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Putnam
1966 Putnam
A1
Putnam 1966 A1
Putnam 1966 A1
Source:
April 6, 2022
college contests
Problem Statement
Let
f
(
n
)
f(n)
f
(
n
)
be the sum of the first
n
n
n
terms of the sequence
0
,
1
,
1
,
2
,
2
,
3
,
3
,
4
,
…
,
0,1,1,2,2,3,3,4, \dots,
0
,
1
,
1
,
2
,
2
,
3
,
3
,
4
,
…
,
where the
n
n
n
th term is given by
a
n
=
{
n
/
2
if
n
is even,
(
n
−
1
)
/
2
if
n
is odd.
a_n= \begin{cases} n/2 & \text{if } n \text{ is even,} \\ (n-1)/2 & \text{if } n \text{ is odd.} \end{cases}
a
n
=
{
n
/2
(
n
−
1
)
/2
if
n
is even,
if
n
is odd.
Show that if
x
x
x
and
y
y
y
are positive integers and
x
>
y
x>y
x
>
y
then
x
y
=
f
(
x
+
y
)
−
f
(
x
−
y
)
xy=f(x+y)-f(x-y)
x
y
=
f
(
x
+
y
)
−
f
(
x
−
y
)
.
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