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Page 1 of 4

Mathematical Miniatures
  • Svetoslav Savchev, Titu Andreescu
  • Published by:
    Mathematical Association of America
    Published online:
    26 October 2011
    Print publication:
    03 January 2003
  • Mathematical Miniatures is a problem collection of arresting mathematical insight and ingenuity. The authors brought together materials from mathematical competitions, books, research papers, discussions, and their own work. Such mathematical substance went far beyond the purposes of a traditional problem-solving book. The most attractive results refused to fit into the schemes of an instruction manual meant to exemplify typical problem solving techniques. A broader interpretation of these problems had to be identified, and this book is the fruit of that effort. Savchev and Andreescu detach certain statements or groups of related statements into independent sections. Treating these gems separately, in self-contained essays, emphasizes the source of their natural charm---connections with genuine mathematical experience. The essays are of impressive diversity, enlivened by fresh and original ideas. They involve concepts not only useful but also beautiful and nonstandard, with lots of esthetic appeal. The book is thus not so much a mathematical toolchest: it is an anthology of mathematical verse.

  • Coffee Break 7

  • Svetoslav Savchev, Matematika, Titu Andreescu, University of Texas at Dallas
  • Book:
    Mathematical Miniatures
    Published by:
    Mathematical Association of America
    Published online:
    26 October 2011
    Print publication:
    03 January 2003, pp 146-148

  • Summary

    1. Six points are given in space such that the pairwise distances between them are all distinct. Consider the triangles with vertices at these points. Prove that the longest side in one of these triangles is at the same time the shortest side in another triangle.

    2. A circle is inscribed in a given circular segment, touching its arc and chord at A and B, respectively. Prove that the line AB passes through a constant point.

    3. The number xn is defined as the last digit in the decimal representation of the integer ⌊(√2)n⌋ (n =1, 2,…). Determine whether or not the sequence x1, x2,…,xn,…is periodic.

    Solutions

    1. Color the shortest side in each triangle black; some of the line segments may be colored more than once. If the remaining edges are colored white, then the complete graph with six vertices has all its edges colored in two colors. As is well known, any such coloring contains a triangle T whose sides are of the same color. But T must necessarily be black, because it has at least one black side—the shortest one. The longest side of T is black; it is therefore the shortest side in some other triangle.

    2. Denote the circle containing the arc of the segment by c, the endpoints of the chord by M and N, and the given circle by c1. The point A is the center of a dilatation h taking c1 to c.

  • Coffee Break 9

  • Svetoslav Savchev, Matematika, Titu Andreescu, University of Texas at Dallas
  • Book:
    Mathematical Miniatures
    Published by:
    Mathematical Association of America
    Published online:
    26 October 2011
    Print publication:
    03 January 2003, pp 190-192

  • Summary

    1. The Big Apple got wormy. A worm dug a tunnel of length 101 miles inside it, and went out (starting and ending at the surface). Assuming that The Big Apple is an ideal sphere of radius 51 miles, prove that it can be cut into two congruent pieces one of which is not wormy.

    2. Each vertex of a convex polyhedron is the endpoint of an even number of edges. Prove that any plane section of the polyhedron not containing a vertex is a polygon with an even number of sides.

    3. Asterisks are placed in some cells of an m by n rectangular table, where m < n, so that there is at least one asterisk in each column. Prove that there exists an asterisk such that there are more asterisks in its row than in its column.

    Solutions

    1. Let A and B be the beginning and the end of the tunnel. Consider the set of points X such that AX + BX ≤ 101. It is an ellipsoid of rotation E with foci A and B. Each point X of the tunnel is contained in E. Indeed, since AX andXB do not exceed the lengths of the worm's routes from A to X and from X to B, respectively, we have AX + XB ≤ 101. On the other hand, the center O of the Big Apple is outside E,because AO + BO equals 51+51 = 102 > 101. Everything is clear now. Because E is convex and O is outside it, there is a plane through O that does not intersect E.

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