Difference between revisions of "1973 USAMO Problems/Problem 4"
(New page: ==Problem== Determine all the roots, real or complex, of the system of simultaneous equations <center><math>x+y+z=3</math>, <math>x^2+y^2+z^2=3</math>, <math>x^3+y^3+z^3=3<...) |
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| Line 6: | Line 6: | ||
==Solution== | ==Solution== | ||
| − | Let <math>x</math>, <math>y</math>, and <math>z</math> be the roots of the | + | Let <math>x</math>, <math>y</math>, and <math>z</math> be the roots of the cubic <math>x^3+ax^2+bx+c</math>. Let <math>S_1=x+y+z=3</math>, <math>S_2=x^2+y^2+z^2=3</math>, and <math>S_3=x^3+y^3+z^3=3</math>. From this, <math>S_1+a=0</math>, <math>S_2+aS_1+b=0</math>, and <math>S_3+aS_2+bS_1+c=0</math>. Solving each of these, <math>a=-3</math>, <math>b=3</math>, and <math>c=-1</math>. Thus <math>x</math>, <math>y</math>, and <math>z</math> are the roots of the polynomial <math>x^3-3x^2+3x-1=(x-1)^3</math>. Thus <math>x+y+z=1</math>, and there are no other solutions. |
==See also== | ==See also== | ||
[[Newton's Sums]] | [[Newton's Sums]] | ||
| + | {{USAMO newbox|year=1973|num-b=3|num-a=5}} | ||
| + | |||
| + | [[Category:Olympiad Algebra Problems]] | ||
Revision as of 13:28, 4 October 2008
Problem
Determine all the roots, real or complex, of the system of simultaneous equations
,
,
.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 • 6 | ||
| All USAMO Problems and Solutions | ||
