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Poland Contests
Polish MO Finals
1996 Polish MO Finals
3
two n-tuples and inequality
two n-tuples and inequality
Source:
October 30, 2005
inequalities
LaTeX
inequalities unsolved
Problem Statement
a
i
,
x
i
a_i, x_i
a
i
,
x
i
are positive reals such that
a
1
+
a
2
+
.
.
.
+
a
n
=
x
1
+
x
2
+
.
.
.
+
x
n
=
1
a_1 + a_2 + ... + a_n = x_1 + x_2 + ... + x_n = 1
a
1
+
a
2
+
...
+
a
n
=
x
1
+
x
2
+
...
+
x
n
=
1
. Show that
2
∑
i
<
j
x
i
x
j
≤
n
−
2
n
−
1
+
∑
a
i
x
i
2
1
−
a
i
2 \sum_{i<j} x_ix_j \leq \frac{n-2}{n-1} + \sum \frac{a_ix_i ^2}{1-a_i}
2
i
<
j
∑
x
i
x
j
≤
n
−
1
n
−
2
+
∑
1
−
a
i
a
i
x
i
2
When do we have equality?
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