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Polish MO Finals
1994 Polish MO Finals
3
n distinct reals and a,b,c,d
n distinct reals and a,b,c,d
Source: Problem 6, Polish NO 1994
October 7, 2005
inequalities unsolved
inequalities
Problem Statement
The distinct reals
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ... , x_n
x
1
,
x
2
,
...
,
x
n
,(
n
>
3
n > 3
n
>
3
) satisfy
∑
i
=
1
n
x
i
=
0
\sum_{i=1}^n x_i = 0
∑
i
=
1
n
x
i
=
0
,
∑
i
=
1
n
x
i
2
=
1
\sum_{i=1}^n x_i ^2 = 1
∑
i
=
1
n
x
i
2
=
1
. Show that four of the numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
must satisfy:
a
+
b
+
c
+
n
a
b
c
≤
∑
i
=
1
n
x
i
3
≤
a
+
b
+
d
+
n
a
b
d
a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd
a
+
b
+
c
+
nab
c
≤
i
=
1
∑
n
x
i
3
≤
a
+
b
+
d
+
nab
d
.
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