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IMO Longlists
1978 IMO Longlists
25
25
Part of
1978 IMO Longlists
Problems
(1)
Two real roots of magnitude less than 1
Source:
10/28/2010
Consider a polynomial
P
(
x
)
=
a
x
2
+
b
x
+
c
P(x) = ax^2 + bx + c
P
(
x
)
=
a
x
2
+
b
x
+
c
with
a
>
0
a > 0
a
>
0
that has two real roots
x
1
,
x
2
x_1, x_2
x
1
,
x
2
. Prove that the absolute values of both roots are less than or equal to
1
1
1
if and only if
a
+
b
+
c
≥
0
,
a
−
b
+
c
≥
0
a + b + c \ge 0, a -b + c \ge 0
a
+
b
+
c
≥
0
,
a
−
b
+
c
≥
0
, and
a
−
c
≥
0
a - c \ge 0
a
−
c
≥
0
.
algebra
polynomial
Vieta
algebra unsolved