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PrintSouth African Mathematics Olympiad Third Round
South Africa algebra
Problem
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 (11 players per team) were all the numbers from to . At half-time, the coach substituted her goal-keeper (who had the number on her jersey) for a reserve player. The coach then noticed that after the substitution, no two players on the field had the same jersey number and that the sum of the jersey numbers of each of the teams were exactly equal.
a) What is the smallest (positive) possible jersey number of the reserve player?
b) What is the greatest (positive) possible jersey number of the reserve player?
[10]
a) What is the smallest (positive) possible jersey number of the reserve player?
b) What is the greatest (positive) possible jersey number of the reserve player?
[10]
Solution
The sum from to is .
Therefore the new number must be even or otherwise the sum can't be exactly divisible by .
The smallest possible even number we can use is . To see that this is indeed possible take
To find the greatest number possible, we try and put the smallest numbers with the greatest number. The sum of the numbers from to is
Hence the largest total can be . Subtracting the numbers to from this gives us which is an even number. Thus is the largest possible jersey number to achieve this.
Therefore the new number must be even or otherwise the sum can't be exactly divisible by .
The smallest possible even number we can use is . To see that this is indeed possible take
To find the greatest number possible, we try and put the smallest numbers with the greatest number. The sum of the numbers from to is
Hence the largest total can be . Subtracting the numbers to from this gives us which is an even number. Thus is the largest possible jersey number to achieve this.
Final answer
a) 24; b) 122
Techniques
Sums and products