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Vojtěch Jarník IMC
2006 VJIMC
Problem 1
inequality over (0,1), with x_(i+1)-x_i<=h
inequality over (0,1), with x_(i+1)-x_i<=h
Source: VJIMC 2006 1.1
June 28, 2021
inequalities
Problem Statement
Given real numbers
0
=
x
1
<
x
2
<
…
<
x
2
n
<
x
2
n
+
1
=
1
0=x_1<x_2<\ldots<x_{2n}<x_{2n+1}=1
0
=
x
1
<
x
2
<
…
<
x
2
n
<
x
2
n
+
1
=
1
such that
x
i
+
1
−
x
i
≤
h
x_{i+1}-x_i\le h
x
i
+
1
−
x
i
≤
h
for
1
≤
i
≤
2
n
1\le i\le2n
1
≤
i
≤
2
n
, show that
1
−
h
2
<
∑
i
=
1
n
x
2
i
(
x
2
i
+
1
−
x
2
i
−
1
)
<
1
+
h
2
.
\frac{1-h}2<\sum_{i=1}^nx_{2i}(x_{2i+1}-x_{2i-1})<\frac{1+h}2.
2
1
−
h
<
i
=
1
∑
n
x
2
i
(
x
2
i
+
1
−
x
2
i
−
1
)
<
2
1
+
h
.
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