One Hundred Problems in Elementary Mathematics
192
One Hundred Problems in Elementary Mathematics
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Overview
Are the problems in this book really elementary? Perhaps not in the lay reader’s sense, for anyone who desires to solve these problems must know a fair amount of mathematics, up to calculus. Nevertheless, Professor Steinhaus has given complete, detailed solutions to every one of his 100 problems, and anyone who works through the solutions will painlessly learn an astonishing amount of mathematics. A final chapter provides a true test for the most proficient readers: 13 additional unsolved problems, including some for which the author himself does not know the solutions.
Product Details
| ISBN-13: | 9780486811802 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/10/2016 |
| Series: | Dover Math Games & Puzzles |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 192 |
| File size: | 7 MB |
About the Author
Read an Excerpt
One Hundred Problems in Elementary Mathematics
By Hugo Steinhaus
Dover Publications, Inc.
Copyright © 1964 Basic Books, Inc.All rights reserved.
ISBN: 978-0-486-81180-2
CHAPTER 1
ROBLEMS ON NUMBERS, EQUATIONS AND INEQUALITIES
1. Exercise on the multiplication table
We construct a sequence of numbers as follows : The first number is 2, the next is 3,
2 . 3 = 6,
the third number of the sequence is 6,
3 . 6 = 18,
the fourth number is 1, and the fifth is 8,
6 . 1 = 6,1 . 8 = 8,
the sixth number is 6, then follows 8, etc.
This is the sequence which we obtain:
2 3 6 1 8 6 8 ...
The little arcs under the numbers denote the multiplication carried out, the result of which follows the last digit of the sequence. For example, we ought to multiply now 6 by 8 and write down the numbers of the result, namely 4, 8. There will never be a shortage of numbers for multiplication, since the number of arcs is increased by one with each multiplication and the result will yield at least one and often two digits, so that there always appears at least one new digit.
Prove that numbers 5, 7 and 9 never appear in this sequence.
2. An interesting property of numbers
Let us first write an arbitrary natural number (for example, 2583), and then add the squares of its digits (22+52+82+32 = 102). Next, we do the same with the number obtained (12+02+22 = 5), and proceed in the same way (52 = 25, 22+52 = 29, 22+92 = 85, ...).
Prove that unless this procedure leads to the number 1 (in which case the number 1 will of course recur indefinitely, it must lead to the number 145, and the following cycle will occur again and again:
145, 42, 20, 4, 16, 37, 58, 89.
3. Division by 11
Prove that the number
55k+1+45k+2+35k
is divisible by 11 for every natural k.
4. The divisibility of numbers
The number
3105+4105
is divisible by 13, 49, 181 and 379, and is not divisible either by 5 or by 11.
How can this result be confirmed?
5. A simplified form of Fermat's theorem
If x, y, z and n are natural numbers, and n [??] z, then the relation xn+yn = zn does not hold.
6. Distribution of numbers
Find ten numbers x,1x,2 ..., x10 such that
(i) the number x is contained in the closed interval <0, 1>,
(ii) the numbers x1 and x2 lie in different halves of the interval <0, 1>,
(iii) the numbers x,1x2 and x3 lie each in different thirds of the interval <0, 1>,
(iv) the numbers x1, x2, x3 and x4 lie each in a different quarter of the interval, etc. and, finally
(v) the numbers x1, x2, ..., x10 lie each in a different tenth of the closed interval <0, 1>.
7. Generalization
Is the above problem solvable if instead of 10 numbers and 10 conditions there are nnumbers and n conditions (where n is an arbitrary natural number)?
8. Proportions
Consider numbers A, B, C, p, q, r whose mutual dependence is expressed as follows:
A: B = p, B: C = q, C: A = r.
Write the proportion
A: B: C = [] : [] : []
is such a way that in the empty squares there will appear expressions constructed from p, q and r, these expressions being obtained from one another by a cyclic permutation of p, q and r. (By the above we mean the following: if instead of p we write q, instead of q we write r, and instead of r we write p, then the first expression will become the second, the second expression will become the third, and the third expression will become the first.)
9. Irrationality of the root
Give an elementary proof that the positive root of the equation
x5 + x = 10
is irrational.
10. Inequality
Prove the inequality
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
in which all letters denote positive numbers.
11. A sequence of numbers
Find a sequence a0, a1, a2, ... whose elements are positive and such that a0 = 1 and an - an = an for n]I TL = 0, 1, 2, ... Show that there is only one such sequence.
CHAPTER 2PROBLEMS ON POINTS, POLYGONS, CIRCLES AND ELLIPSES
12. Points in a plane
Consider several points lying in a plane. We connect each point to the nearest point by means of a straight line. Since we assume all distances to be different, there is no doubt as to which point is the nearest one. Prove that the resulting figure does not contain any closed polygons or intersecting segments.
13. Examination of an angle
Let x1, x2, ..., xn be positive numbers. We choose in a plane a ray OX, and we lay off on it a segment OP1 = x1. Then we draw a segment P1P2 = x2 perpendicular to OP1 and next a segment P2P3 = x3 perpendicular to OP2. We continue in this way up to PITLn_1 TL[Pn =xn. The right angles. are directed in such a way that their left arms pass through O. We can consider the ray OX to rotate around O from the initial position through points P1, P2, ... up to the final position OPn; in doing so it sweeps out a certain angle. Prove that for given numbers xi this angle is smallest when the numbers xi decrease (i.e., x1 [??] x2 [??] ... [??] xn), and largest when the numbers increase (i.e., x1 [??] x2 [??] ... [??] xn).
14. Area of a triangle
Prove, without the help of trigonometry, that in a triangle with one angle A = 60° the area S of the triangle is given by the formula
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
and if A = 120°, then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
15. Triple halving of the perimeter of a triangle
Consider an arbitrary triangle. We can, of course, cut the triangle with a straight line to halve its perimeter. We can even impose the direction of the cutting line. If we repeat this operation twice, using two different directions, the straight lines will intersect at a certain point Q. Then, two straight lines halving the perimeter will pass through the point Q.
Does there exist a point through which three such lines pass? If so, how can we find it?
16. Division of a triangle
Divide a triangle into 19 triangles in such a way that at each vertex of the newly formed figure (and also at the vertices of the original triangle) the same number of sides meet.
In this problem the number 19 cannot be replaced by a larger number, but it can be replaced by smaller ones. Which ones?
17. Triangles
In this problem n denotes a natural number. We have 3n points in a plane, no three of which lie on a straight line. Can we form from these points — taking them as vertices — n triangles which do not overlap and do not embrace one another?
A similar problem for 4n points can be formulated for quadrangles, and in the case of 5n points for pentagons, etc. Are they all positively soluble?
18. Triangular network (I)
It is well known how to cover the whole plane with a network of equilateral triangles.
Is it possible to put at each of the nodes one of the signs plus and minus so that in each of the component triangles of the network the following condition will be satisfied: if two vertices of a triangle have the same sign, then the third sign is plus, and when the signs are opposite, then the third is minus?
Obviously, of course, plus signs can be given everywhere, but we exclude this trivial solution.
19. Triangular network (H)
Prove that it is impossible to cover the whole plane with a network of triangles in such a way that at every vertex five triangles would meet.
20. What is left from a rectangle?
A golden rectangle is a rectangle whose sides have the golden ratio, i.e. they fulfill
a: b = b: (a-b).
Let us imagine that this rectangle is cut out from a piece of paper and laid on the table with the longer side turned toward us. We cut from the left side of the rectangle the largest possible square, and what remains is again a golden rectangle. We go over to the left side of the table to have again the longer side before us, and we do the same with the new rectangle. In this way we go clockwise around the table, and we cut out the consecutive squares. Every point of the rectangle, with only one exception, will sooner or later be cut out. Find the location of this exceptional point.
21. Division of a square
Let us divide a square of area 1 km2 into three parts A, B, C. Whatever this division may be, there always exists at least one pair of points P, Q belonging both to the same part, the distance PQ being greater than 1.00778 km.
How can we prove it?
22. Square network
We can cover the whole plane with a network of congruent squares. The nodes of this network are called by mathematicians the lattice of integral numbers. Is it possible to mark the nodes by letters a, b, c, d so that each component square has all four letters at its vertices, and so that four letters appear in every row and column of the lattice?
23. Lattice points
For the definition of a lattice we refer the reader to problem 22. Prove that a circle with centre ([square root of 2], [square root of 3]) may, by a suitable choice of the radius, be made to pass through any point of the lattice, but that there are no circles with this centre passing through two or more points of the lattice.
24. Lattice points inside a circle
In this problem we shall deal with lattice points inside a circle K, that is, with points enclosed by the circle K. We do not include here the lattice points on the circle itself.
Prove that there exist circles containing zero lattice points, one lattice point, two lattice points, etc. We can associate with every number n (natural or zero) a circle containing exactly n points.
25.14 = 15
In Wroclaw in 1952, during a meeting of the participants of the Mathematical Olympiad, Dr. J. Mikusinski demonstrated a division of the whole plane into heptagons in such a way that at every vertex three heptagons meet. From this we shall deduce that 14 = 15. Let us denote by P the angle 180°. The sum of the angles in a heptagon is 5P; thus the average size of an angle in a heptagon is 5P/7. As the whole plane is covered with heptagons, it follows that the average angle in this mosaic is 5P/7. But at each vertex three such angles meet, whence the average size of an angle at every vertex is 2P/3. From this it follows that the average size of an angle in the mosaic is 2P/3, because every angle belongs to a certain vertex; thus 2P/3 = 5P/7, 2/3 = 5/7, 14 = 15, which was to be proved.
Find the error in the above argument.
26. Polygon
There are n points lying in a plane, no three of them lying on the same straight line. Is it always possible to find a closed polygon with n non-intersecting sides whose vertices are these n points?
27. Points and a circle
We have 4 points in a plane which do not lie on a common circle or a common straight line. Is it always possible to mark these points A, B, C, D in such a way that the circle passing through the points A, B, C contains the point D?
28. Geometrical problem
Given an ellipse with major axis of length 2a, and minor axis of length 2b, draw a closed curve of the same length as the length of the ellipse such that the closed curve encloses an area greater than the area of the ellipse by (a–b)2.
CHAPTER 3PROBLEMS ON SPACE, POLYHEDRA AND SPHERES
29. Division of space
Through a fixed point in space we pass planes to divide the space into the greatest possible number of parts. One plane will divide the space into two parts, two intersecting planes into four, and three planes intersecting at the fixed point and having no other common points divide the space into eight parts. What is the number of parts obtained with four planes? What number do we get in the case of n planes?
30. Two projections
Let us imagine a plane Π1, tangent to the globe at the north pole N, and a plane Π1 tangent to the globe at the south pole S. We can draw a map by projecting each point on the surface of the globe from N to Π2, and another map by projecting each point on the surface of the globe from S to Π1; ITL these are the so-called stereographic projections. Now, we can superimpose the two planes on each other so that the meridians agree. Each town on one map appears also on the second one. Thus we have defined a certain transformation of the plane into itself. How can we define this transformation directly?
31. Cube
Holding in the hand a model of a cube so that it can rotate about the longest axis (i.e. around the line joining two opposite vertices), we can wind black yarn closely around the cube. This yarn will cover only half of this cube (why?). The same can be done with other axes; there are four of them, and each time we take a yarn of different colour (black, red, blue and yellow). The whole cube will be covered with overlapping colours, producing mixed shades (the model of the cube is white, which we do not consider as a colour). How many shades will result and which ones?
32. Geodesics
This problem does not require much mathematics. Let us place on a smooth rigid cube a rubber band, such as are used in chemists' shops to fasten packages, in such a way that it stays on the cube but does not cross itself. We call the line defined by the rubber band a geodesic and we draw all such geodesics on our model.
1. How many times will the geodesies cover the surface of the cube (that is, how many geodesies pass through each point of the surface of the cube)?
2. How many different families of geodesies have the property of covering the surface of the cube?
33. Motion of a particle
Inside a cubic box there is moving a material particle free from the action of external forces; it is reflected from the walls of the box according to the classical law (the angle of reflection is equal to the angle of incidence, and the perpendicular to the wall at the reflection point is the symmetry line of the angle made by the path of the particle hitting the wall). Is it possible that such a particle runs incessantly around a closed hexagon, hitting on every trip all the walls of the box in turn? Define the points of reflection, and examine whether or not this hexagon is knotted.
34. Diagrams of the cube
Models of polyhedra are made from flat diagrams drawn on pasteboard. On the diagram the faces are adjacent, and one makes a model of it by folding the pasteboard along the lines of the diagram. A regular tetrahedron has two different diagrams.
How many has the cube?
35. Cubes
As we know, the whole of space can be filled with cubes adjoining one another. At every vertex eight cubes meet. It is possible, therefore, by cutting off the corners of those cubes in a suitable way and glueing together the eight adjacent fragments thus obtained (which gives us a new polyhedron and a regular octahedron) to fill the space with regular octahedra, and with the solids that remain of the original cubes. What kinds of solids will these be? If we enlarge the octahedra as much as possible, what part of the space will they occupy? What will be the size of the remaining solids? And, how many solids will meet at every vertex?
36. Hexahedron
Does there exist a hexahedron which is limited by congruent rhombi but is not a cube in the usual sense, i.e. not a die?
37. Tetrahedra
We have six sticks of different lengths, and they are such that in every permutation they can be considered to be edges of a tetrahedron. How many tetrahedra can be constructed from these sticks?
38. Tetrahedron with congruent faces
Is it possible to construct a tetrahedron whose faces are mutually congruent triangles with sides of arbitrary lengths a, b and c?
If so, compute the volume of the tetrahedron.
(Continues...)
Excerpted from One Hundred Problems in Elementary Mathematics by Hugo Steinhaus. Copyright © 1964 Basic Books, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Contents
Foreword,Preface,
Chapter I PROBLEMS ON NUMBERS, EQUATIONS AND INEQUALITIES,
Chapter II PROBLEMS ON POINTS, POLYGONS, CIRCLES AND ELLIPSES,
Chapter III PROBLEMS ON SPACE, POLYHEDRA AND SPHERES,
Chapter IV PRACTICAL AND NON-PRACTICAL PROBLEMS,
Chapter V PROBLEMS ON CHESS, VOLLEYBALL AND PURSUIT,
Chapter VI MATHEMATICAL ADVENTURES OF DR. ABRACADABRUS,
Chapter VII PROBLEMS WITHOUT SOLUTION,