Maths Puzzles & Problems
with JMC's attempt at answers
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Maths Puzzles & Problems with JMC's attempt at answers |
On this page I share worked solutions of some mathematical puzzles which I have thought about. To date I have included only a few problems, but will add to them steadily.
Some of the questions were collected by maths lecturers at the University of Manchester and used within an undergraduate problem solving course -- many thanks for these. A few problems are from maths olympiad competitions, as organised by the Mathematical Association of America. Others are from English A and S Levels maths examination papers dating from the1950s, when questions were quite hard! Others still are my own generalisations of more elementary questions.
Where the questions come from public maths competitions, such as the International Mathematical Olympiad (IMO) and William Lowell Putnam competitions, solutions are published. For instance, John Scholes of Kalva has an extensive web site of problems and solutions. If you want a good book on how to go about solving competition problems, try 'The art and craft of problem solving' by Paul Zeitz, published John Wiley, 1999.
In writing my own solutions I have NOT looked at any of these published answers -- that would be cheating! Even now I have still not looked at these publshed solutions. It is quite possible, therefore, that my solutions are distinctly inferior or even downright wrong. But that is part of the challenge of maths problems, and each of us can bring something new as we rise to the challenge. It is rather like learning a new piece of classical music on the piano, violin, etc. -- it may have been recorded 20 times already by great artists, but it is still a challenge and a pleasure for each amateur musician.
Maths problems seem almost alive. I do like the quip by a famous mathematician -- G. H. Hardy I think -- that a good problem always bites back!
The questions are listed below is no particular order. Some are quite straightfowards, other tricky. You can download my attempts at answers in pdf format. (Each answer is headed by a restatement of the question using maths symbols not available in HTML.)
Q1: Evaluate the indefinite integral sqrt[tan x] dx, and its definite integrals over the intervals [0, pi/4] and [0,pi/2]. (Answer)
Note that this problem caused considerable discussion in the UK in the journal of the Institute of Maths and its Applications, and several solutions were given over a few months, I think in 2004 or 2005. You can now put this integral into a symbolic maths package such as Mathematica and get the answer in a second, so perhaps no one works out integrals by hand these days. Nevertheless, it is part of a mathematician's training.
Q2: Evaluate the integral of log(x+1)/(x^2+1) over the range [0,1].
This problem appears in the 2005 Putnam competition. It can be solved elegantly by appeal to symmetry of the integrand, once converted to a trigonometric expression. Click here for this solution and for integrals of similar combinations of logarithm and polynomials.
Q3: Show that every positive integer has a multiple whose decimal representation contains only the digits 0 and 1. (Answer)
Q4: a) From the first 200 natural numbers, 101 of them are arbitrarily chosen. Prove that among the numbers chosen there exists a pair such that one divides the other. b) Prove that if 100 numbers are chosen from the first 200 natural numbers and include a number less than 16, then one of them is divisible by another. c) Generalise this to choosing n+1 numbers from 1, 2, ..... 2n. (Answer)
I extended the question to try to determine the largest set of mutually indivisible integers < 200 for a given lowest integer. My result for the lowest being 8 has these 97 integers : 8 12 18 20 27 28 30 42 44 45 50 52 63 66 67 68 70 71 73 75 76 78 79 83 89 92 97 98 99 101 102 103 105 106 107 109 110 111 113 114 115 116 117 118 119 121 122 123 124 125 127 129 130 131 133 137 138 139 141 143 145 147 148 149 151 153 154 155 157 159 161 163 164 165 167 169 170 171 172 173 174 175 177 179 181 182 183 185 186 187 188 190 191 193 195 197 199. Can you do better?
Q5: Which positive integers can be expressed as the sum of three or more consecutive positive integers? (Answer)
This is a question from a maths problems class, but I did not find it very interesting. Can you see a way to generalise it, to spice it up?
Q6: Which is greater, cos(sin x) or sin(cos x)? (Answer)
Q7: Find the smallest integer N with initial digit 1 such that, if the initial digit is moved to the end, the resulting integer is 3N. Find all possible initial digits for which this can occur. (Answer)
Q8 : Find an integer N with digits abcabd, with d = c+1, such that N is a perfect square. (Answer)
Q9 : Let a and b be positive integers such that ab+1 divides a^2+b^2. Show that the quotient (a^2+b^2) / (ab+1) is always a perfect square. (Answer)
Note that the pdf file giving my solution refers to a doument on continued fractions which I have not yet completed. I will add it to this web site soon.