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Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2008 Switzerland - Final Round
10
10
Part of
2008 Switzerland - Final Round
Problems
(1)
x + y^2 + z^3 + w^6 >=a (xyzw)^b
Source: Switzerland - 2008 Swiss MO Final Round p10
12/26/2022
Find all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive real numbers with the following properties: (i) For all positive real numbers
x
,
y
,
z
,
w
x, y, z,w
x
,
y
,
z
,
w
holds
x
+
y
2
+
z
3
+
w
6
≥
a
(
x
y
z
w
)
b
x + y^2 + z^3 + w^6 \ge a (xyzw)^{b}
x
+
y
2
+
z
3
+
w
6
≥
a
(
x
yz
w
)
b
. (ii) There is a quadruple
(
x
,
y
,
z
,
w
)
(x, y, z,w)
(
x
,
y
,
z
,
w
)
of positive real numbers such that in equality (i) applies.
algebra
inequalities