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2018 Pan African
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5
Part of
2018 Pan African
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(1)
PAMO Problem 5: Inequality involving fractions with strange condition
Source: 2018 Pan-African Mathematics Olympiad
7/3/2018
Let
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
be non-zero pairwise different real numbers such that
a
b
+
b
c
+
c
d
+
d
a
=
4
and
a
c
=
b
d
.
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd.
b
a
+
c
b
+
d
c
+
a
d
=
4
and
a
c
=
b
d
.
Show that
a
c
+
b
d
+
c
a
+
d
b
≤
−
12
\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12
c
a
+
d
b
+
a
c
+
b
d
≤
−
12
and that
−
12
-12
−
12
is the maximum.
inequalities
algebra
AMGM