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[SPAM] [obm-l] Problema de Desigualdade
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- Subject: [SPAM] [obm-l] Problema de Desigualdade
- From: Lucas Daniel <lucasdgf@xxxxxxxxxxxx>
- Date: Sat, 5 May 2007 18:25:27 -0300 (ART)
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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> + <span style="text-decoration: underline;">2y (1 - y²)</span> + <span
style="text-decoration: underline;">2z (1 - z²)</span> <span style="text-decoration: underline;"><</span> <span style="text-decoration: underline;"> x </span> + <span style="text-decoration: underline;"> y </span> + <span style="text-decoration: underline;"> z </span><br></div><div class="MsoBodyTextIndent" style="text-indent: 0cm; font-family: times new roman;"> (1+x²)² (1+y²)² (1+z²)² 1+x² 1+y² 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: "Times New Roman";" 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: "Times New Roman";" lang="EN-US"></span> <p> __________________________________________________<br>Fale com seus amigos de graça com o novo Yahoo! Messenger <br>http://br.messenger.yahoo.com/
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