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2000 Croatia National Olympiad
Problem 1
log inequality
log inequality
Source: Croatia 2000 2nd Grade P1
May 8, 2021
inequalities
Problem Statement
Let
a
>
0
a>0
a
>
0
and
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
be real numbers with
x
1
+
x
2
+
x
3
=
0
x_1+x_2+x_3=0
x
1
+
x
2
+
x
3
=
0
. Prove that
log
2
(
1
+
a
x
1
)
+
log
2
(
1
+
a
x
2
)
+
log
2
(
1
+
a
x
3
)
≥
3.
\log_2\left(1+a^{x_1}\right)+\log_2\left(1+a^{x_2}\right)+\log_2\left(1+a^{x_3}\right)\ge3.
lo
g
2
(
1
+
a
x
1
)
+
lo
g
2
(
1
+
a
x
2
)
+
lo
g
2
(
1
+
a
x
3
)
≥
3.
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