UNEB UACE MATHEMATICS QUESTION BANK 1993- (original) (raw)
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2013
"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) "
A level past examination paper, 2020
This paper is based on A level past examination paper and it is meant for the learners' revision as they prepare for there examinations.
ELEMENTARY PROBLEMS AND SOLUTIONS
The Fibonacci numbers F n and the Lucas numbers L n satisfy F n+2 = F n+l +F n, F Q = °> ^1 = ^ A7+2-AH-I + A i > A>-A L X — i. Also, a = (l + V5)/2, /? = (l-V 5) / 2 , F n = (a n-/3 n)/^md L n = a n + j3 n. Let a Q-a l-l and let a n = 5a n _ 1-a n _ 2 for n > 2. Prove that a^ +1 +a^+3 is a multiple of a n a n+l f° F all W > 1. If F^ +2k = aF* +2 + hF^ +c(-l) n , where a, b, and c depend only on k but not on n, find a, b, andc. B-787 Proposed by H.-J. Seiffert, Berlin, Germany For n > 0 and k > 0, it is known that F kn l F k and P kn I P k are integers. Show that these two integers are congruent modulo R k-L k. [Note: P n and R^ = 2Q n are the Pell and Pell-Lucas numbers, respectively, defined by P n+ 2-2P n+1 + P n , P 0 = 0, /} = 1 md.Q n+2 = 2Q n+l + Q n , Q 0 = 1, fl= 1.]
407 Polynomials & Trigonometry Review Problems
Here are 407 missing problems that did not fit into the pool of 1220 problems in the first volume of The Olympiad Algebra Book dedicated to Polynomials and Trigonometry. The majority of the questions are chosen from American competitions such as AIME (American Invitational Mathematics Examination), HMMT (Harvard-MIT Math Tournament), CHMMC (Caltech Harvey Mudd Math Competition), and PUMaC (Princeton University Math Competition). All of the AIME problems are copyright © Mathematical Association of America, and they can be found on the Contests page on the Art of Problem Solving website. In this document, the links to the problems posted on AoPS forums are embedded (if existent).