Difference between revisions of "1989 USAMO Problems/Problem 3"
m (→Resources) |
|||
Line 23: | Line 23: | ||
Since <math>P(a+bi) = P(z_i) = 0</math>, these real numbers <math>a,b</math> satisfy the problem's conditions. <math>\blacksquare</math> | Since <math>P(a+bi) = P(z_i) = 0</math>, these real numbers <math>a,b</math> satisfy the problem's conditions. <math>\blacksquare</math> | ||
− | == | + | == See Also == |
{{USAMO box|year=1989|num-b=2|num-a=4}} | {{USAMO box|year=1989|num-b=2|num-a=4}} |
Latest revision as of 19:11, 18 July 2016
Problem
Let be a polynomial in the complex variable
, with real coefficients
. Suppose that
. Prove that there exist real numbers
and
such that
and
.
Solution
Let be the (not necessarily distinct) roots of
, so that
Since all the coefficients of
are real, it follows that if
is a root of
, then
, so
, the complex conjugate of
, is also a root of
.
Since
it follows that for some (not necessarily distinct) conjugates
and
,
Let
and
, for real
. We note that
Thus
Since
, these real numbers
satisfy the problem's conditions.
See Also
1989 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.