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CIIM
2021 CIIM
4
4
Part of
2021 CIIM
Problems
(1)
function, inequality, function, ine...
Source: CIIM 2021 #4
10/31/2021
Let
Z
+
\mathbb{Z}^{+}
Z
+
be the set of positive integers. a) Prove that there is only one function
f
:
Z
+
→
Z
+
f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}
f
:
Z
+
→
Z
+
, strictly increasing, such that
f
(
f
(
n
)
)
=
2
n
+
1
f(f(n))=2n+1
f
(
f
(
n
))
=
2
n
+
1
for every
n
∈
Z
+
n\in \mathbb{Z}^{+}
n
∈
Z
+
. b) For the function in a. Prove that for every
n
∈
Z
+
n\in \mathbb{Z}^{+}
n
∈
Z
+
4
n
+
1
3
≤
f
(
n
)
≤
3
n
+
1
2
\frac{4n+1}{3}\leq f(n)\leq \frac{3n+1}{2}
3
4
n
+
1
≤
f
(
n
)
≤
2
3
n
+
1
c) Prove that in each inequality side of b the equality can reach by infinite positive integers
n
n
n
.
function
inequalities