2sqrt{1-x^2}ge x$$
$$2xsqrt{1-x^2}ge 1$$
$$4x^2-4x^4-1ge 0$$
$$-(2x^2-1)^2ge 0$$
$$2x^2-1=0$$
$$x=frac{1}{sqrt{2}}$$
By the same way , after remplacing $x$ by its value
we will find that $x=y=z=frac{1}{sqrt{2}}$
The inequality is symetric.
so we can suppose that $xge y ge z$
the second inequality becomes
$$2x^2sqrt{1-x^2}ge x$$
$$2xsqrt{1-x^2}ge 1$$
$$4x^2-4x^4-1ge 0$$
$$-(2x^2-1)^2ge 0$$
$$2x^2-1=0$$
$$x=frac{1}{sqrt{2}}$$
By the same way , after remplacing $x$ by its value
we will find that $x=y=z=frac{1}{sqrt{2}}$
edited Nov 23 at 21:14
answered Nov 23 at 21:09
user600785
2610
2610
Also there is the case $x=0$, which gives $x=y=z=0$. Nice! +1.
– Michael Rozenberg
Nov 23 at 21:27
but I wrotte $]0;1[$ , sooo we can't have $0$ as a solution.
– user600785
Nov 23 at 21:56
But by the given it gives a solution. You can't change the given.It's not fair.
– Michael Rozenberg
Nov 23 at 22:07
add a comment |
Also there is the case $x=0$, which gives $x=y=z=0$. Nice! +1.
– Michael Rozenberg
Nov 23 at 21:27
but I wrotte $]0;1[$ , sooo we can't have $0$ as a solution.
– user600785
Nov 23 at 21:56
But by the given it gives a solution. You can't change the given.It's not fair.
– Michael Rozenberg
Nov 23 at 22:07
Also there is the case $x=0$, which gives $x=y=z=0$. Nice! +1.
– Michael Rozenberg
Nov 23 at 21:27
Also there is the case $x=0$, which gives $x=y=z=0$. Nice! +1.
– Michael Rozenberg
Nov 23 at 21:27
but I wrotte $]0;1[$ , sooo we can't have $0$ as a solution.
– user600785
Nov 23 at 21:56
but I wrotte $]0;1[$ , sooo we can't have $0$ as a solution.
– user600785
Nov 23 at 21:56
But by the given it gives a solution. You can't change the given.It's not fair.
– Michael Rozenberg
Nov 23 at 22:07
But by the given it gives a solution. You can't change the given.It's not fair.
– Michael Rozenberg
Nov 23 at 22:07
add a comment |
up vote
2
down vote
You were very close, but you made a mistake. You should have
$$frac{1}{1+tan^2 a}+frac{1}{1+tan^2b}geq frac{1}{tan c}$$
instead. Here is a similar approach.
First, we assume that $x,y,z>0$. So, we can define $p,q,rgeq 0$ to be $frac{sqrt{1-x^2}}{x}$, $frac{sqrt{1-y^2}}{y}$, and $frac{sqrt{1-z^2}}{z}$, respectively. The three inequalities become
$$frac{1}{1+p^2}+frac{1}{1+q^2}geq frac1rwedge frac{1}{1+q^2}+frac{1}{1+r^2}geq frac1p wedge frac{1}{1+r^2}+frac{1}{1+p^2}geq frac1q.$$
Adding all of these and then dividing the result by $2$ yield
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}geq frac{1}{2p}+frac{1}{2q}+frac{1}{2r}.tag{1}$$
However, by AM-GM, $1+p^2ge 2p$, $1+q^2ge 2q$, and $1+r^2ge 2r$. That is,
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}leq frac{1}{2p}+frac{1}{2q}+frac{1}{2r}.tag{2}$$
From (1) and (2), we must have
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}= frac{1}{2p}+frac{1}{2q}+frac{1}{2r},$$
which implies $1+p^2=2p$, $1+q^2=2q$, $1+r^2=2r$, so $p=q=r=1$ and $x=y=z=frac1{sqrt{2}}$.
Now, WLOG, if $x=0$, then $y^2sqrt{1-z^2}geq z$ and $z^2sqrt{1-y^2}geq y$. Multiplying the two inequalities gives
$$y^2z^2sqrt{1-y^2}sqrt{1-z^2}geq yz.$$
If $yzneq 0$, then dividing by $yz$, we have
$$yzsqrt{1-y^2}sqrt{1-z^2}geq 1.$$
But by AM-GM, $ysqrt{1-y^2}=sqrt{y^2(1-y^2)}leq frac{y^2+(1-y^2)}{2}=frac12$ and similarly, $zsqrt{1-z^2}leq frac12$. So,
$$yzsqrt{1-y^2}sqrt{1-z^2}leq frac12cdotfrac12=frac14<1.$$
This is a contradiction, so $yz=0$, so $y=0$ or $z=0$, so $x=y=z=0$. Therefore, if any of the variables is $0$, all of them are $0$. There are then two solutions $$x=y=z=0 wedge x=y=z=1/sqrt{2}.$$
add a comment |
up vote
2
down vote
You were very close, but you made a mistake. You should have
$$frac{1}{1+tan^2 a}+frac{1}{1+tan^2b}geq frac{1}{tan c}$$
instead. Here is a similar approach.
First, we assume that $x,y,z>0$. So, we can define $p,q,rgeq 0$ to be $frac{sqrt{1-x^2}}{x}$, $frac{sqrt{1-y^2}}{y}$, and $frac{sqrt{1-z^2}}{z}$, respectively. The three inequalities become
$$frac{1}{1+p^2}+frac{1}{1+q^2}geq frac1rwedge frac{1}{1+q^2}+frac{1}{1+r^2}geq frac1p wedge frac{1}{1+r^2}+frac{1}{1+p^2}geq frac1q.$$
Adding all of these and then dividing the result by $2$ yield
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}geq frac{1}{2p}+frac{1}{2q}+frac{1}{2r}.tag{1}$$
However, by AM-GM, $1+p^2ge 2p$, $1+q^2ge 2q$, and $1+r^2ge 2r$. That is,
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}leq frac{1}{2p}+frac{1}{2q}+frac{1}{2r}.tag{2}$$
From (1) and (2), we must have
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}= frac{1}{2p}+frac{1}{2q}+frac{1}{2r},$$
which implies $1+p^2=2p$, $1+q^2=2q$, $1+r^2=2r$, so $p=q=r=1$ and $x=y=z=frac1{sqrt{2}}$.
Now, WLOG, if $x=0$, then $y^2sqrt{1-z^2}geq z$ and $z^2sqrt{1-y^2}geq y$. Multiplying the two inequalities gives
$$y^2z^2sqrt{1-y^2}sqrt{1-z^2}geq yz.$$
If $yzneq 0$, then dividing by $yz$, we have
$$yzsqrt{1-y^2}sqrt{1-z^2}geq 1.$$
But by AM-GM, $ysqrt{1-y^2}=sqrt{y^2(1-y^2)}leq frac{y^2+(1-y^2)}{2}=frac12$ and similarly, $zsqrt{1-z^2}leq frac12$. So,
$$yzsqrt{1-y^2}sqrt{1-z^2}leq frac12cdotfrac12=frac14<1.$$
This is a contradiction, so $yz=0$, so $y=0$ or $z=0$, so $x=y=z=0$. Therefore, if any of the variables is $0$, all of them are $0$. There are then two solutions $$x=y=z=0 wedge x=y=z=1/sqrt{2}.$$
add a comment |
up vote
2
down vote
up vote
2
down vote
You were very close, but you made a mistake. You should have
$$frac{1}{1+tan^2 a}+frac{1}{1+tan^2b}geq frac{1}{tan c}$$
instead. Here is a similar approach.
First, we assume that $x,y,z>0$. So, we can define $p,q,rgeq 0$ to be $frac{sqrt{1-x^2}}{x}$, $frac{sqrt{1-y^2}}{y}$, and $frac{sqrt{1-z^2}}{z}$, respectively. The three inequalities become
$$frac{1}{1+p^2}+frac{1}{1+q^2}geq frac1rwedge frac{1}{1+q^2}+frac{1}{1+r^2}geq frac1p wedge frac{1}{1+r^2}+frac{1}{1+p^2}geq frac1q.$$
Adding all of these and then dividing the result by $2$ yield
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}geq frac{1}{2p}+frac{1}{2q}+frac{1}{2r}.tag{1}$$
However, by AM-GM, $1+p^2ge 2p$, $1+q^2ge 2q$, and $1+r^2ge 2r$. That is,
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}leq frac{1}{2p}+frac{1}{2q}+frac{1}{2r}.tag{2}$$
From (1) and (2), we must have
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}= frac{1}{2p}+frac{1}{2q}+frac{1}{2r},$$
which implies $1+p^2=2p$, $1+q^2=2q$, $1+r^2=2r$, so $p=q=r=1$ and $x=y=z=frac1{sqrt{2}}$.
Now, WLOG, if $x=0$, then $y^2sqrt{1-z^2}geq z$ and $z^2sqrt{1-y^2}geq y$. Multiplying the two inequalities gives
$$y^2z^2sqrt{1-y^2}sqrt{1-z^2}geq yz.$$
If $yzneq 0$, then dividing by $yz$, we have
$$yzsqrt{1-y^2}sqrt{1-z^2}geq 1.$$
But by AM-GM, $ysqrt{1-y^2}=sqrt{y^2(1-y^2)}leq frac{y^2+(1-y^2)}{2}=frac12$ and similarly, $zsqrt{1-z^2}leq frac12$. So,
$$yzsqrt{1-y^2}sqrt{1-z^2}leq frac12cdotfrac12=frac14<1.$$
This is a contradiction, so $yz=0$, so $y=0$ or $z=0$, so $x=y=z=0$. Therefore, if any of the variables is $0$, all of them are $0$. There are then two solutions $$x=y=z=0 wedge x=y=z=1/sqrt{2}.$$
You were very close, but you made a mistake. You should have
$$frac{1}{1+tan^2 a}+frac{1}{1+tan^2b}geq frac{1}{tan c}$$
instead. Here is a similar approach.
First, we assume that $x,y,z>0$. So, we can define $p,q,rgeq 0$ to be $frac{sqrt{1-x^2}}{x}$, $frac{sqrt{1-y^2}}{y}$, and $frac{sqrt{1-z^2}}{z}$, respectively. The three inequalities become
$$frac{1}{1+p^2}+frac{1}{1+q^2}geq frac1rwedge frac{1}{1+q^2}+frac{1}{1+r^2}geq frac1p wedge frac{1}{1+r^2}+frac{1}{1+p^2}geq frac1q.$$
Adding all of these and then dividing the result by $2$ yield
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}geq frac{1}{2p}+frac{1}{2q}+frac{1}{2r}.tag{1}$$
However, by AM-GM, $1+p^2ge 2p$, $1+q^2ge 2q$, and $1+r^2ge 2r$. That is,
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}leq frac{1}{2p}+frac{1}{2q}+frac{1}{2r}.tag{2}$$
From (1) and (2), we must have
$$frac{1}{1+p^2}+frac{1}{1+q^2}+frac{1}{1+r^2}= frac{1}{2p}+frac{1}{2q}+frac{1}{2r},$$
which implies $1+p^2=2p$, $1+q^2=2q$, $1+r^2=2r$, so $p=q=r=1$ and $x=y=z=frac1{sqrt{2}}$.
Now, WLOG, if $x=0$, then $y^2sqrt{1-z^2}geq z$ and $z^2sqrt{1-y^2}geq y$. Multiplying the two inequalities gives
$$y^2z^2sqrt{1-y^2}sqrt{1-z^2}geq yz.$$
If $yzneq 0$, then dividing by $yz$, we have
$$yzsqrt{1-y^2}sqrt{1-z^2}geq 1.$$
But by AM-GM, $ysqrt{1-y^2}=sqrt{y^2(1-y^2)}leq frac{y^2+(1-y^2)}{2}=frac12$ and similarly, $zsqrt{1-z^2}leq frac12$. So,
$$yzsqrt{1-y^2}sqrt{1-z^2}leq frac12cdotfrac12=frac14<1.$$
This is a contradiction, so $yz=0$, so $y=0$ or $z=0$, so $x=y=z=0$. Therefore, if any of the variables is $0$, all of them are $0$. There are then two solutions $$x=y=z=0 wedge x=y=z=1/sqrt{2}.$$
answered Nov 23 at 20:47
Zvi
4,335429
4,335429
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add a comment |
up vote
2
down vote
The hint:
$$(x^2+y^2)sqrt{1-z^2}geq z$$ it's
$$(x^2+y^2)^2(1-z^2)geq z^2$$ or
$$(x^2+y^2)^2geq((x^2+y^2)^2+1)z^2,$$ which gives
$$z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$$
By the same way $$y^2leqfrac{x^2+z^2}{2}$$ and $$x^2leqfrac{y^2+z^2}{2},$$ which gives
$$x=y=z$$ and all inequalities the are equalities.
why it gives $z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$
– user600785
Nov 24 at 10:16
@user600785 I used AM-GM: $(x^2+y^2)^2+1geq2sqrt{(x^2+y^2)^2cdot1}=2(x^2+y^2).$ See also my post, I added something.
– Michael Rozenberg
Nov 24 at 12:20
add a comment |
up vote
2
down vote
The hint:
$$(x^2+y^2)sqrt{1-z^2}geq z$$ it's
$$(x^2+y^2)^2(1-z^2)geq z^2$$ or
$$(x^2+y^2)^2geq((x^2+y^2)^2+1)z^2,$$ which gives
$$z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$$
By the same way $$y^2leqfrac{x^2+z^2}{2}$$ and $$x^2leqfrac{y^2+z^2}{2},$$ which gives
$$x=y=z$$ and all inequalities the are equalities.
why it gives $z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$
– user600785
Nov 24 at 10:16
@user600785 I used AM-GM: $(x^2+y^2)^2+1geq2sqrt{(x^2+y^2)^2cdot1}=2(x^2+y^2).$ See also my post, I added something.
– Michael Rozenberg
Nov 24 at 12:20
add a comment |
up vote
2
down vote
up vote
2
down vote
The hint:
$$(x^2+y^2)sqrt{1-z^2}geq z$$ it's
$$(x^2+y^2)^2(1-z^2)geq z^2$$ or
$$(x^2+y^2)^2geq((x^2+y^2)^2+1)z^2,$$ which gives
$$z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$$
By the same way $$y^2leqfrac{x^2+z^2}{2}$$ and $$x^2leqfrac{y^2+z^2}{2},$$ which gives
$$x=y=z$$ and all inequalities the are equalities.
The hint:
$$(x^2+y^2)sqrt{1-z^2}geq z$$ it's
$$(x^2+y^2)^2(1-z^2)geq z^2$$ or
$$(x^2+y^2)^2geq((x^2+y^2)^2+1)z^2,$$ which gives
$$z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$$
By the same way $$y^2leqfrac{x^2+z^2}{2}$$ and $$x^2leqfrac{y^2+z^2}{2},$$ which gives
$$x=y=z$$ and all inequalities the are equalities.
edited Nov 24 at 12:22
answered Nov 23 at 20:47
Michael Rozenberg
95k1588183
95k1588183
why it gives $z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$
– user600785
Nov 24 at 10:16
@user600785 I used AM-GM: $(x^2+y^2)^2+1geq2sqrt{(x^2+y^2)^2cdot1}=2(x^2+y^2).$ See also my post, I added something.
– Michael Rozenberg
Nov 24 at 12:20
add a comment |
why it gives $z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$
– user600785
Nov 24 at 10:16
@user600785 I used AM-GM: $(x^2+y^2)^2+1geq2sqrt{(x^2+y^2)^2cdot1}=2(x^2+y^2).$ See also my post, I added something.
– Michael Rozenberg
Nov 24 at 12:20
why it gives $z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$
– user600785
Nov 24 at 10:16
why it gives $z^2leqfrac{(x^2+y^2)^2}{(x^2+y^2)^2+1}leqfrac{(x^2+y^2)^2}{2(x^2+y^2)}=frac{x^2+y^2}{2}.$
– user600785
Nov 24 at 10:16
@user600785 I used AM-GM: $(x^2+y^2)^2+1geq2sqrt{(x^2+y^2)^2cdot1}=2(x^2+y^2).$ See also my post, I added something.
– Michael Rozenberg
Nov 24 at 12:20
@user600785 I used AM-GM: $(x^2+y^2)^2+1geq2sqrt{(x^2+y^2)^2cdot1}=2(x^2+y^2).$ See also my post, I added something.
– Michael Rozenberg
Nov 24 at 12:20
add a comment |
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