We consider two variants of the secretary problem, the Best-or-Worst and the Postdoc problems, which are closely related. First, we prove that both variants, in their standard form with binary payoff 1 or 0, share the same optimal stopping rule. We also consider additional cost/perquisites depending on the number of interviewed candidates. In these situations the optimal strategies are very different. Finally, we also focus on the Best-or-Worst variant with different payments depending on whether the selected candidate is the best or the worst.

En este trabajo realizamos un estudio detallado de los problemas de proporcionalidad compuesta de doce libros de texto españoles de segundo curso de Educación Secundaria Obligatoria (13-14 años). En concreto, se realiza un análisis de contenido textual y a priori, clasificando los problemas atendiendo a su contexto, su estructura, su posición y papel dentro de la Unidad Didáctica correspondiente y a la tipología de magnitudes utilizadas. Entre otros resultados se concluye que, aunque la presencia de problemas varía ligeramente en cuanto a número entre los distintos textos, el tratamiento es bastante homogéneo respecto a su contexto, estructura y magnitudes implicadas: la mayoría de los problemas son de contexto realista, de valor perdido y con cinco cantidades de magnitud extensivas. También se detecta poca presencia de problemas de comparación cuantitativa y de situaciones de tipo inversa - inversa, así como poca presencia y variedad de magnitudes intensivas.

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Lehmer's totient problem consists of determining the set of positive integers n such that '.n/ j .n 1/ where ' is Euler's totient function. In this paper we introduce the concept of k-Lehmer number. A k-Lehmer number is a composite number such that '.n/ j .n 1/ k . The relation between k-Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also k-Lehmer numbers.

In this paper, we characterize the odd positive integers n satisfying the congruence n−1 j=1 j n−1 2 ≡ 0 (mod n). We show that the set of such positive integers has an asymptotic density which turns out to be slightly larger than 3/8.

In this paper we compute the sum of the k-th powers of all the elements of a finite commutative unital ring, thus generalizing known results for finite fields, the rings of integers modulo n or the ring of Gaussian integers modulo n. As an application, we focus on quotient rings of the form (Z/nZ)[x]/(f (x)) for a polynomial f ∈ Z[x].