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2022 CHMMC Winter (2022-23)
7
CHMMC 2022 Winter / 2022-23 Team #7
CHMMC 2022 Winter / 2022-23 Team #7
Source:
August 10, 2023
number theory
algebra
Problem Statement
Let
N
0
N_0
N
0
be the set of all non-negative integers. Let
f
:
N
0
×
N
0
→
N
0
f : N_0 \times N_0 \to N_0
f
:
N
0
×
N
0
→
N
0
be a function such that for all non-negative integers
a
,
b
a,b
a
,
b
:
f
(
a
,
b
)
=
f
(
b
,
a
)
,
f(a,b) = f(b,a),
f
(
a
,
b
)
=
f
(
b
,
a
)
,
f
(
a
,
0
)
=
0
,
f(a,0) = 0,
f
(
a
,
0
)
=
0
,
f
(
a
+
b
,
b
)
=
f
(
a
,
b
)
+
b
.
f(a+b,b) = f(a,b) +b.
f
(
a
+
b
,
b
)
=
f
(
a
,
b
)
+
b
.
Compute
∑
i
=
0
30
∑
j
=
0
2
i
−
1
f
(
2
i
,
j
)
\sum_{i=0}^{30}\sum_{j=0}^{2^i-1}f(2^i, j)
i
=
0
∑
30
j
=
0
∑
2
i
−
1
f
(
2
i
,
j
)
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