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National and Regional Contests
Austria Contests
Austrian MO National Competition
2005 Federal Competition For Advanced Students, Part 2
2
2
Part of
2005 Federal Competition For Advanced Students, Part 2
Problems
(2)
Austrian MO 2005 inequality
Source: Austrian MO 2005 round 2
6/27/2005
Prove that for all positive reals
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
, we have
a
+
b
+
c
+
d
a
b
c
d
≤
1
a
3
+
1
b
3
+
1
c
3
+
1
d
3
\frac{a+b+c+d}{abcd}\leq \frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}+\frac{1}{d^{3}}
ab
c
d
a
+
b
+
c
+
d
≤
a
3
1
+
b
3
1
+
c
3
1
+
d
3
1
inequalities
inequalities unsolved
(a,b,c,d,e,f)
Source: Austrian MO 2005 round 2
6/27/2005
Find all real
a
,
b
,
c
,
d
,
e
,
f
a,b,c,d,e,f
a
,
b
,
c
,
d
,
e
,
f
that satisfy the system
4
a
=
(
b
+
c
+
d
+
e
)
4
4a = (b + c + d + e)^4
4
a
=
(
b
+
c
+
d
+
e
)
4
4
b
=
(
c
+
d
+
e
+
f
)
4
4b = (c + d + e + f)^4
4
b
=
(
c
+
d
+
e
+
f
)
4
4
c
=
(
d
+
e
+
f
+
a
)
4
4c = (d + e + f + a)^4
4
c
=
(
d
+
e
+
f
+
a
)
4
4
d
=
(
e
+
f
+
a
+
b
)
4
4d = (e + f + a + b)^4
4
d
=
(
e
+
f
+
a
+
b
)
4
4
e
=
(
f
+
a
+
b
+
c
)
4
4e = (f + a + b + c)^4
4
e
=
(
f
+
a
+
b
+
c
)
4
4
f
=
(
a
+
b
+
c
+
d
)
4
4f = (a + b + c + d)^4
4
f
=
(
a
+
b
+
c
+
d
)
4
algebra unsolved
algebra