Problem of the Week 986

Not A Putnam Problem

In the venerable Putnam competition a score can be any number from 0 to 120. This year the scores ranged from 0 to 116 and each score earned a ranking as indicated in the data set below. The rank is computed as follows: Suppose k students achieve score s. The rank of s is the average of what the ranks would be if the students all scored slightly different scores very near s. Example: If 20 students scored more than s, and if 5 students had score s, the the rank of s is the average of 21, 22, 23, 24, and 25, which is 23.

In the 2002 event Macalester's top student was Michael Decker, who scored 40.9 points. How many students achieved a score of 40.9?

The scores and ranks from 2002 are as follows.

Score    Rank

116       1
108       2
106       3
96       4.5
95       6
94       7
91       8
87       9
85      10
82      11
81      12
80      13.5
79      15
78      17.5
75      20
74      21
73      22
72      23
70      26
69      29
68      31.5
67      35.5
66      40.5
65      44.5
64      46
62      48.5
61      53
60      61
59      71.5
58      82.5
57      92
56      99
55     104
53     107.5
52     111
51     116.5
50     127.5
49     138.5
48     146
47     157
46     167.5
45     171.5
44     173
43     174.5
42     180.5
41     189.5
40.9   198.5
40.8   204.5
40.7   206
40.6   207
40.5   208
40.4   209
40.3   210
40     222.5
39     246
38     264
37     276
36     283.5
35     286.5
34     292
33     299
32     314
31     335.5
30     370.5
29     415
28     447
27     469
26     481.5
25     489
24     497
23     512.5
22     545
21     595
20     676
19     758.5
18     809.5
17     836
16     848.5
15     859.5
14     875.5
13     910
12     996
11    1113.5
10    1288
 9    1453.5
 8    1528.5
 7    1559.5
 6    1566
 5    1578
 4    1611.5
 3    1672.5
 2    1859.5
 1    2100
 0    2768.5

Source: Keith Brandt and Donald Vestal An inversion formula for Putnam data Crux Mathematicorum 29:2 (March 2003) 106-109