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Brazil National Olympiad
2019 Brazil National Olympiad
2
Inequation gcd
Inequation gcd
Source: Brazil National Olympiad 2019 - level - #2
November 23, 2019
number theory
inequalities
Problem Statement
Let
a
,
b
a, b
a
,
b
and
k
k
k
be positive integers with
k
>
1
k> 1
k
>
1
such that
l
c
m
(
a
,
b
)
+
g
c
d
(
a
,
b
)
=
k
(
a
+
b
)
lcm (a, b) + gcd (a, b) = k (a + b)
l
c
m
(
a
,
b
)
+
g
c
d
(
a
,
b
)
=
k
(
a
+
b
)
. Prove that
a
+
b
≥
4
k
a + b \geq 4k
a
+
b
≥
4
k
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