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. 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
Given the roots of the equation
.
First, .
Then and
.
Remains or
.
Let and
, so
(1).
Second, is a root,
and
is a root,
.
Multiplying: or
.
Solving .
In (1): .
or
.
Conclusion: is a root of
.
See Also
1977 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.