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Benelux Olympiad 2018, Problem 1
Benelux Olympiad 2018, Problem 1
Source: Benelux 2018, Problem 1
April 28, 2018
BxMO
Benelux
inequalities
algebra
Problem Statement
(a) Determine the minimal value of
(
x
+
1
y
)
(
x
+
1
y
−
2018
)
+
(
y
+
1
x
)
(
y
+
1
x
−
2018
)
,
\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}-2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}-2018\right),
(
x
+
y
1
)
(
x
+
y
1
−
2018
)
+
(
y
+
x
1
)
(
y
+
x
1
−
2018
)
,
where
x
x
x
and
y
y
y
vary over the positive reals.(b) Determine the minimal value of
(
x
+
1
y
)
(
x
+
1
y
+
2018
)
+
(
y
+
1
x
)
(
y
+
1
x
+
2018
)
,
\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}+2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}+2018\right),
(
x
+
y
1
)
(
x
+
y
1
+
2018
)
+
(
y
+
x
1
)
(
y
+
x
1
+
2018
)
,
where
x
x
x
and
y
y
y
vary over the positive reals.
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