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Soros Olympiad in Mathematics
VI Soros Olympiad 1999 - 2000 (Russia)
11.3
sum a^4 + 2(sum ab^2 )^2 < 1 - VI Soros Olympiad 1999-00 Round 1 11.3
sum a^4 + 2(sum ab^2 )^2 < 1 - VI Soros Olympiad 1999-00 Round 1 11.3
Source:
May 21, 2024
algebra
inequalities
Problem Statement
The numbers
a
,
b
a, b
a
,
b
and
c
c
c
are such that
a
2
+
b
2
+
c
2
=
1
a^2 + b^2 + c^2 = 1
a
2
+
b
2
+
c
2
=
1
. Prove that
a
4
+
b
4
+
c
4
+
2
(
a
b
2
+
b
c
2
+
c
a
2
)
2
≤
1.
a^4 + b^4 + c^4 + 2(ab^2 + bc^2 + ca^2)^2\le 1.
a
4
+
b
4
+
c
4
+
2
(
a
b
2
+
b
c
2
+
c
a
2
)
2
≤
1.
At what
a
,
b
a, b
a
,
b
and
c
c
c
does inequality turn into equality?
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