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. It has also been translated into Portuguese (as «Como resolver problemas matemáticos», by Paulo Ventura Araújo for the Sociedade Portuguesa de Matemática), Chinese («解题·成长·快乐——陶哲轩教你学数学», ISBN 9787301154472, by Qinglin Yu (于青林) for Peking University Press), and Italian («Risolvere problemi matematici. Il mio punto di vista», ISBN 8896973880, by Samuele Maschio and Carlo Càssola for Scienza Express Edizioni). It has been reviewed for the Notices of the American Mathematical Society by Loren Larson.
p. 2“Problem 1.1 question” should just be “Problem 1.1”.
p. 6second to last paragraph, “once can avoid” should be “one can avoid”.
p. 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 $192t^2$.
p. 9example (e), “876” should be “376”. “which are exactly” should be “which have exactly”. In example (d), $n+1$ should be $n(n+1)/2$.
p. 13“which was weaker” should be “which was stronger”.
p. 16bottom, “217” should be “$2^{17}$“. “if this question is true, then the original question is true” should be “if the answer to the original question is “yes”, then so is the answer to this question”.
p. 25third paragraph: one of the “n”s should be in math mode.
p. 33Exercise 2.5: For an additional challenge, prove this exercise without using Bertrand’s postulate.
p. 35In the quote, “that was originally” should be “than was originally”.
p. 37second display: $f(f(2)-1)$ should be $f(f(f(2)-1)))$.
p. 40“smells heavily on” should be “smells heavily of”.
p. 42$t^m$ should be $t^n$ in one of the displays.
p. 445ab should be (13) (two occurrences)
p. 45Problem 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.
p. 46in the sentence “But polynomials only have as many degrees of freedom as their degree”, insert “with leading coefficient 1” after “polynomials”.
p. 47Exercise 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”.
p. 50the intersection of BC and AI should be labeled D.
p. 52In the second to last line on page 52, “either $\beta = 60^\circ$ or $\alpha = \gamma$” should be “either $\alpha = 60^\circ$ or $\beta = \gamma$“.
p. 53In the diagram on page 53, the angle at D should be $\gamma + \beta/2$, and the angle at E should be $\beta + \gamma/2$.
p. 58one of the instances of $ABEF$ should be in math mode (like all the other instances). “on the same side of $AB$” should be “on the same side of $AC$“.
p. 63third line, “inner square” should be “inner rectangle”.
p. 65-66the 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.)
p. 66problem 44, there is a $=2$ missing at the end of the string of equations, thus $x^p+y^q=y^r+z^p=z^q+x^r=2$.
p. 74second paragraph: “this can be true is of” should be “this can be true is if”
p. 76-77The 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.)
p. 77second-to-last paragraph, “it seems that the assertion is plausible” should be “it seems plausible that the procedure always terminates”.
p. 79In the diagrams on page 79 and page 82, the labels C and D should be switched.
p. 82“X is a quarter-length or less from M” should be “X is a quarter-length or more from M”.
p. 87second-to-last paragraph, “eliminated (c)” should be “eliminated is (c)”.
p. 90second paragraph: “ths game” should be “this game”.
p. 92third paragraph: “that it is a sure” should be “that is a sure”. In the second to last line, “all sure losers, as we have shown above” should be “all sure winners, as we have shown above”.
p. 95change all occurrences of “rouble” to “ruble” (for consistency). In the third paragraph, “in terms of equation” should be “in terms of equations”.
p. 96last line: “restricted to between” should be “restricted to lie between”.
p. 97second paragraph: “$s = \pm 4 (\hbox{mod }20)$” should be “$s = \pm 4 (\hbox{mod }20)$ or $s = \pm 6 (\hbox{mod }20)$“, and so the example values of s should now read $s = 4, 6, 14, 16, 24, 26, 34, 36, \ldots$.
In the references, “G.A. Hardy” should be “G.H. Hardy”.
p. 40(d), “如果k是正奇數，則 1^k+2^k+…+n^k 可被 n+1 整除”, n+1 should be n(n+1)/2.
p. 76“幸好第一個因數（p-1)i 和 p^2 是互質的（因為（p-1)i 和 p 是互質的）”, the i’s should be factorial symbols.
p. 146“與一些三角形和圓有關的∠APF，看起來似乎是比較「主流」的角”, APF should be ABF.
p. 154“若 n 為偶數，令 m=(n+2)/1”, (n+2)/1 should be n/2+1.
p. 202” 我們只能說 s = ±4 (mod 20)，因此羊的數量可能是 4、16、24、36、44、56 等”, $s = \pm 4 (\hbox{mod }20)$” should be “$s = \pm 4 (\hbox{mod }20)$ or $s = \pm 6 (\hbox{mod }20)$“, and so the example values of s should now read $s = 4, 6, 14, 16, 24, 26, 34, 36, \ldots$.
Contributors
Thanks to all those who have contributed corrections. Corrections received on or before 2026-07-09 were reported by the readers listed below over the years; individual per-erratum attributions for these legacy entries were not preserved in migration.