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Austria 2010 algebra
Problem
We consider a group of trees in a nature reserve, all of which have a positive integral age. The average age is 41 years. After destruction of a tree with an age of 2010 years by lightning, the average age of the remaining trees is 40 years.
Determine the original number of trees in the group. What is the maximal number of trees of an age of 2010 years in the original group?
Determine the original number of trees in the group. What is the maximal number of trees of an age of 2010 years in the original group?
Solution
The original number of trees is denoted by and the sum of their ages by . Then we have On the other hand, we have This immediately yields As , there are at most 39 trees of an age of 2010 years. The maximum is for instance realised for 1930 trees of one year, one tree of 450 years and 39 trees of 2010 years. qed
Final answer
Original number of trees: 1970; Maximal number of 2010-year-old trees: 39
Techniques
Simple EquationsIntegers