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Putnam
1965 Putnam
B2
B2
Part of
1965 Putnam
Problems
(1)
Putnam 1965 B2
Source:
9/28/2020
In a round-robin tournament with
n
n
n
players
P
1
P_1
P
1
,
P
2
P_2
P
2
,
…
\ldots
…
,
P
n
P_n
P
n
(where
n
>
1
n > 1
n
>
1
), each player plays one game with each of the other players and the rules are such that no ties can occur. Let
w
r
w_r
w
r
and
l
r
l_r
l
r
be the number of games won and lost, respectively, by
P
r
P_r
P
r
. Show that
∑
r
=
1
n
w
r
2
=
∑
r
=
1
n
l
r
2
.
\sum_{r=1}^nw_r^2 = \sum_{r=1}^nl_r^2.
r
=
1
∑
n
w
r
2
=
r
=
1
∑
n
l
r
2
.
Putnam