1973 USAMO Problems/Problem 4
Problem
Determine all the roots, real or complex, of the system of simultaneous equations
![$x+y+z=3$](http://latex.artofproblemsolving.com/a/d/5/ad5ff13ebc979a03c144ba96c88e6281c5556894.png)
,
![$x^3+y^3+z^3=3$](http://latex.artofproblemsolving.com/e/4/7/e47fe3685e8d2812abd631eb34e54bcffd0d2643.png)
Solution
Let ,
, and
be the roots of the cubic
. Let
,
, and
. From this,
,
, and
. Solving each of these,
,
, and
. Thus
,
, and
are the roots of the polynomial
. Thus
, and there are no other solutions.
See also
1973 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |