Question: Given positive integers a1, a2, ..., an. If (a1 + a2 + ... + an) - n + 1 pigeons are placed into n pigeonholes, then it is guaranteed that for some i, the i-th pigeonhole contains at least ai pigeons.

Pigeonhole Principle in Action: A Mathematical Proof

The pigeonhole principle is a fundamental concept in mathematics that states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. However, the given problem takes this principle to the next level by introducing a new condition: if (a1 + a2 + ... + an) - n + 1 pigeons are placed into n pigeonholes, then it is guaranteed that for some i, the i-th pigeonhole contains at least ai pigeons. In this article, we will explore how to prove this mathematical statement.

To begin with, let's understand the given condition. We have n pigeonholes and a total of (a1 + a2 + ... + an) - n + 1 pigeons. The key here is to recognize that the number of pigeons is not just a simple sum of the individual values ai, but rather a modified sum that takes into account the number of pigeonholes. The expression (a1 + a2 + ... + an) - n + 1 can be seen as a way to "normalize" the sum of the individual values, making it comparable to the number of pigeonholes.

Now, let's consider the scenario where each pigeonhole contains exactly ai pigeons. In this case, the total number of pigeons would be n * ai, which is equal to (a1 + a2 + ... + an). However, we are given that the total number of pigeons is (a1 + a2 + ... + an) - n + 1, which is less than n * ai. This implies that at least one pigeonhole must contain more than ai pigeons.

To formalize this argument, let's assume that each pigeonhole contains exactly ai pigeons. Then, the total number of pigeons would be n * ai, which is equal to (a1 + a2 + ... + an). However, we are given that the total number of pigeons is (a1 + a2 + ... + an) - n + 1, which is less than n * ai. This is a contradiction, as we assumed that each pigeonhole contains exactly ai pigeons.

Therefore, our assumption that each pigeonhole contains exactly ai pigeons must be false. This means that at least one pigeonhole must contain more than ai pigeons, which proves the given statement.

In conclusion, the given problem is a clever application of the pigeonhole principle, and the proof involves a simple yet elegant argument. By recognizing the relationship between the number of pigeons and the number of pigeonholes, we can derive a contradiction that leads to the desired result.