The Toughest and the Easiest Math Problems Asked in the IMO
“There should be no such thing as boring mathematics.”
I have been a part of several physics and math competitions during my high school years. One of the best things that I like about studying mathematics is that there is no limit to the complexity and I think that is what makes it extremely beautiful. Just like the unanswered questions help science to move forward, the complexity in mathematics is what makes it so interesting and drives it forward. Complexity is a subjective term. Something Complex for one individual might not necessarily be complex for the other depending upon the extent of the expertise that one has in that field.
International Mathematical Olympiad (IMO is an annual competition that takes place every year in a different nation organized by the Science Olympiad Foundation (SOF). According to the official website of IMO, the first International Mathematical Olympiad was held in 1959 in Romania. Ever since then the number of countries participating in the IMO has been increasing. One of the best things about such Olympiads is that they bring together teams from different levels and different countries and different backgrounds together on one platform where they work together in collaboration and against one another on many grounds. This allows them to know their levels of understanding and problem-solving skills. You get to meet and compete with like-minded intellectuals and learn from them.
In this article, I shall share the hardest and the easiest problems ever asked in International Mathematics Olympiad. As mentioned earlier, the difficulty is a subjective concept. The complexity of these problems has been measured based on scores secured by the participants at the IMO.
“There should be no such thing as boring mathematics.”
- Edsger W. Dijkstra.
P6, IMO 1988: the hardest problem
The toughest problem ever asked in any International Mathematical Olympiad competition hands down has to be problem 6 of IMO 1988. Before explaining why this problem drags the credit of being the most complicated problem ever, let’s first understand what the problem was.
Why was this problem so tough? One of the reasons why this question was so hard is because the question tries to play with your mind. It might seem easy at first but when you dig into it, it becomes increasingly complicated, even for mathematicians. This problem was first given to the six members of the Australian problem committee by someone from West Germany. The committee included some of the most prolific mathematicians and problem-solvers of the time. None of the committee members could solve the problem.
After submitting the problem to the Jury committee of IMO, the Jury finally decided to add it to the IMO question paper on number 6. Fields medalist Ngô Bảo Châu, and mathematician Ravi Vakil were two of the eleven participants who were able to score the highest for the problem. Fields medalist and math prodigy Terence Tao scored a 1 out of 7 for this problem. Though at the time he was only 13 years old.
A solution to the problem can be studied here:
“Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.”
― Deepak Chopra
P1, IMO 1959: the easiest problem
Now talking about the easiest problem ever asked in International Mathematical Olympiad was problem 1 of 1959. This was the first-ever question of the first-ever IMO and the question was this:
Even though the questions asked at the IMO every year are quite complex for any general public and even for people with a mathematical background, this one particular was comparatively easy for anyone having a fundamental knowledge of algebra. The question wasn’t perfectly solved by all the participants when it was first asked in 1959, perhaps because it was the first-ever Olympiad of Mathematics at an International level and the participants weren’t quite acquainted with the sort of questions and the complexity of questions that come in the IMO. Although if today this question is asked to even anyone with fundamental knowledge of algebra, they will solve it pretty easily. Such easy questions aren’t, however, asked anymore. At least not to my knowledge.
Three different solutions to this problem have been uploaded by
mathmemo:
IMO brings some of the most challenging problems every year for the participants and it tests the level of complexity that we can have in the field of mathematics. Certain problems have also helped me discover new and new aspects of several disciplines in the field of math. These competitions must be encouraged and the students in schools must be allowed and encouraged to participate in these competitions as they allow them to explore more and more areas in the field of mathematics.
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