A Potpourri of a Diverse Variety of Algebra Problems(including a Crux Mathematicorum Olympiad Corner): Fourteen problems in total (original) (raw)

"As the title of this article points out, there is indeed a diverse variety of algebra problems in this paper. This is a compilation of fourteen algebra problems from four different sources. Problem14, is an Olympiad Corner Problem, OC91, published in the September 2012 issue of the journal, Crux Mathematicorum with Mathematical mayhem(see reference [1]). Problem11 can be found in the linear algebra text entitled, 'Elementary Linear Algebra Applications Version", by Howard Anton(see reference [2]). Problem5 is of this author's own making. The rest of the problems were originally found in an obscure algebra book, published circa 1971 in Athens, Greece. However, some of those problems have been modified by this author. In Section2, we list all fourteen problems. In Section3, the reader will find detailed solutions to these problems, by this author. Here are three of these problems: Problem4: Let d and k be given positive real numbers. Consider a triangle that satisfies the following two conditions. 1. The three side lengths of the triangle are consecutive terms of an arithmetic progression with common difference d. 2. A/E=k; where A is the area of the triangle, and E being the area of the rectangle whose two pairs of side lengths are the two smallest side lengths(among the three) of the triangle. For which values of k does this problem no solution? For which values of k does this problem have a unique solution? Find the three side lengths in this case. For which values of k does the problem have two solutions. Find the three side lengths in this case. Problem6: Find the integer values of the constant k, for which the equation, x^3-(k+1)x +2k= 0, has a rational root. Problem13: Find all the positive integer solutions to the 4-variable system of equations, x+y=100, xy = z^2, and (1/x)+ (1/y )= (2/w) "