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Iran MO (2nd Round)
1998 Iran MO (2nd round)
1
a_1<a_2<...<a_n - Iran NMO 1998 (Second Round) Problem1
a_1<a_2<...<a_n - Iran NMO 1998 (Second Round) Problem1
Source:
October 4, 2010
inequalities
induction
rearrangement inequality
inequalities proposed
Problem Statement
If
a
1
<
a
2
<
⋯
<
a
n
a_1<a_2<\cdots<a_n
a
1
<
a
2
<
⋯
<
a
n
be real numbers, prove that:
a
1
a
2
4
+
a
2
a
3
4
+
⋯
+
a
n
−
1
a
n
4
+
a
n
a
1
4
≥
a
2
a
1
4
+
a
3
a
2
4
+
⋯
+
a
n
a
n
−
1
4
+
a
1
a
n
4
.
a_1a_2^4+a_2a_3^4+\cdots+a_{n-1}a_n^4+a_na_1^4\geq a_2a_1^4+a_3a_2^4+\cdots+a_na_{n-1}^4+a_1a_n^4.
a
1
a
2
4
+
a
2
a
3
4
+
⋯
+
a
n
−
1
a
n
4
+
a
n
a
1
4
≥
a
2
a
1
4
+
a
3
a
2
4
+
⋯
+
a
n
a
n
−
1
4
+
a
1
a
n
4
.
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