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Turkey Team Selection Test
2020 Turkey Team Selection Test
4
Functional inequality
Functional inequality
Source: 2020 Turkey TST P4
March 14, 2020
function
Inequality
inequalities
Problem Statement
Let
Z
+
Z^+
Z
+
be positive integers set.
f
:
Z
+
→
Z
+
f:\mathbb{Z^+}\to\mathbb{Z^+}
f
:
Z
+
→
Z
+
is a function and we show
f
∘
f
∘
.
.
.
∘
f
f \circ f \circ ...\circ f
f
∘
f
∘
...
∘
f
with
f
l
f_l
f
l
for all
l
∈
Z
+
l\in \mathbb{Z^+}
l
∈
Z
+
where
f
f
f
is repeated
l
l
l
times. Find all
f
:
Z
+
→
Z
+
f:\mathbb{Z^+}\to\mathbb{Z^+}
f
:
Z
+
→
Z
+
functions such that
(
n
−
1
)
2020
<
∏
l
=
1
2020
f
l
(
n
)
<
n
2020
+
n
2019
(n-1)^{2020}< \prod _{l=1}^{2020} {f_l}(n)< n^{2020}+n^{2019}
(
n
−
1
)
2020
<
l
=
1
∏
2020
f
l
(
n
)
<
n
2020
+
n
2019
for all
n
∈
Z
+
n\in \mathbb{Z^+}
n
∈
Z
+
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