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STEMS 2023 Math Cat A
4
Sum of possible function values
Sum of possible function values
Source: STEMS 2023 Maths CAT A Part A P4
January 8, 2023
function
Computational
STEMS
algebra
Problem Statement
Let
f
:
N
→
N
f : \mathbb{N} \to \mathbb{N}
f
:
N
→
N
be a function such that the following conditions hold:
(
1
)
f
(
1
)
=
1.
\qquad\ (1) \; f(1) = 1.
(
1
)
f
(
1
)
=
1.
(
2
)
(
x
+
y
)
2
<
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
∀
x
,
y
∈
N
.
\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.
(
2
)
2
(
x
+
y
)
<
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
∀
x
,
y
∈
N
.
(
3
)
f
(
4
n
+
1
)
<
2
f
(
2
n
+
1
)
∀
n
≥
0.
\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.
(
3
)
f
(
4
n
+
1
)
<
2
f
(
2
n
+
1
)
∀
n
≥
0.
(
4
)
f
(
4
n
+
3
)
≤
2
f
(
2
n
+
1
)
∀
n
≥
0.
\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.
(
4
)
f
(
4
n
+
3
)
≤
2
f
(
2
n
+
1
)
∀
n
≥
0.
Find the sum of all possible values of
f
(
2023
)
f(2023)
f
(
2023
)
.
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