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2013 Irish Math Olympiad
10
Irish Mathematical Olympiad 2013 (2)
Irish Mathematical Olympiad 2013 (2)
Source: Paper 2 , Problem 10
September 6, 2014
inequalities
inequalities proposed
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers and let
x
=
a
+
b
+
c
,
y
=
a
2
+
b
2
+
c
2
,
z
=
a
3
+
b
3
+
c
3
x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3
x
=
a
+
b
+
c
,
y
=
a
2
+
b
2
+
c
2
,
z
=
a
3
+
b
3
+
c
3
and
S
=
2
x
3
−
9
x
y
+
9
z
.
S=2x^3-9xy+9z .
S
=
2
x
3
−
9
x
y
+
9
z
.
(
a
)
(a)
(
a
)
Prove that
S
S
S
is unchanged when
a
,
b
,
c
a,b,c
a
,
b
,
c
are replaced by
a
+
t
,
b
+
t
,
c
+
t
a+t,b+t,c+t
a
+
t
,
b
+
t
,
c
+
t
, respectively , for any real number
t
t
t
.
(
b
)
(b)
(
b
)
Prove that
(
3
y
−
x
2
)
3
≥
S
2
.
(3y-x^2)^3\ge S^2 .
(
3
y
−
x
2
)
3
≥
S
2
.
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