Maths Puzzles & Problems

with JMC's answers

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Here are almost 30 maths and mechanics problems and my attempts at answers.

Some are my own invention, some were collected by maths lecturers at the University of Manchester and used within an undergraduate problem solving course.  A few 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! 

In writing my own solutions I have NOT looked at any published answers -- that would be cheating!   It is quite possible, therefore, that my solutions are distinctly inferior or even downright wrong. 

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Q1: Evaluate the indefinite integral of sqrt(tan x), , and its definite integrals over the intervals [0, pi/2] and [0, pi/4], namely .   (Answer)

This problem led to 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 an instant, so perhaps no one works out integrals by hand these days.  Nevertheless, it is part of a mathematician's training.

 

Q2: Evaluate the integral   (Answer)

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. This solution is given on my web page about Logarithms.  However, I took this as the starting point for a more wide ranging investigation of integrals involving integrals of similar combinations of logarithm and polynomials -- click here for the full document.

 

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. Initially I did not find it very interesting, but later discovered some merit in it.

 

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 divides.   Show that the quotient 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.

Q10 : Evaluate , the 8th root of the continued fraction 2207 - 1/(2207 - 1/(2207 - .... )).     (Answer)

Q11 : Rationalise the denominator of the surd fraction .    (Answer)

Q12 : Prove that    (Answer)

This challenging integral was posed to me by a reader from Belgium. I managed to get the answer, but by a round about route involving term by term integration and summation of an infinite series expansion of the integrand. Proving the validity of this has required Lebesgues' dominated convergence theorem. Perhaps you can see a more direct way of proving this integral?

Q13 : Evaluate .     (Answer)

Q14 : Evaluate where and (Answer)

This is another Putnam competition question. My Answer contains a longish exploration of various ramifications of this problem. It has led me to tabulate the integral of the modified Bessel function as a function of its upper limit X. Also to explore an interesting function defined as a finite sum of binomial coefficients : [ nCr / 2^r r!] summed from r=0 to r=n.

Q15 : Prove that there are unique integers a, n such that (Answer)

The solution is quite easy to find -- the challenge is mainly in showing uniqueness.

Q16 : Consider the power series expansion Prove that for each integer n >= 0 there is another integer m such that   (Answer)

Another Putnam question from 1999. Do-able by straightforward, standard methods, though perhaps I missed a clever trick?

Q17 : Let N be the positive integer with 1998 decimal digits, all of them 1.  That is N = 111111 .... 111.  Find the thousandth digit after the decimal point of the square root of N.   (Answer)

 

Q18 : Let a and b be positive integers. Show that (a+b)! /(a+b)^(a+b) is less than (a!/a^a)( b!/b^b).   That is     (Answer)

Another Putnam question. I have written this as a case study in how to solve, and how not to solve, a problem. If you spot the key, its solution is immediate. If not,........

 

Q19 : A dart, thrown at random, hits a square target. Find the probability that the point hit is nearer to the centre than to any edge.     (Answer)

 

Q20 : Show that the curve x^3 + 3xy + y^3 =1 contains only one set of three distinct points which are the vertices of an equilateral triangle, and find its area.     (Answer)

 

Q21 : A rounding-up process with integers which tends to PI. Choose an integer N, e.g 10. Round 10 up to the next multiple of 10-1=9. You get 2*9=18. Now round 18 up to the next multiple of 9-1=8. You get 3*8=24. Round 24 up to 4*(8-1)=28, then to 5*6=30. Now 6-1=5, and 5 divides 30, so leave it and go on to divide by 5-1=4. The next multiple of 4 is 32. Then round to 3*11=33 and finally ro 17*2=34. Call this F for final. The contention is that as N get large, N^2/F tend to pi.   (Answer)

 

Q22 : A revolving mass on an elastic chord. This is a question in classical dynamics. A mass is revolving at speed on a smooth horizontal table. It is attached to the centre of the circle of motion by a spring or elastic chord. The mass is suddenly given an extra displacement radially outwards then released, causing it to oscillate radially. Describe the subsequent motion. (Answer)

 

Q23 : Prove that every non-zero coefficient of the Taylor series of (1-x-x^2) exp(x)about 0 is a rational number whose numerator in lowest terms is either 1 or a prime. A Putnam competition question from 2014. (Answer)

 

Q24 : Define the exponential function and derive its relation to the logarithm from first principles. This is revisiting very old ground; a piece of analysis using series. I use it to demonstrate the binomial theorem with general exponent (not just integer). (Answer)

 

Q25 : A new continued fraction for e = exp(1). The problem is to prove that it does in fact converge to e and not to some other number close to e = 2.71828.... (Answer)

 

Q26 : Curve fitting and musical phrase shapes. Finding mathematical formulae to model the shapes of typical musical phrases..... (Answer)

 

Q27 : Find a formula for the sum of n consecutive integers j each raised to the power k: Sum j^k. This is an investigation of a classic problem that was solved 300 years ago by Bernoulii, Euler and Maclaurin. I compare numerical integration schemes with the Euler-Maclaurin formula. (Answer)

Q28 : Why does a mandarin orange (and any other object with a flat bottom) oscillate with a shudder when tilted and released? Have you noticed this amusing motion at the breakfast table? I have attempted to model the dynamics, studying some simpler cases first, so there is a full solution of a cylindrical tube with an eccentric weight inside it. This is compared with experiment and is in good agreement. However, when generalising to an ellipse, I found the maths becoming too involved. Perhaps if you are interested, you can take this further. (Answer)