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2012 APMO
5
APMO 2012 #5
APMO 2012 #5
Source: APMO 2012 #5
April 2, 2012
inequalities
LaTeX
function
inequalities proposed
Problem Statement
Let
n
n
n
be an integer greater than or equal to
2
2
2
. Prove that if the real numbers
a
1
,
a
2
,
⋯
,
a
n
a_1 , a_2 , \cdots , a_n
a
1
,
a
2
,
⋯
,
a
n
satisfy
a
1
2
+
a
2
2
+
⋯
+
a
n
2
=
n
a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n
a
1
2
+
a
2
2
+
⋯
+
a
n
2
=
n
, then
∑
1
≤
i
<
j
≤
n
1
n
−
a
i
a
j
≤
n
2
\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2}
1
≤
i
<
j
≤
n
∑
n
−
a
i
a
j
1
≤
2
n
must hold.
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