At the start of the Mighty Mathematicians Football Team's first game of the season, their coach noticed that the jersey numbers of the 22 players on the field were all the numbers from 1 to 22. At halftime, the coach substituted her goal-keeper, with jersey number 1, for a reserve player. No other substitutions were made by either team at or before halftime. The coach noticed that after the substitution, no two players on the field had the same jersey number and that the sums of the jersey numbers of each of the teams were exactly equal. Determine * the greatest possible jersey number of the reserve player, * the smallest possible (positive) jersey number of the reserve player.
Problem
Source: SAMO 2016 Q1
Tags: combinatorics
Nicio9
18.04.2024 11:36
If we leave out the reserve player, the greatest possible difference between the jersey
numbers of the two teams is obtained when the reserve player’s team has jersey numbers 2–11,
while the other team has numbers 12–22. The difference in this case is
(12 + 13 + ··· + 22) − (2 + 3 + ··· + 11) = 122,
which is therefore the greatest possible jersey number of the reserve player. On the other hand,
since
2 + 3 + 4 + ··· + 21 + 22 = 252
and the total sum of all jersey numbers must be even for the two teams to have the same sum,
the reserve player must have an even jersey number. The smallest positive even number that
is not already taken by another player is 24, and indeed it is possible that the sums of the two
teams are the same in this case, for example:
2, 5, 6, 9, 10, 13, 14, 17, 18, 20, 24 vs. 3, 4, 7, 8, 11, 12, 15, 16, 19, 21, 22.