MATHEMATICS–PHYSICS–INFORMATICS

Butterly wings in a trapezoid

Lenka Juklová, Jaroslav Švrček — Sat, 01 Mar 2025 00:00:00 +0000

This article relates to the topic of solving some plane geometry problems using of so-called "butterfly wings in a trapezoid", i.e. using the same area of two triangles having exactly one common vertex at the point of intersection of diagonals of a trapezoid and with remaining two vertices of these two triangles coinciding with endpoints of its legs. There are given here some interesting applications of this statement.

Gambler's ruin in the field of gambling

Pavel Tlustý — Sat, 01 Mar 2025 00:00:00 +0000

The article addresses the issue of probabilistic thinking in the field of gambling. It shows that a correct understanding of the basic principles of gambling is one way to protect against gambling addiction. It allows the player to think rationally and return to reality.

About bound extrema of functions

Dag Hrubý — Sat, 01 Mar 2025 00:00:00 +0000

In the article, simpler problems on bound extrema of functions are solved. Methods that do not use differential calculus are used to solve problems. The property of the discriminant of the quadratic equation is significantly used here.

Výpočty a obrázky

František Kuřina, Jana Cachová — Sat, 01 Mar 2025 00:00:00 +0000

The article deals with the role of figures in problem-solving. In the second part, there are several problems and their solutions using figures.

On a Number Theory Problem

Martin Melicher, Michal Janík — Sat, 01 Mar 2025 00:00:00 +0000

The article presents proof of a problem from number theory that appeared in this journal more than two years ago and to this day no solution has been presented. The proof is based on a reformulation of the problem into a question of rational approximation of numbers of a certain form. The problem is then solved using Dirichlet's approximation theorem, which deals precisely with rational approximations. The paper also includes a proof of this useful theorem using the pigeonhole principle.

Interesting math problems

Editor MFI — Sat, 01 Mar 2025 00:00:00 +0000

We continue to publish other problems of our traditional section.

Minimal triangulation of a polygon (Problems from MO category P, part 50)

Pavel Töpfer — Sat, 01 Mar 2025 00:00:00 +0000

With today's 50th anniversary part, we conclude the long-lasting series of articles in which we have gradually introduced you to selected competition problems from the Mathematical Olympiad of category P (programming). These 50 articles have been published on the pages of our magazine from 2000 to the present day, i.e. for a full 25 years. This time, we will show one classical combinatorial problem given in the national round of the 39th MO category P in the school year 1989/90. It deals with the so-called triangulation of a polygon, which means dividing its area into triangles by diagonals that do not cross each other anywhere. As we will see, an efficient solution to this problem will be based on the idea of dynamic programming, a method very often used not only in various competition problems but also in programming in general.

Generativní umělá inteligence – Díl první: příliš velká očekávání

Eduard Bartl — Sat, 01 Mar 2025 00:00:00 +0000

Artificial Intelligence is not only a topic of discussion in the professional community but is also known to virtually everyone accessing the media, especially the Internet. Indeed, the impact of AI on most areas of human activity is unprecedented. Most of the public has high expectations. Some people see AI as a cure for many of the world's problems, others are afraid of it and would prefer to see it banned. The series of articles starting with this piece will look at one of the most debated subfields of artificial intelligence, which is now called generative artificial intelligence, and into which large language models such as ChatGPT, in particular, belong.

Measuring the shear modulus of nitinol wire with shape memory

Adéla Richterová, Vladimíra Erhartová, Jiří Erhart — Sat, 01 Mar 2025 00:00:00 +0000

Using a shape memory nitinol wire as an example, we present a simple classroom method for measuring shear modulus by stretching a spring using Vernier equipment. Nitinol has significantly different elastic moduli in the martensitic (below activation temperature) and austenitic (above activation temperature) phases due to the structural phase transformation of the crystal lattice. Thus, students can be shown an example of a transformation with a change in material properties as well as a general method for determining the shear modulus by stretching a spring.

A realistic model of the general solution of the frequency of oscillations of an electrostatic pendulum

Čeněk Kodejška — Sat, 01 Mar 2025 00:00:00 +0000

The electrostatic pendulum, like the Van de Graaff generator (VdGG), is an essential demonstration tool for physics teachers when explaining electrostatic phenomena. It is usually used to demonstrate the work done by the electric field of a plate capacitor in transferring a charged ball to a table tennis table. In the introduction of this paper, we briefly mention various theories of electrostatic pendulum dynamics involving either the influence of the electric field only or also gravitational forces or other influences. In this paper, we first compare the different models with each other with real experimental results of oscillation frequency measurements and then propose a realistic model of the physical pendulum oscillation, which provides a general solution including both the electric and gravitational forces acting on the pendulum and the restitution coefficient. In the second part of the paper, we attempt to explain the value of the coefficient, which represents the fraction of charge transferred when the pendulum contacts the capacitor plate, by comparing the discharge time constant of the VdGG with a surrogate RC circuit. In explaining the theory describing the motion of a real physical pendulum, we simultaneously differentiate the relationships for the frequency of the pendulum at each level from high school to college course. We will also show how the results of this work can be applied in high school laboratory exercises to measure VdGG as a high-voltage source in an electrostatic pendulum experiment. The conclusions, however, apply in general to any high-voltage source used.