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Đây là cuốn sách tiếng anh trong bộ sưu tập "Mathematics Olympiads and Problem Solving Ebooks Collection",là loại sách giải các bài toán đố,các dạng toán học, logic,tư duy toán học.Rất thích hợp cho những người đam mê toán học và suy luận logic.
[...]... Inequations: Traveling in Groups 7 25 Equations and Inequations: Traveling in Groups In one way or another, the problems in this section concern systems of equations or inequalities (or both!) Two unusual topics included are the notion of an approximate solution for an inconsistent system of equations and linear programming 7-1 Let the lines 15x + lOy = -2 and x - y = -2 intersect in point P Find all values... shown in Fig 6-7, so that the distance from 0 to A is 3 Starting at 0, how many points can you reach if the distance, as here defined, is n, where n is a positive integer? y ' 3 - 2 A - 1 0 1 2 3 6·7 x- f - - 24 PROBLEMS 6-8 Given n straight lines in a plane such that each line is infinite in extent in both directions, no two lines are parallel (fail to meet), and no three lines are concurrent (meet in. .. are concurrent (meet in one point), into how many regions do the n lines separate the plane? Let there be n = r + k lines in the plane (infinite in both directions) such that no three of the n lines are concurrent, but k lines are parallel (but no others) Find the number of partitions of the plane Challenge 1 Challenge 2 Let there be n straight lines in the plane (infinite in both directions) such that... such that no two are paranel Find the number of plane separations Challenge 3 A set of k 1 parallel lines in the plane is intersected by another set of k 2 parallel lines, all infinite in extent Find the number of plane separations Challenge 4 In Challenge 3, introduce an additional line not parallel to any of the given lines, and not passing through any of the klk2 points of intersection How many additional... numbers is 53 Find the A.M of the numbers in the sets combined Challenge 1 Change the A.M of the second set to -53, and solve Challenge 2 Change the number of elements in each set to I, and solve Challenge 3 Find the point-average of a student with A in mathematics, A in physics, B in chemistry, B in English, and C in history - using the scale: A, 5 points; B, 4 points; C, 3 points; D, I point - when (a)... We then add the entries in column II, omitting those that are associated with the even entries in column I 4-2 If x = {O, 1,2, , n, }, find the possible terminating digits of x 2 + x in base 2 Challenge 1 If x = {O, 1,2, ,n, }, find the possible terminating digits of x 2 + x + 1 in base 2 Challenge 2 If x = {O, I, 2, , n, }, find the possible terminating digits of x 2 + x in base 5 Challenge 3 If... Equations Inequations and Pitfalls 5 17 Equations, Inequations, and Pitfalls Equations and inequalities play a double role on the problemsolving stage They may be the stars of the show if they are tricky to solve or especially interesting More often, they are in the supporting cast, serving as the indispensable tool for expressing the data in a problem Both roles are explored in this section 5-1 Find the... ensure that the line 2x + 3y = P goes through point P 7-2 Let (x, y) be the coordinates of point P in the xy-plane, and let (X, Y) be the coordinates of point Q (the image of point P) in the XY-plane If X = x + y and Y = x - y, find the simplest equation for the set of points in the XY-plane which is the image of the set of points x 2 + y2 = I in the xy-plane 7-3 The numerator and the denominator of a fraction... SI lattice points in the base and 52 lattice points in the altitude The vertices are lattice points (a) Find the number of interior lattice points, N(I) (b) Find the number of boundary lattice points, N(B) (c) Find the total number of lattice points, N Challenge Suppose the word "rectangle" is changed to "square"; find N(I), N(B), and N 3-3 An approximate formula for a barometric reading, p(millimeters),... digit." For example, 4 5 T04 Find the terminal digit of (a) 212 (b) 2 30 (c) 77 (d) 8 10 (e) 8 10 • 7 11 Challenge Find the terminating digit of (a) G) 5 (b) G) 5 2-7 If N = I· 2· 3 100 (more conveniently written l00!), find the number of terminating zeros when the multiplications are carried out 8 PROBLEMS Challenge Find the number of terminating zeros in D = 36! - 24! 2-8 Find the maximum value of x . THE BOOK Challenging Problems in Algebra is organized into three main parts: " ;Problems, " "Solutions," and "Answers." Unlike. the introductory material. Library of Congress Cataloging -in- Publication Data Posamentier. Alfred S. Challenging problems in algebra! Alfred S. Posamentier,