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Francophone Mathematical Olympiad
2023 Francophone Mathematical Olympiad
1
4 b_{n+1} <= P(1)^2
4 b_{n+1} <= P(1)^2
Source: P1 Francophone Math Olympiad Senior 2023
May 2, 2023
polynomial
algebra
inequalities
Problem Statement
Let
P
(
X
)
=
a
n
X
n
+
a
n
−
1
X
n
−
1
+
⋯
+
a
1
X
+
a
0
P(X) = a_n X^n + a_{n-1} X^{n-1} + \cdots + a_1 X + a_0
P
(
X
)
=
a
n
X
n
+
a
n
−
1
X
n
−
1
+
⋯
+
a
1
X
+
a
0
be a polynomial with real coefficients such that
0
⩽
a
i
⩽
a
0
0 \leqslant a_i \leqslant a_0
0
⩽
a
i
⩽
a
0
for
i
=
1
,
2
,
…
,
n
i = 1, 2, \ldots, n
i
=
1
,
2
,
…
,
n
. Prove that, if
P
(
X
)
2
=
b
2
n
X
2
n
+
b
2
n
−
1
X
2
n
−
1
+
⋯
+
b
n
+
1
X
n
+
1
+
⋯
+
b
1
X
+
b
0
P(X)^2 = b_{2n} X^{2n} + b_{2n-1} X^{2n-1} + \cdots + b_{n+1} X^{n+1} + \cdots + b_1 X + b_0
P
(
X
)
2
=
b
2
n
X
2
n
+
b
2
n
−
1
X
2
n
−
1
+
⋯
+
b
n
+
1
X
n
+
1
+
⋯
+
b
1
X
+
b
0
, then
4
b
n
+
1
⩽
P
(
1
)
2
4 b_{n+1} \leqslant P(1)^2
4
b
n
+
1
⩽
P
(
1
)
2
.
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