Here are about
problems and my attempts at answers.
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! Others still
are my own generalisations of more elementary questions.
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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Evaluate the indefinite integral of sqrt(tan x), ,
and its definite integrals over the intervals [0, pi/2] and [0,
pi/4], namely .
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.
Evaluate the integral (Answer)
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.
Show that every positive integer has a multiple whose decimal representation
contains only the digits 0 and 1. (Answer)
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)
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?
Which positive integers can be expressed as the sum of three or
more consecutive positive integers?
is a question from a maths problems class. Initially I did not find
it very interesting, but later discovered some merit in it.
Which is greater, cos(sin x) or sin(cos x)? (Answer)
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)
: Find an integer N with digits abcabd, with d = c+1,
such that N is a perfect square.
: Let a and b be positive integers such that divides.
Show that the quotient is
always a perfect square. (Answer)
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.
: Evaluate ,
the 8th root of the continued fraction 2207 - 1/(2207 - 1/(2207
- .... )).
: Rationalise the denominator of the surd fraction .
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?
: Evaluate .
: Evaluate where
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.
: Prove that there are unique integers a, n such that
solution is quite easy to find -- the challenge is mainly in showing
: Consider the power series expansion
Prove that for each integer n >=
0 there is another integer m such that (Answer)
Putnam question from 1999. Do-able by straightforward, standard
methods, though perhaps I missed a clever trick?
: 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.
: 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
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,........
: 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.
: 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.
: 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.
: 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.
: 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.
: 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).
: 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....
: Curve fitting and musical phrase shapes. Finding
mathematical formulae to model the shapes of typical mausical phrases.....
: 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.