1977 USAMO Problems/Problem 3

Problem

If and are two of the roots of $x^4\plus{}x^3\minus{}1\equal{}0$ (Error compiling LaTeX. Unknown error_msg), prove that 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 (Problems • Resources)
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