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Putnam
1958 February Putnam
B3
B3
Part of
1958 February Putnam
Problems
(1)
Putnam 1958 February B3
Source: Putnam 1958 February
7/18/2022
In a round-robin tournament with
n
n
n
players in which there are no draws, the numbers of wins scored by the players are
s
1
,
s
2
,
…
,
s
n
s_1 , s_2 , \ldots, s_n
s
1
,
s
2
,
…
,
s
n
. Prove that a necessary and sufficient condition for the existence of three players
A
,
B
,
C
A,B,C
A
,
B
,
C
such that
A
A
A
beats
B
B
B
,
B
B
B
beats
C
C
C
, and
C
C
C
beats
A
A
A
is
s
1
2
+
s
2
2
+
…
+
s
n
2
<
(
2
n
−
1
)
(
n
−
1
)
n
6
.
s_{1}^{2} +s_{2}^{2} + \ldots +s_{n}^{2} < \frac{(2n-1)(n-1)n}{6}.
s
1
2
+
s
2
2
+
…
+
s
n
2
<
6
(
2
n
−
1
)
(
n
−
1
)
n
.
Putnam
graph