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1
Romanian National Olympiad 2018 - Grade 10 - problem 1
Romanian National Olympiad 2018 - Grade 10 - problem 1
Source: Romania NMO - 2018
April 12, 2018
Problem Statement
Let
n
∈
N
≥
2
n \in \mathbb{N}_{\geq 2}
n
∈
N
≥
2
and
a
1
,
a
2
,
…
,
a
n
∈
(
1
,
∞
)
.
a_1,a_2, \dots , a_n \in (1,\infty).
a
1
,
a
2
,
…
,
a
n
∈
(
1
,
∞
)
.
Prove that
f
:
[
0
,
∞
)
→
R
f:[0,\infty) \to \mathbb{R}
f
:
[
0
,
∞
)
→
R
with
f
(
x
)
=
(
a
1
a
2
.
.
.
a
n
)
x
−
a
1
x
−
a
2
x
−
.
.
.
−
a
n
x
f(x)=(a_1a_2...a_n)^x-a_1^x-a_2^x-...-a_n^x
f
(
x
)
=
(
a
1
a
2
...
a
n
)
x
−
a
1
x
−
a
2
x
−
...
−
a
n
x
is a strictly increasing function.
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