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Taiwan National Olympiad
1996 Taiwan National Olympiad
4
4
Part of
1996 Taiwan National Olympiad
Problems
(1)
not all the roots are real
Source: 5-th Taiwanese Mathematical Olympiad 1996
1/12/2007
Show that for any real numbers
a
3
,
a
4
,
.
.
.
,
a
85
a_{3},a_{4},...,a_{85}
a
3
,
a
4
,
...
,
a
85
, not all the roots of the equation
a
85
x
85
+
a
84
x
84
+
.
.
.
+
a
3
x
3
+
3
x
2
+
2
x
+
1
=
0
a_{85}x^{85}+a_{84}x^{84}+...+a_{3}x^{3}+3x^{2}+2x+1=0
a
85
x
85
+
a
84
x
84
+
...
+
a
3
x
3
+
3
x
2
+
2
x
+
1
=
0
are real.
algebra
polynomial
Vieta
algebra proposed