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Korea Junior Mathematics Olympiad
2024 Korea Junior Math Olympiad
8
Dirty function(not dirty)
Dirty function(not dirty)
Source: 2024 KJMO P8
November 9, 2024
function
algebra
Problem Statement
f
f
f
is a function from the set of positive integers to the set of all integers that satisfies the following.
⋅
\cdot
⋅
f
(
1
)
=
1
,
f
(
2
)
=
−
1
f(1)=1, f(2)=-1
f
(
1
)
=
1
,
f
(
2
)
=
−
1
⋅
\cdot
⋅
f
(
n
)
+
f
(
n
+
1
)
+
f
(
n
+
2
)
=
f
(
⌊
n
+
2
3
⌋
)
f(n)+f(n+1)+f(n+2)=f(\left\lfloor\frac{n+2}{3}\right\rfloor)
f
(
n
)
+
f
(
n
+
1
)
+
f
(
n
+
2
)
=
f
(
⌊
3
n
+
2
⌋
)
Find the number of positive integers
k
k
k
not exceeding
1000
1000
1000
such that
f
(
3
)
+
f
(
6
)
+
⋯
+
f
(
3
k
−
3
)
+
f
(
3
k
)
=
5
f(3)+f(6)+\cdots+f(3k-3)+f(3k)=5
f
(
3
)
+
f
(
6
)
+
⋯
+
f
(
3
k
−
3
)
+
f
(
3
k
)
=
5
.
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