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Greece Contests
Greece National Olympiad
1994 Greece National Olympiad
3
(a-b)^2+(b-c)^2+(c-d)^2+(a-c)^2+(a-d)^2+(b-d)^2<= 4
(a-b)^2+(b-c)^2+(c-d)^2+(a-c)^2+(a-d)^2+(b-d)^2<= 4
Source: 1993 Greece MO p3
September 6, 2024
algebra
inequalities
Problem Statement
If
a
2
+
b
2
+
c
2
+
d
2
=
1
a^2+b^2+c^2+d^2=1
a
2
+
b
2
+
c
2
+
d
2
=
1
, prove that
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
d
)
2
+
(
a
−
c
)
2
+
(
a
−
d
)
2
+
(
b
−
d
)
2
≤
4
(a-b)^2+(b-c)^2+(c-d)^2+(a-c)^2+(a-d)^2+(b-d)^2\leq 4
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
d
)
2
+
(
a
−
c
)
2
+
(
a
−
d
)
2
+
(
b
−
d
)
2
≤
4
When does equality holds?
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