Question
What is the minimum number of times an integer must be removed from a set of 100 integers to guarantee the removal of an even integer?
Answers
GPT-4o mini
To determine the minimum number of integers that must be removed from a set of 100 integers to ensure at least one even integer is removed, we begin by considering the worst-case scenario regarding the composition of the set.
1. **Composition of the Integer Set**: The set can contain a mix of even and odd integers. In the worst-case scenario for our goal (removing an even integer), we consider the situation where we maximize the number of odd integers in the set.
2. **Maximizing Odd Integers**: Suppose we have a set of 100 integers consisting entirely of odd integers. Therefore:
- Total odd integers in the set = 100
- Total even integers in the set = 0
3. **Removal Strategy**: If we begin removing integers from this set and only odd integers are present, we can continue to remove up to 100 odd integers without removing any even integers.
4. **Need to Guarantee Removal of an Even Integer**: To guarantee that we remove at least one even integer, we can add one more removal after potentially removing all the unusual odds:
- If I remove all 100 odd integers (removing 100 integers total), I still have not removed an even integer because there were none present in the set originally.
5. **Determining Total Removals**: Thus to ensure that we are guaranteed to remove an even integer, we can consider a case where we have up to 99 odd integers remaining or any set configurations. As a strategy:
- After removing 99 integers (the maximum number of odd integers), if there is at least one even integer in the original set, it remains.
6. **Conclusion**: Therefore, to guarantee that we will remove at least one even integer, it suffices to remove one more integer beyond 99 odd integers, totaling:
\[
\text{Minimum removals} = 99 + 1 = 100
\]
Thus the minimum number of integers that must be removed from the set to guarantee that at least one even integer is removed is:
\[
\boxed{100}
\]
1. **Composition of the Integer Set**: The set can contain a mix of even and odd integers. In the worst-case scenario for our goal (removing an even integer), we consider the situation where we maximize the number of odd integers in the set.
2. **Maximizing Odd Integers**: Suppose we have a set of 100 integers consisting entirely of odd integers. Therefore:
- Total odd integers in the set = 100
- Total even integers in the set = 0
3. **Removal Strategy**: If we begin removing integers from this set and only odd integers are present, we can continue to remove up to 100 odd integers without removing any even integers.
4. **Need to Guarantee Removal of an Even Integer**: To guarantee that we remove at least one even integer, we can add one more removal after potentially removing all the unusual odds:
- If I remove all 100 odd integers (removing 100 integers total), I still have not removed an even integer because there were none present in the set originally.
5. **Determining Total Removals**: Thus to ensure that we are guaranteed to remove an even integer, we can consider a case where we have up to 99 odd integers remaining or any set configurations. As a strategy:
- After removing 99 integers (the maximum number of odd integers), if there is at least one even integer in the original set, it remains.
6. **Conclusion**: Therefore, to guarantee that we will remove at least one even integer, it suffices to remove one more integer beyond 99 odd integers, totaling:
\[
\text{Minimum removals} = 99 + 1 = 100
\]
Thus the minimum number of integers that must be removed from the set to guarantee that at least one even integer is removed is:
\[
\boxed{100}
\]