International Mathematical Olympiad 2013
Understanding the Problem
The opening problem of the International Mathematical Olympiad is often designed to be accessible while still containing a clever idea. IMO 2013 Problem 1 is a perfect example. At first glance, the statement looks harmless. We are given an increasing sequence of positive integers and asked to show that there is exactly one place where a certain average falls between two consecutive terms of the sequence.
Many students who first encounter this problem immediately start experimenting with examples. This is a natural reaction. If we take a simple increasing sequence such as 1, 2, 3, 4, 5, … and compute the averages described in the problem, a pattern quickly appears. The average seems to move through the sequence in a predictable way. This observation suggests that the statement is not a coincidence but rather the consequence of a deeper property shared by all increasing sequences.
The challenge is that the average depends on many terms at once. Every time we move from one index to the next, both the sum and the denominator change. Following the average directly soon becomes difficult. This is where experienced olympiad contestants begin to think differently. Instead of studying the average itself, they look for another quantity that contains the same information but is easier to analyze.
A useful habit in olympiad mathematics is to ask what the problem is really trying to tell us. The statement compares an average with consecutive terms of a sequence. Whenever averages and inequalities appear together, there is often some hidden monotonic behaviour waiting to be discovered. In other words, there may be a quantity that steadily increases or steadily decreases as we move along the sequence.
Another clue comes from the word “unique.” Existence and uniqueness together are often strong indicators that some quantity changes sign exactly once. If a number starts positive and eventually becomes negative while moving in only one direction, then there must be a unique moment when the transition occurs. Many olympiad problems are solved by identifying precisely such a quantity.
What makes this problem elegant is that the sequence is completely arbitrary apart from being strictly increasing. There is no explicit formula and no recurrence relation. This means that any successful solution must use only the most fundamental property of increasing sequences. The proof cannot rely on special tricks or computations. Instead, it must uncover a structural fact that is true for every increasing sequence of positive integers.
This problem is also a good lesson in mathematical maturity. A beginner might spend a long time manipulating the given inequality directly. An experienced solver, however, quickly realizes that the inequality is not the real object of study. The real task is to find the right perspective from which the inequality becomes natural. Once that perspective is found, the proof becomes surprisingly short and clean.
Looking back after solving the problem, many students are surprised by how little calculation is actually required. The solution is driven almost entirely by a single idea. This is one of the hallmarks of a good olympiad problem. The difficulty lies not in carrying out complicated algebra but in discovering the viewpoint that reveals the hidden structure.
For anyone training for mathematical olympiads, this problem is valuable far beyond the competition itself. It teaches a recurring lesson: when a quantity is difficult to study directly, search for another quantity that captures the same information in a simpler form. Learning to make that transition is one of the most important skills in advanced problem solving.