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VTRMC
2017 VTRMC
5
2017 VTRMC #5
2017 VTRMC #5
Source:
August 8, 2018
algebra
Problem Statement
Let
f
(
x
,
y
)
=
(
x
+
y
)
/
2
,
g
(
x
,
y
)
=
x
y
,
h
(
x
,
y
)
=
2
x
y
/
(
x
+
y
)
f ( x , y ) = ( x + y ) / 2 , g ( x , y ) = \sqrt { x y } , h ( x , y ) = 2 x y / ( x + y )
f
(
x
,
y
)
=
(
x
+
y
)
/2
,
g
(
x
,
y
)
=
x
y
,
h
(
x
,
y
)
=
2
x
y
/
(
x
+
y
)
, and let
S
=
{
(
a
,
b
)
∈
N
×
N
∣
a
≠
b
and
f
(
a
,
b
)
,
g
(
a
,
b
)
,
h
(
a
,
b
)
∈
N
}
S = \{ ( a , b ) \in \mathrm { N } \times \mathrm { N } | a \neq b \text { and } f( a , b ) , g ( a , b ) , h ( a , b ) \in \mathrm { N } \}
S
=
{(
a
,
b
)
∈
N
×
N
∣
a
=
b
and
f
(
a
,
b
)
,
g
(
a
,
b
)
,
h
(
a
,
b
)
∈
N
}
where
N
\mathbb{N}
N
denotes the positive integers. Find the minimum of
f
f
f
over
S
S
S
.
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