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Putnam
1979 Putnam
A5
A5
Part of
1979 Putnam
Problems
(1)
Putnam 1979 A5
Source:
4/8/2022
Denote by
⌈
x
⌉
\lceil x \rceil
⌈
x
⌉
the greatest integer less than or equal to
x
x
x
and by
S
(
x
)
S(x)
S
(
x
)
the sequence
⌈
x
⌉
,
⌈
2
x
⌉
,
⌈
3
x
⌉
,
…
.
\lceil x \rceil, \lceil 2x \rceil, \lceil 3x \rceil, \dots.
⌈
x
⌉
,
⌈
2
x
⌉
,
⌈
3
x
⌉
,
…
.
Prove that there are distinct real solutions
α
\alpha
α
and
β
\beta
β
of the equation
x
3
−
10
x
2
+
29
x
−
25
=
0
x^3-10x^2+29x-25=0
x
3
−
10
x
2
+
29
x
−
25
=
0
such that infinitely many positive integers appear both in
S
(
α
)
S(\alpha)
S
(
α
)
and in
S
(
β
)
.
S(\beta).
S
(
β
)
.
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