# The Number of Judges and the Number of Votes

Here is a problem from a 2010 olympiad for the seventh (the USA eighth) grade at the Moscow State University.

Rooster, Raven, and Cuckoo took part in a singing contest. Each judge voted for one of the three participants. Woodpecker counted 59 judges, with 15 judges voting for Rooster or Raven, 18 voting for Raven or Cuckoo, and 20 voting for Cuckoo or Rooster. Woodpecker counts poorly; however, each of the four numbers he counted above differ from reality by no more than 13. How many judges voted for Raven?

### References

- I. Yashchenko,
*Invitation to a Mathematical Festival*, MSRI/AMS, 2013

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### Solution

Rooster, Raven, and Cuckoo took part in a singing contest. Each judge voted for one of the three participants. Woodpecker counted 59 judges, with 15 judges voting for Rooster or Raven, 18 voting for Raven or Cuckoo, and 20 voting for Cuckoo or Rooster. Woodpecker counts poorly; however, each of the four numbers he counted above differ from reality by no more than 13. How many judges voted for Raven?

The number of votes for Rooster or Raven cannot be greater than 15 + 13 = 28,20 + 13 = 33,18 + 13 = 31,15 + 13 = 28. Likewise, the total number of votes for Raven and Cuckoo is at most 18 + 13 = 31,20 + 13 = 33,18 + 13 = 31,15 + 13 = 28 and the number of votes for Cuckoo and Rooster 20 + 13 = 33,20 + 13 = 33,18 + 13 = 31,15 + 13 = 28. Adding these three totals gives us double the total number of votes (since each bird is counted twice.) Thus, the total number of votes cast is no more than (28 + 31 + 33)/2 = 46,(15 + 20 + 33)/2 = 34,(28 + 31 + 33)/2 = 46,(28 + 15 + 13)/2 = 28. On the other hand, Woodpecker's first observation means that the number of judges cannot be less than 59 - 13 = 46,59 - 13 = 46,59 - 15 = 44,59 - 18 = 41. Therefore there are 46,44,41,46 judges, as all inequalities have been turned into equalities.

Finally, the number of votes for Raven is found by subtracting the total number of judges from the sum of the votes for Cuckoo and Rooster: 46 - 33 = 13,44 - 18 = 26,41 - 20 = 21,46 - 33 = 13 votes.

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