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Algebra Difficulty 5.5 AIME, harder Prove it Austria

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?

Solution

The original number of trees is denoted by $n$ and the sum of their ages by $s$ .
Then we have
$s = 41n.$
On the other hand, we have
$s - 2010 = 40(n - 1).$
This immediately yields
$n = 1970 \quad \text{and} \quad s = 80770.$
As $80770 = 2010 \cdot 40 + 370 = 2010 \cdot 39 + 2380$ , 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

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