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2000 USAMO
6
The hardest USAMO inequality problem
The hardest USAMO inequality problem
Source: USAMO 2000/6
August 11, 2004
USAMO
inequalities
quadratics
vector
calculus
induction
Problem Statement
Let
a
1
,
b
1
,
a
2
,
b
2
,
…
,
a
n
,
b
n
a_1, b_1, a_2, b_2, \dots , a_n, b_n
a
1
,
b
1
,
a
2
,
b
2
,
…
,
a
n
,
b
n
be nonnegative real numbers. Prove that
∑
i
,
j
=
1
n
min
{
a
i
a
j
,
b
i
b
j
}
≤
∑
i
,
j
=
1
n
min
{
a
i
b
j
,
a
j
b
i
}
.
\sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}.
i
,
j
=
1
∑
n
min
{
a
i
a
j
,
b
i
b
j
}
≤
i
,
j
=
1
∑
n
min
{
a
i
b
j
,
a
j
b
i
}
.
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