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[SPAM] [obm-l] Problema de Desigualdade



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Olá.

Sou aluno do 1.º ano do Ensino Médio e ontem meu professor de Matemática para a OBM me passou um problema sobre desigualdade o qual ainda não consegui resolver. Seria possível me passar a resolução?

Obrigado, 

Lucas.


O problema é o seguinte:

  Sejam x, y, z reais positivos tais que xy + yz + zx = 1. Prove que:
2x (1 - x²) +  2y (1 - y²) + 2z (1 - z²)  <     x    +     y    +    z   

  (1+x²)²          (1+y²)²         (1+z²)²         1+x²     1+y²      1+z²









Obrigado!

                                                    
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Olá.<br><br>Sou aluno do 1.º ano do Ensino Médio e ontem meu professor de Matemática para a OBM me passou um problema sobre desigualdade o qual ainda não consegui resolver. Seria possível me passar a resolução?<br><br>Obrigado, <br><br>Lucas.<br><br><br style="font-family: times new roman;"><span style="font-family: times new roman;">O problema é o seguinte:</span><br style="font-family: times new roman;"><br style="font-family: times new roman;">  <div class="MsoBodyTextIndent" style="text-indent: 0cm; font-family: times new roman;"><span style="font-size: 11pt;">Sejam <i style="">x</i>, <i style="">y</i>,<i style=""> z</i> reais positivos tais que <i style="">xy</i> + <i style="">yz</i> + <i style="">zx</i> = 1. Prove que:</span></div><div class="MsoBodyTextIndent" style="text-indent: 0cm; font-family: times new roman;"><span style="text-decoration: underline;">2x (1 - x²)</span> +&nbsp; <span style="text-decoration: underline;">2y (1 - y²)</span> + <span
 style="text-decoration: underline;">2z (1 - z²)</span>&nbsp; <span style="text-decoration: underline;">&lt;</span> &nbsp;<span style="text-decoration: underline;"> &nbsp; x&nbsp;&nbsp;&nbsp;</span> +&nbsp;<span style="text-decoration: underline;">&nbsp;&nbsp;&nbsp; y&nbsp; &nbsp;</span> + <span style="text-decoration: underline;">&nbsp;&nbsp; z&nbsp;&nbsp; </span><br></div><div class="MsoBodyTextIndent" style="text-indent: 0cm; font-family: times new roman;">&nbsp; (1+x²)²&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (1+y²)²&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (1+z²)²&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1+x²&nbsp;&nbsp;&nbsp;&nbsp; 1+y²&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1+z²</div><br><div class="MsoBodyTextIndent" style="text-indent: 0cm;"><br></div><br><div class="MsoBodyTextIndent" style="text-indent: 0cm;"><br></div><br><div class="MsoBodyTextIndent" style="text-indent: 0cm;"><br></div><div class="MsoBodyTextIndent" style="text-indent:
 0cm;">Obrigado!<br><span style="font-size: 11pt;"></span></div><span style="font-size: 11pt; font-family: &quot;Times New Roman&quot;;" lang="EN-US"><span style="position: relative; top: 14pt;"><!--[if gte vml 1]><v:shapetype  id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t"  path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">  <v:stroke joinstyle="miter"/>  <v:formulas>   <v:f eqn="if lineDrawn pixelLineWidth 0"/>   <v:f eqn="sum @0 1 0"/>   <v:f eqn="sum 0 0 @1"/>   <v:f eqn="prod @2 1 2"/>   <v:f eqn="prod @3 21600 pixelWidth"/>   <v:f eqn="prod @3 21600 pixelHeight"/>   <v:f eqn="sum @0 0 1"/>   <v:f eqn="prod @6 1 2"/>   <v:f eqn="prod @7 21600 pixelWidth"/>   <v:f eqn="sum @8 21600 0"/>   <v:f eqn="prod @7 21600 pixelHeight"/>   <v:f eqn="sum @10 21600 0"/>  </v:formulas>  <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/>  <o:lock v:ext="edit" aspectratio="t"/> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75"
 style='width:273pt;  height:33.75pt' o:ole="" fillcolor="window">  <v:imagedata src="file:///C:\DOCUME~1\Casa\CONFIG~1\Temp\msohtml1\06\clip_image001.wmz"   o:title=""/> </v:shape><![endif]--><!--[if !vml]--></span></span><span style="font-size: 11pt; font-family: &quot;Times New Roman&quot;;" lang="EN-US"></span>  <p>&#32;__________________________________________________<br>Fale com seus amigos  de graça com o novo Yahoo! Messenger <br>http://br.messenger.yahoo.com/ 
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