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1969 IMO Shortlist
66
Polynomial inequality
Polynomial inequality
Source:
October 4, 2010
algebra
polynomial
Inequality
Taylor series
IMO Shortlist
IMO Longlist
Problem Statement
(
U
S
S
3
)
(USS 3)
(
U
SS
3
)
(
a
)
(a)
(
a
)
Prove that if
0
≤
a
0
≤
a
1
≤
a
2
,
0 \le a_0 \le a_1 \le a_2,
0
≤
a
0
≤
a
1
≤
a
2
,
then
(
a
0
+
a
1
x
−
a
2
x
2
)
2
≤
(
a
0
+
a
1
+
a
2
)
2
(
1
+
1
2
x
+
1
3
x
2
+
1
2
x
3
+
x
4
)
(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)
(
a
0
+
a
1
x
−
a
2
x
2
)
2
≤
(
a
0
+
a
1
+
a
2
)
2
(
1
+
2
1
x
+
3
1
x
2
+
2
1
x
3
+
x
4
)
(
b
)
(b)
(
b
)
Formulate and prove the analogous result for polynomials of third degree.
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