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2022 Benelux
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Polynomial bounded by sum of coefficients (BxMO 2022, Problem 1)
Source: BxMO 2022, Problem 1
5/1/2022
Let
n
⩾
0
n\geqslant 0
n
⩾
0
be an integer, and let
a
0
,
a
1
,
…
,
a
n
a_0,a_1,\dots,a_n
a
0
,
a
1
,
…
,
a
n
be real numbers. Show that there exists
k
∈
{
0
,
1
,
…
,
n
}
k\in\{0,1,\dots,n\}
k
∈
{
0
,
1
,
…
,
n
}
such that
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n
⩽
a
0
+
a
1
+
⋯
+
a
k
a_0+a_1x+a_2x^2+\cdots+a_nx^n\leqslant a_0+a_1+\cdots+a_k
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n
⩽
a
0
+
a
1
+
⋯
+
a
k
for all real numbers
x
∈
[
0
,
1
]
x\in[0,1]
x
∈
[
0
,
1
]
.
BxMO
algebra
polynomial