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Swedish Mathematical Competition
1989 Swedish Mathematical Competition
5
1/(x_1 +x_3)^a+1/(x_2 +x_4)^a+1/(x_2 +x_5)^a < ...
1/(x_1 +x_3)^a+1/(x_2 +x_4)^a+1/(x_2 +x_5)^a < ...
Source: 1989 Swedish Mathematical Competition p5
March 28, 2021
algebra
inequalities
Problem Statement
Assume
x
1
,
x
2
,
.
.
,
x
5
x_1,x_2,..,x_5
x
1
,
x
2
,
..
,
x
5
are positive numbers such that
x
1
<
x
2
x_1 < x_2
x
1
<
x
2
and
x
3
,
x
4
,
x
5
x_3,x_4, x_5
x
3
,
x
4
,
x
5
are all greater than
x
2
x_2
x
2
. Prove that if
a
>
0
a > 0
a
>
0
, then
1
(
x
1
+
x
3
)
a
+
1
(
x
2
+
x
4
)
a
+
1
(
x
2
+
x
5
)
a
<
1
(
x
1
+
x
2
)
a
+
1
(
x
2
+
x
3
)
a
+
1
(
x
4
+
x
5
)
a
\frac{1}{(x_1 +x_3)^a}+ \frac{1}{(x_2 +x_4)^a}+ \frac{1}{(x_2 +x_5)^a} <\frac{1}{(x_1 +x_2)^a}+ \frac{1}{(x_2 +x_3)^a}+ \frac{1}{(x_4 +x_5)^a}
(
x
1
+
x
3
)
a
1
+
(
x
2
+
x
4
)
a
1
+
(
x
2
+
x
5
)
a
1
<
(
x
1
+
x
2
)
a
1
+
(
x
2
+
x
3
)
a
1
+
(
x
4
+
x
5
)
a
1
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