Problem 4.8. In a grove, there are trees of four species: birches, firs, pines, and aspens. There are a total of 100 trees. It is known that among any 85 trees, there will be trees of all four species. Among what minimum number of any trees in this grove will there definitely be trees of at least three species?
Answer: 69.

Solution. Suppose there are no more than 15 birches in the grove. Then there are at least 85 other trees, and according to the problem's condition, among them, there must be trees of all four types. This is a contradiction. Therefore, there must be at least 16 birches in the grove. Similarly, we can conclude that there are at least 16 trees of each type.

We will prove that among any 69 trees, there will definitely be three different types of trees. Suppose this is not the case, and among some 69 trees, there are only two different types. Then among the remaining 31 trees, there are trees of the other two types, but as we have proven earlier, each type is represented by at least 16 trees. This is a contradiction.

Now, let's provide an example where among some 68 trees, there are no more than two types (i.e., 68 trees may "not be enough"). Suppose in the grove, there are 34 birches, 34 firs, 16 pines, and 16 aspens. From the previous reasoning, we understand that among any 85 trees, there will be trees of all four types (since among the missing 15 trees, no type can be entirely absent). However, if we take all the birches and firs, a total of 68 trees, then among them, there will be no trees of three types.

## 5th grade