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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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










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    up vote
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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










    share|cite|improve this question


























      up vote
      3
      down vote

      favorite
      2









      up vote
      3
      down vote

      favorite
      2






      2





      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










      share|cite|improve this question















      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






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      edited Nov 23 at 20:54









      Michael Rozenberg

      95k1588183




      95k1588183










      asked Nov 23 at 19:20









      user600785

      2610




      2610






















          3 Answers
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          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 $x=y=z=frac{1}{sqrt{2}}$






          share|cite|improve this answer























          • 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




















          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 $$x=y=z=0 wedge x=y=z=1/sqrt{2}.$$






          share|cite|improve this answer




























            up vote
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            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.






            share|cite|improve this answer























            • 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 $x=y=z=frac{1}{sqrt{2}}$






            share|cite|improve this answer























            • 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

















            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 $x=y=z=frac{1}{sqrt{2}}$






            share|cite|improve this answer























            • 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















            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 $x=y=z=frac{1}{sqrt{2}}$






            share|cite|improve this answer














            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 $x=y=z=frac{1}{sqrt{2}}$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            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




















            • 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












            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 $$x=y=z=0 wedge x=y=z=1/sqrt{2}.$$






            share|cite|improve this answer

























              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 $$x=y=z=0 wedge x=y=z=1/sqrt{2}.$$






              share|cite|improve this answer























                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 $$x=y=z=0 wedge x=y=z=1/sqrt{2}.$$






                share|cite|improve this answer












                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 $$x=y=z=0 wedge x=y=z=1/sqrt{2}.$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 23 at 20:47









                Zvi

                4,335429




                4,335429






















                    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.






                    share|cite|improve this answer























                    • 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

















                    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.






                    share|cite|improve this answer























                    • 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















                    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.






                    share|cite|improve this answer














                    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.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    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




















                    • 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




















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