O pewnej implikacji

The purpose of this paper is to show how the process of provinga theorem in different ways or proving generalized versions of the theorem,after learning one its proofs, influences the development of the skills of provingtheorems and analysing proofs by the students of mathematics. Toillustrate this process we use an elementary theorem about numbers and itsgeneralizations, giving fourteen proofs. Proving theorems we use methodsand facts which are available to high school students

O nieskończonych ciągach liczb naturalnych, parami względnie pierwszych

In the first part of the paper the authors, using general formulas, determine and describe a class of infinite series of natural numbers pairs of which are relatively prime. The second part of the paper contains - as a proposition - a set of problems concerning prime numbers and pairs of relatively prime numbers suggested for use during the process of work with Mathematics students, as well as some didactic comments concerning these problems

Cechy równoboczności trójkątów

The paper presents some non-standard (in the sense: not yet presented in the literature), necessary and sufficient conditions for a triangle to be isosceles

On some intger triangles with a rational median

We use the recurrence relations and the Pell equations to determineall integer triangles whose lengths are consecutive integers and the length ofa fixed median is a rational number

Wielomiany Fibonacciego stopnia k

In this paper, we give formulas determining the Fibonacci polynomials of order k using the so-called generalized Newton symbols, i.e., the coefficients in the expansion of (1+z +z^2 +. . .+z^{k−1})n with respect to the powers of z

Średnie quasi-arytmetyczne

We present a list of geometric problems with solutions that lead to knownor less known means. We also prove, by elementary means, some property for so-calledquasi-arithmetic means. We use the proved result to justify some inequalities betweenthe means

Wokół twierdzenia Wilsona

In this paper some known conditions and new congruences characterising prime numbers are given. Some of them are obtained by the generalised Wilson theorem given by Gauss. The elementary proof of this theorem is also presented

Simple proofs of some generalizations of the Wilson’s theorem

In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group

A few generalizations of some strategic game

In the paper a set of strategy games is presented. It is shown how the “manipulative” developing of the winning strategy of a known and simple game can lead to conceptual reasoning based on reduction; then it is suggested how to formalise and generalise such a game, similar games and the procedure of finding the winning strategy