## Problem of the Week 986## Not A Putnam ProblemIn 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© Copyright 2003 Stan Wagon. Reproduced with permission. |

22 April 2003