Category: Exercises and dialogues

16 November 2023
Not exceedingly complex
Putnam competition questions have often represented for me a very good motivation to train and further develop my knowledge in real analysis. In this post I want to show you...
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5 January 2022
A bumpy ride
We take again inspiration from the Putnam competition, for an exercise on continuity and differentiability of real fuctions obtained by "stretching" a basic replica. Let...
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5 December 2021
Problem unraveling
Let us go back to Terence Tao's Solving Mathematical Problems, for another exercise. As done in a previous post, I would like to show an alternative solution to a problem...
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2 December 2021
Extreme derivatives
Theorem. Let \(f\) be a real function, which is differentiable up to order \(n\) in  \(c \in \Bbb R\), for some positive integer \(n\). Suppose that \(f^{(n)}(c) \neq 0\),...
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30 October 2021
Euclid strikes back
I found the following problem on Terenece Tao's "Solving Mathematical Problems". I propose, here, a synthetic approach based on simple Euclidean geometry concepts, instead of...
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17 July 2021
Close enough?
While solving some exercises for my students, I bumped into the following sub-problem: given a divergent sequence \((a_n)\), such that \((a_{n+1} - a_n)\) converges to \(1\), can...
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25 September 2020
The power of powers
In this post we make use of Taylor polynomial expansions to efficiently solve one of 2018 Putnam competition problems. Here is the problem statement. Let \(f:\Bbb R \to...
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9 July 2020
Uneven integration
Very seldom does one need to make use of the actual definition of Riemann integral. Most of the times, in fact, we are dealing with very "regular" functions for which a whole...
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20 January 2020
Euler's number \(e\), the base of the natural logarithm, manifests itself in various mathematical contexts, sometimes unexpectedly. In the following you will read an example of...
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23 December 2019
Analysis and synthesis
While solving some geometrical problems, I have been investigating on the possible alternatives between a trigonometric approach versus an approach based on congruences and...
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25 November 2019
Breaking the harmony
I found this curious exercise on the book Problems and Theorems in Analysis I, by Pólya and Szegö, and I thought it was a nice problem to propose here, since its solution...
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14 November 2019
Non sequitur
In this post the behavior of sequences obtained by convolving two sequences is analyzed. In particular, we will give sufficient conditions for the convolution to be a null...
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22 October 2019
The revenge of Menelaus
This post is devoted to Menelaus's Theorem, named for the astronomer Menelaus of Alexandria. It is a very imporant theorem in geometry, relating the ratios of the segments...
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