Last updated: Jan 6, 2009

Solving Mathematical Problems (Second edition)
Terence Tao
Oxford University Press, Oxford, England: 2006
Paper, 150 pages. ISBN13: 9780199205608
ISBN10: 0199205604

This book discusses various Olympiad level problems and how one can go about trying to solve them. It is the second edition of an earlier first-edition run.  It has also been translated into Portugese as “Como resolver problemas matemáticos” by Paulo Ventura Araújo for the Sociedade Portuguesa de Matemática.

— Errata —

  • In page 2, “Problem 1.1 question” should just be “Problem 1.1”.
  • In page 6, second to last paragraph, “once can compute” should be “one can compute”.
  • In page 7, “ompute” should be “compute”, and “put clear denominators” should be “clear denominators”. On the fourth displayed equation, 4d^2 should be 4d^4, and 196t^2 should be 192t^2.
  • In page 9, example (e), “876” should be “376”.  “which are exactly” should be “which have exactly”.
  • In page 16, bottom, “217″ should be “2^{17}“.
  • In page 25, third paragraph: one of the “n”s should be in math mode.
  • In page 33, Exercise 2.5: For an additional challenge, prove this exercise without using Bertrand’s postulate.
  • In page 35: In the quote, “that was originally” should be “than was originally”.
  • In page 37, second display: f(f(2)-1) should be f(f(f(2)-1))).
  • In page 40, “smells heavily on” should be “smells heavily of”.
  • In page 44, 5ab should be (13) (two occurrences)
  • In page 45, Problem 3.4, there should be no commas between (x-a_1)^2 and (x-a_n)^2. “are all integers” should be “are all distinct integers”. One should delete all references to a_0, for instance deleting the factor (x-a_0)^2 in the problem box, and also p(a_0) and q(a_0) in the next page.
  • In page 46, in the sentence “But polynomials only have as many degrees of freedom as their degree”, insert “with leading coefficient 1″ after “polynomials”.
  • In page 47, Exercise 3.7, the +1 should be a -1, one should look at p(x)+q(x) rather than p(x)-q(x), and “are integers” should be “are distinct integers”.
  • On page 50, the intersection of BC and AI should be labeled D.
  • In the second to last line on page 52, “either \beta = 60^\circ or \alpha = \gamma” should be “either \alpha = 60^\circ or \beta = \gamma“.
  • In the diagram on page 53, the angle at D should be \gamma + \beta/2, and the angle at E should be \beta + \gamma/2.
  • In page 58, one of the instances of ABEF should be in math mode (like all the other instances).
  • In page 63, third line, “inner square” should be “inner rectangle”.
  • In page 65-66, the informal geometric argument given is incomplete, the issue being that just because the sum of side lengths of R_1 is (say) bigger than 1, it is not immediately obvious that the same is true for, say, R_2. But one can check using algebra that if a+b > 1, then (1-a) + \frac{ab}{1-a} > 1, and similarly with the inequalities reversed; this allows the argument as stated to be made rigorous. (One can also argue by considering the rectangle with the narrowest side, and showing that it is adjacent to one which is even narrower if its sides do not add up to length 1.)
  • In page 66, problem 44, there is a =2 missing at the end of the string of equations, thus x^p+y^q=y^r+z^p=z^q+x^r=2.
  • Page 74, second paragraph: “this can be true is of” should be “this can be true is if”
  • Page 76-77: The informal topological argument here does not quite work as stated, for if two rectangles with integer horizontal lengths (say) are connected by a common horizontal line segment rather than a common vertical one, then the lengths do not add together as suggested in the argument. To fix this, one needs a more complicated colouring scheme. Namely, one colours the interiors of rectangles with integer horizontal lengths green, and those with integer vertical lengths red. As for the edges, one colours the (open) vertical edges green and the (open) horizontal edges red. There are a few remaining corners which are not on any open edge that remain to be coloured; these can be assigned either red or green arbitrarily. With this colouring, any green path between the two horizontal edges of the big rectangle can be used to deduce the integer horizontal length of that rectangle, and similarly for a red path between the two vertical edges. (There are also several other proofs, for instance one can induct on the number of rectangles.)
  • In page 77, second-to-last paragraph, “it seems that the assertion is plausible” should be “it seems plausible that the procedure always terminates”.
  • In the diagrams on page 79 and page 82, the labels C and D should be switched.
  • In page 82, “X is a quarter-length or less from M” should be “X is a quarter-length or more from M”.
  • In page 87, second-to-last paragraph, “eliminated (c)” should be “eliminated is (c)”.
  • In page 90, second paragraph: “ths game” should be “this game”.
  • In page 92, third paragraph: “that it is a sure” should be “that is a sure”.
  • In page 95: change all occurrences of “rouble” to “ruble” (for consistency).  In the third paragraph, “in terms of equation” should be “in terms of equations”.
  • In page 96, last line: “restricted to between” should be “restricted to lie between”.
  • In the references, “G.A. Hardy” should be “G.H. Hardy”.

Thanks to Paulo Ventura Araújo, Thomas Drucker, Percival Li, Cecil Rousseau, Naoki Sato, dsilvestre, Tom Verhoeff, and Weiyu Xu for corrections.

[Update, July 31 2007: bad link corrected.]