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

(See Also)
Line 9: Line 9:
  
 
[[Category:Olympiad Algebra Problems]]
 
[[Category:Olympiad Algebra Problems]]
 +
a,b,c,d are roots of equation  <math> x^4\plus{}x^3\minus{}1\equal{}0</math> 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 <math> x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0</math>.
 +
 +
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

Revision as of 04:15, 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

This problem needs a solution. If you have a solution for it, please help us out by adding it.

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