The problem-solving assumes theoretical and analytical skills, as
well as algorithmic skills, coupled with a basic mathematical
intuition. The concept of this problem book successfully supports
the development of these skills of the solver and meanwhile offers
mathematics instructors models for teaching problem-solving as an
integral part of the mathematics learning process. The topics of
the problems belong to algebra II and algebra of the first two
years of college, passing through fields of integer and real
numbers, equations, inequalities, powers, logarithms, divisibility,
polynomials, and combinatorics. This work is the first of a series
that will also operate in other domains and subdomains of
mathematics. The problem book is structured in four separate and
independent sections, namely Problems, Hints, Algorithms, and
Proofs, in this order. The Problems section consists of over 100
problems themselves, which are of medium to advanced difficulty
level. The problems can be presented and discussed at mathematics
workshops and sessions of preparation for contests and olympiads as
well as in the classroom, using problems of different difficulty
levels with separate groups of students; for advanced groups, an
instructor can use this problem book in its entirety. The Hints
offer groups of keywords that suggest to the solver - intuitively,
as well as analytically - an initial approach to the problem,
important observations upon which the solution is based, categories
of theoretical results applied when solving, and specific
theoretical results. The hints also suggest indirectly the solving
algorithm (found in the next section), but without exposing or
synthesizing it. In the Algorithms section, the solving algorithms
provide chronological groups of steps necessary for generating the
complete solution. The algorithm is presented as a brief list of
tasks; it does not reveal the complete solution to the problem, but
only points out the partial tasks whose results will finally yield
the logical construction of the solution. The Proofs represent the
complete integral solutions of the problems, unfolded according to
the solving algorithm. This comprehensive presentation includes the
detailed steps to be executed, the observations that precede the
deductions, and the entire logical motivation. No partial results
are left unproved, neither as an exercise nor as being obvious or
easily deduced. The sections described previously are separated in
this book so that the solver can explore the problem and search for
solving paths independently, consulting the next section only when
he or she has exhausted, with no success, his or her own approach
and individual study methods. As the solver moves progressively
from a partial solution to a more complete one, this additional
effort itself becomes a useful mathematical exercise. Moreover, the
process of moving successively through the indications of the
problems together with individual investigation and autocorrection
of a wrong approach, stimulates and motivates the solver toward a
solution. All these elements give this type of problem book a truly
interactive character.
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