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Sweden Contests
Swedish Mathematical Competition
1985 Swedish Mathematical Competition
4
4
Part of
1985 Swedish Mathematical Competition
Problems
(1)
p(x)+ p'(x)+ p''(x)+...+ p^{(n)}(x) >= 0 if p(x) >= 0
Source: 1985 Swedish Mathematical Competition p4
3/28/2021
Let
p
(
x
)
p(x)
p
(
x
)
be a polynomial of degree
n
n
n
with real coefficients such that
p
(
x
)
≥
0
p(x) \ge 0
p
(
x
)
≥
0
for all
x
x
x
. Prove that
p
(
x
)
+
p
′
(
x
)
+
p
′
′
(
x
)
+
.
.
.
+
p
(
n
)
(
x
)
≥
0
p(x)+ p'(x)+ p''(x)+...+ p^{(n)}(x) \ge 0
p
(
x
)
+
p
′
(
x
)
+
p
′′
(
x
)
+
...
+
p
(
n
)
(
x
)
≥
0
.
analysis
algebra
polynomial