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Turkey Contests
Turkey Team Selection Test
1996 Turkey Team Selection Test
3
Turkey TST 1996 Problem 3
Turkey TST 1996 Problem 3
Source: Turkey TST 1996 Problem 3
September 28, 2011
inequalities proposed
inequalities
Problem Statement
If
0
=
x
1
<
x
2
<
.
.
.
<
x
2
n
+
1
=
1
0=x_{1}<x_{2}<...<x_{2n+1}=1
0
=
x
1
<
x
2
<
...
<
x
2
n
+
1
=
1
are real numbers with
x
i
+
1
−
x
i
≤
h
x_{i+1}-x_{i} \leq h
x
i
+
1
−
x
i
≤
h
for
1
≤
i
≤
2
n
1 \leq i \leq 2n
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}^{n}{x_{2i}(x_{2i+1}-x_{2i-1})}\leq \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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