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Caucasus Mathematical Olympiad
2024 Caucasus Mathematical Olympiad
1
Easy algebra on 4 positive reals
Easy algebra on 4 positive reals
Source: Caucasus MO 2024, Seniors P1
March 15, 2024
algebra
Problem Statement
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be positive real numbers. It is given that at least one of the following two conditions holds:
a
b
>
min
(
c
d
,
d
c
)
,
c
d
>
min
(
a
b
,
b
a
)
.
ab >\min(\frac{c}{d}, \frac{d}{c}), cd >\min(\frac{a}{b}, \frac{b}{a}).
ab
>
min
(
d
c
,
c
d
)
,
c
d
>
min
(
b
a
,
a
b
)
.
Show that at least one of the following two conditions holds:
b
d
>
min
(
c
a
,
a
c
)
,
c
a
>
min
(
d
b
,
b
d
)
.
bd>\min(\frac{c}{a}, \frac{a}{c}), ca >\min(\frac{d}{b}, \frac{b}{d}).
b
d
>
min
(
a
c
,
c
a
)
,
c
a
>
min
(
b
d
,
d
b
)
.
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