Difference between revisions of "1977 USAMO Problems/Problem 3"

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[[Category:Olympiad Algebra Problems]]
 
[[Category:Olympiad Algebra Problems]]
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a,b,c,d are roots of equation  <math> x^4\plus{}x^3\minus{}1\equal{}0</math> then by vietas relation
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ab +bc+cd+da+ac+bd=c/a = 0
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let us suppose ab,bc,cd,da,ac,bd are roots of <math> x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0</math>.
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then sum of roots = ab +bc+cd+da+ac+bd=c/a = -b/a=0
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sum taken two at a time= abxbc + bcxca +..........=c/a=1
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similarly we prove for the roots taken three four five and six at a time
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to prove ab,bc,cd,da,ac,bd are roots of second equation

Revision as of 05:15, 6 November 2012

Problem

If $a$ and $b$ are two of the roots of $x^4\plus{}x^3\minus{}1\equal{}0$ (Error compiling LaTeX. Unknown error_msg), prove that $ab$ is a root of $x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0$ (Error compiling LaTeX. Unknown error_msg).

Solution

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See Also

1977 USAMO (ProblemsResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5
All USAMO Problems and Solutions

a,b,c,d are roots of equation $x^4\plus{}x^3\minus{}1\equal{}0$ (Error compiling LaTeX. Unknown error_msg) then by vietas relation ab +bc+cd+da+ac+bd=c/a = 0 let us suppose ab,bc,cd,da,ac,bd are roots of $x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0$ (Error compiling LaTeX. Unknown error_msg).

then sum of roots = ab +bc+cd+da+ac+bd=c/a = -b/a=0 sum taken two at a time= abxbc + bcxca +..........=c/a=1 similarly we prove for the roots taken three four five and six at a time to prove ab,bc,cd,da,ac,bd are roots of second equation