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National and Regional Contests
PEN Problems
PEN K Problems
24
K 24
K 24
Source:
May 25, 2007
function
induction
Functional Equations
Problem Statement
A function
f
f
f
is defined on the positive integers by
{
f
(
1
)
=
1
,
f
(
3
)
=
3
,
f
(
2
n
)
=
f
(
n
)
,
f
(
4
n
+
1
)
=
2
f
(
2
n
+
1
)
−
f
(
n
)
,
f
(
4
n
+
3
)
=
3
f
(
2
n
+
1
)
−
2
f
(
n
)
,
\left\{\begin{array}{rcl}f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.
⎩
⎨
⎧
f
(
1
)
f
(
3
)
f
(
2
n
)
f
(
4
n
+
1
)
f
(
4
n
+
3
)
=
=
=
=
=
1
,
3
,
f
(
n
)
,
2
f
(
2
n
+
1
)
−
f
(
n
)
,
3
f
(
2
n
+
1
)
−
2
f
(
n
)
,
for all positive integers
n
n
n
. Determine the number of positive integers
n
n
n
, less than or equal to 1988, for which
f
(
n
)
=
n
f(n) = n
f
(
n
)
=
n
.
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