Gerry Myerson

SOME PROBLEMS CONCERNING RECURRENCE SEQUENCES

There are questions about recurrence sequences that seem to crop up again and again. Plainly, though their answers are well known they are not known well. We endeavour to explain these answers in context so that they may become more widely known. The sequence 0,1,-1,2,-2,..., in which each integer occurs exactly once, is a recurrence sequence; that is, it satisfies a linear, homogeneous recurrence relation with constant coefficients, namely,

MARRIAGE, MAGIC, AND SOLITAIRE

It turns out that it is always possible to make such a selection. The proof is a simple application of Halls Marriage Theorem, as we show in Example 1 in the next section. In Sections 3 and 4, we identify winning the solitaire game with decomposing a semi-magic square into a linear combination, with positive integer coefficients, of permutation matrices. The remainder of the paper discusses the number of permutation matrices needed to express a given semi-magic square. 2. MARRIAGE. Suppose there are sets A1, A2,. .., An, and you wish to know whether there exist distinct objects xl, x2,.. . , x,, such that xl is in A1, x2 is in

An Impossible Difference Condition: 11074

Costruire una funzione di una variabile definita sui reali che sia derivabile e la cui derivata prima sia discontinua ed a sua volta derivabile su sottoinsiemi densi dei reali

Incongruent restricted disjoint covering systems

We define an incongruent restricted disjoint covering system on [1,n] as a set of congruence classes such that each integer in the interval [1,n] belongs to exactly one class, and each class contains at least two members of the interval. In this paper we report some computational and structural results and present some open problems concerning such systems.

Divisibility Conditions Holding as Many Times as You Want: 10898

Prove that limx_,0 f(x) exists and is finite, and evaluate the limit. 11025. Proposed by Floor van Lamoen, Goes, the Netherlands, and Darij Grinberg, Karlsruhe, Germany. Let four lines a, b, c, and d be given, no two of which are parallel. Let Ta be the triangle bounded by b, c, and d, and let Oa be its circumcenter. Let Tb, Tc, and Td as well as Ob, Oc, and Od be defined similarly. Let Ha be the orthocenter