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IMO Shortlist
1970 IMO Shortlist
9
Inequality on 2n reals - ISL 1970
Inequality on 2n reals - ISL 1970
Source:
September 23, 2010
Inequality
n-variable inequality
IMO Shortlist
Problem Statement
Let
u
1
,
u
2
,
…
,
u
n
,
v
1
,
v
2
,
…
,
v
n
u_1, u_2, \ldots, u_n, v_1, v_2, \ldots, v_n
u
1
,
u
2
,
…
,
u
n
,
v
1
,
v
2
,
…
,
v
n
be real numbers. Prove that
1
+
∑
i
=
1
n
(
u
i
+
v
i
)
2
≤
4
3
(
1
+
∑
i
=
1
n
u
i
2
)
(
1
+
∑
i
=
1
n
v
i
2
)
.
1+ \sum_{i=1}^n (u_i+v_i)^2 \leq \frac 43 \Biggr( 1+ \sum_{i=1}^n u_i^2 \Biggl) \Biggr( 1+ \sum_{i=1}^n v_i^2 \Biggl) .
1
+
i
=
1
∑
n
(
u
i
+
v
i
)
2
≤
3
4
(
1
+
i
=
1
∑
n
u
i
2
)
(
1
+
i
=
1
∑
n
v
i
2
)
.
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