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I arrived in the room a few minutes before the scheduled meeting, but to my surprise, Professor Dr. Weissauer was already there. After briefly greeting me and asking a few preliminary questions, Dr. Tok, a high school teacher and the second professor, joined us. During our beginning conversation, we discussed my CV and the areas of mathematics I have listed as my current interests, which included plane geometry, inequalities, and real analysis. I explained that my inclusion of real analysis was due to the textbook I had written, even though I have not advanced my knowledge and understanding of the subject significantly in the past two years.

My desire to talk about plane geometry was shattered relatively quickly as the prof, as I unfortunately expected, immediatly put his attention towards real analysis, which other professors had already done to other students that had their talks before me. After talking a bit about what the individual chapters of my book contained, we started talking about the convergent sequences. He asked me to prove whether any bounded sequence of real numbers has a convergent subsequence, which is simple by dividing the interval in which the terms lie in 2, considering the one that contains infinitely many points, dividing that into two again etc… We continued our discussion on Cauchy sequences, complete metric spaces, continuity, and Riemann Sums, and I was able to answer most of his questions. However, the main challenge was that he did not ask specific questions, which made it difficult for me to understand what he wanted to know. Eventually I asked: ”I don’t know what direction you are going in, could you please ask a specific question?”, to which he replied that that is something he specifically doesn’t want to. Despite his attempts to steer me in the right direction, I was still unsure of his expectations until he eventually had to explain what he was looking for. In retrospect, I wish I had a clearer understanding of his approach beforehand.

After that, Professor Weissauer turned to Dr. Tok and asked if he had any questions for me. Dr. Tok referred to my mention of being interested in inequalities and asked me to list the inequalities I was familiar with. I listed off several standard ones, including AM-GM, Cauchy, Hölder, and Chebyshev. Dr. Tok then asked if I knew Jensen’s inequality and if I could use it to prove AM-GM.

The night before, I had anticipated the possibility of being asked questions about inequalities and, based on my research from various blog posts, I expected the questions to involve proving theorems rather than using one theorem to prove another. I knew that AM-GM could be proven inductively, but I wanted to explore alternative proof methods. So, I quickly reviewed the proof of AM-GM using Jensen’s inequality, which involved considering the terms in the natural logarithm. However, I did not continue reading about the proofs of other inequalities as I was not interested in memorizing every proof I came across.

During the remaining five minutes, Professor Weissauer brought up my mention of Fourier Series. I initially shook my head in confusion, until I realized there had been a misunderstanding. I corrected him, explaining that the final chapter of my book was on Taylor Series. He asked me to explain what Taylor Series do and the requirements for functions to be approximated by them. I gave the appropriate answers, but when he asked if “infinitely differentiable” was sufficient, I responded with a “No”, but unfortunately, I couldn’t recall a specific counterexample at that moment. He proceeded by giving me an example of such a function, and asked me to prove whether it can be approximated via Taylor Series at x=0, but I wasn’t quick enough to find lim x->0+ for 1/x-ln(x), so time was up.

Natuarlly, as I heard the task he had just given me, I started getting internally excited. ”Sure”, I replied, ‘I may try”. As I started to write down the inequality, I tried making up a motivation of the idea of the solution. ”Basically, we have a factor of 1/n on the left hand side, and a 1/n term in the exponent on the right hand side, which is why we may try to wrap natural logarithms around the the equation.”
After noticing that I must actually wrap ln() around the variables individually for the arithmetic mean and flip the sign because ln(x) is concave, I finished the proof. Hearing ”Yep, that is fully correct”, I felt a bit of relief after feeling like I could have done better a formally expressing what I meant in the discussion with Prof. Weissauer before. Next, Dr. Tok asked me if I could prove Jensen as well, to which I replied: ”I think Jensen can be arrived using Karamata”, but while saying that I realized that I had no idea of how to prove Karamata, so I immediatly said that I could prove Jensen in two variables. I explained that in the convex case, Jensen’s inequality states that the tangent line lies above the curve, and then provided the standard argument for formalizing this result. Both Dr. Tok and Prof. Weissauer seemed satisfied with my explanation.

Although I didn’t come out as the national winner, I am proud to have received a recognition for my efforts. Here are some important lessons I learned from the experience:

  • The professors are, to no suprise, highly knowledgeable in advanced mathematical topics, so it is advisable to specialize in areas like analysis, linear algebra, or group theory, which delve deeper than the mathematics covered in olympiads. You should not expect to talk about 2-D geometry, but usually, if you list a topic that is part of higher mathematics, they are going to talk with you about it.
  • In these areas, it is important to be aware of the main concepts used to solve problems and to have a good understanding of the proofs of key results and their corollaries. This way, you can follow the professor’s line of thinking and know what direction they are trying to steer you towards.
  • Regarding the last point, it is especially important that you are comfortable with being as rigorous as possible with your mathematical commentary.