THIS ERRATA FOR PUTNAM AND BEYOND HAS BEEN WRITTEN WITH THE LATEX CONVENTIONS
(last update: March 27, 2017)

1. page 12 problem 33: replace "100" by "99"

2. page 126 line 3: replace "f(U)\subset V" by "f(U)\backslash\{x_0\})\subset V"

3. Page 303 line 15: replace the display by 1\leq c_0\leq b_1< c_1< b_2< \cdots < b_i

4. Page 303 line 17: replace "E\subset A" by "E=A\cap (\{1,2,\ldots, n\}\backslash \{c_0,b_1,c_1,b_2,\ldots, b_m,c_m\})"

5. Page 405 line 9 from the bottom: repalce "\mbox{Re}(-a_2z^{-2}-\cdots -a_nz^{-n})" by "\mbox{Re}(-a_{n-2}z^{-2}-\cdots -a_0z^{-n})"

6. Page 406 line 8 from the bottom: add "\leq 9(|z|^{-2}+|z|^{-3}+\cdots +|z|^{-n})" at the begining of the line in the second line of the display.

7. Page 263 line 5: replace "4^1+1^2" by "4^1+1^2+1" (add a 1)

8. Page 324 line 21: replace "S_n" by "S_9"

9. page 294 problem 856: replace the second occurence of the word "edges" by "vertices"

10. page 292 first line: after "complete graph" add " with $R(p,q)$ vertices"

11. page 515 line 15: insert "Mathematical" between "Romanian" and "Olympiad"

12. page 68 line 5 from the bottom: replace "positive" by "nonnegative"

13. page 79 line 4 from the bottom: delete the comma between "mechanics" and "observable"

14. page 296 line 4 from the bottom: insert "of" between "and" and "the position"

15. page 312 line 10: delete the word "combinatorial"

16. page 132 line 15: - delete the bar over z in $\psi(z)=f(z)-f(-\bar{z})$ - on next line, delete the entire content of the paranthesis; the phrase should end with the word "continuous" - on next line delete the bar over z_0 - in the display, delete the two bars over z_0 To conclude, in the entire paragraph there should be no bars.

17. page 148 lines 8 and 6 from the bottom: to be very rigorous, the argument of the natural logarithm in the two displays should be taken in absolute value. So we should have "\ln |\sin x+\cos x|" instead of "\ln (\sin x+\cos x)", three times.

18. page 103 line 9: the two exponents in the general term formula should both be $n$, not $m$.

19. Replace the first example from section 2.2.3. page 52 with the following:
\begin{example}
Find all cubic polynomials $P(x)$ with the property that $P(x)$ is a multiple of $P''(x)$.
\end{example}

\bes
Set $P(x)=Q(x)P''(x)$, with $Q(x)$ a quadratic polynomial. Differentiating twice this relation we obtain
\begin{eqnarray*}
P''(x)=Q''(x)P''(x)+Q'(x)P'''(x).
\end{eqnarray*}
Hence $P''(x)(1-Q''(x))=Q'(x)P'''(x)$. We deduce that $P'(x)$ is a constant multiple of $Q'(x)$. Incorporating the constant into $Q(x)$, we see that $P(x)=Q(x)Q'(x)$ for some quadratic polynomial $Q(x)$. It is not hard to check that each polynomial of this form has the desired property.
\ens

20. Page 718 line 1: 7^z should be 7^zx

21. On the back cover there is a repeated "to" on the last line.

22. Page 7 line 1: add "a" before $1\times 1$.

23. Page 93 line 21: in the statement of problem 286 it would look much better if instead of "$(\sin n)_n$" there was "$(\sin n)_{n\geq 1}$"

24. Page 183 last line: replace {\bf i} {\bf j} {\bf k} by \vec{i} \vec{j} \vec{k}

25. Page 102 line 9: replace "$\lambda _{1,2}=\frac{1\pm \sqrt{5}}{2}$" by "$\lambda _1=\frac{1-\sqrt{5}}{2}$ and $\lambda_2=\frac{1+\sqrt{5}}{2}$"

26. Page 206 line 11: "Let $ABC$ be a convex quadrilateral" should be "Let $ABCD$ be a convex quadrilateral"

27. Page 254 line 15: put ( ) around f(2n+1) and f(2n) so that they look like (f(2n+1))^2-(f(2n))^2

28. Page 585 line 11: insert "not" between "is" and "identically"

29. Page 684 line 13: "\lfloor x_1+\frac{1}{2}\rfloor" should read "\lfloor nx_1+\frac{1}{2}\rfloor" an n is missing

30. Page 185 line 4 from the bottom: the third term in the equation should be "\frac{(x+f(0))^2-(x-y)^2-(f(0))^2+y^2}{2x}"

31. Page 112 line 4 from the bottom: replace "\frac{1}{2}(x+1)" and "2(x+1)" by "\frac{1}{2}(x+2)" and "2(x+2)"

32. Page 112 line 2 from the bottom : replace "x\cdot \frac{x+1}{2}" by "x\cdot \frac{x+2}{2}" and "2x(x+1)" by "2x(x+2)"

33. Page 331 line 7: Delete the sentence that starts with "We assume in" and ends with "problem."

34. Page 610 line 4 from the bottom: the vertical bars around the cross product of OM and OP should be doubled, therefore they should be \| and \|

35. Page 610 line 1 from the bottom: in the bracket, the \times between AE and AC should be +

36. Page 199 delete lines 13 and 14, namely delete the sentence that starts with "An alternative approach..."

37. Page 730 line 4 from the bottom: delete the comma between $r-1$ and transpositions.

38. Page 547 line 4 from the bottom: it should be $f'(x)\geq f'(1)$

39. Page 159 line 13: replace $f'(0)>0$ by $f'(1)>0$.

40. The solution to problem 818 at page 279 is wrong and cannot be fixed. Here is a possible replacement:
Find all positive integers $x,y$ such that $15^x-6^y=9$.

Solution (to be replaced at page 722, line 4):
Clearly $x=y=1$ is a solution, while $y=2$ does not yield a solution. If $y>2$, then reducing modulo $4$ we obtain that $x$ is even, $x=2n$. The equation can be rewritten as $15^{2n}-9=6^y$ or
\begin{eqnarray*}
9(5^n\cdot 3^{n-1}+1)(5^n\cdot 3^{n-1}-1)=2^y\cdot 3^y.
\end{eqnarray*}
Because $y>2$, the left-hand side should have an additional factor of $3$. This can only happen when $n=1$, so the only other solution to the given Diophantine equation is $x=2,y=3$.

41. Page 304 problem 882: in the first sum, the summation should be from "i=0" instead of "j=0"

42. Page 761 line 12: $\{0,1\}^n$ should be $\{0,1\}^{p+q+1}$.

43. Page 760 line 9: the expression in the display should be $\sum_{k=0}^m{m\choose k}{{n+k}\choose m}$. The expression written there has nothing to do with the problem.

44. Page 30 line 13: in the display there should be $(n-1)^2$ instead of $(n^2-1)^2$.

45. Page 41: problem 122 is silly, it should be ignored.

46. Page 268 line 2 from the bottom: the $a_n$ in $x\equiv a_n$ should be $a_k$.

47. Page 269 line 8: Warsawa should read Warszawa.

48. Page 711 line 7 from the bottom: Warsawa should read Warszawa.

49. Page 132 in the solution to the example, all the bars over z and z_0 should be ignored (there are 5 of them). As such, "$-z$ is diametrically opposite to $z$".

50. Page 480 the third display: the "+4" in the denominator should be a "+2".

51. Page 255 line 12: the union of sets should be intersection of sets (because of de Morgan's law).

52. Page 373 line 9 from the bottom: In the display, there is an extra bracket in the term |\epsilon(z-\epsilon)|; delete it.

53. Page 330 line 2: replace "x_j>0, x_{j+1}<0", and "x_j+x_{j+1}" by "x_j>0, x_{j+1}\leq 0". The original condition is too strong (e.g 2,3,-4) and not needed.

54. Page 324 lines 3, 5, 6: replace "16" by "15".

55. page 323 line 4 from the bottom: while the solution is not incorrect as is, the number 29 can be replaced by 27 in the argument, namely 21 plus the 6 numbers that can have repeated prime factors.

56. page 84 line 2 from the bottom: replace $n$-dimensional sphere by $n-1$-dimensional sphere.

57. page 86 line 18: replace $K_0$ by $K_1$.

58. page 86 line 19: replace $K$ by $K_1$.

59. page 424 line 12: in the last sum there should be a cosine before $k(m+j)\alpha$.

60. page 555: The solution to problem 497 is wrong. I think the following fix works. Replace replace everything starting with the second line at the bottom of page 555 to the second equation on top of page 556 by
" Similarly
\begin{eqnarray*}
(-1)^m\int_0^\pi f^{(2m)}(x)(n\sin x+sin nx)dx =(na_1-n^{2m}a_n)\frac{\pi}{2}\geq 0.
\end{eqnarray*}"

61. page 239 line 12: in the expression on the left-hand side, the denominator of the first factor in the second term should be 2z not 2x.

62. page 351 line 2: "9/2" should read "13/2".

63. On the front cover of the book, on the last line of the design there is a parenthesis missing in "(x,y)"

64. page 41 line 7: the "2" in the parenthesis should be a "w"

65. page 100, in the determinant at the bottom of the page, on the first row of the determinant there is a "-" missing in front of "a_{k-1}"

66. skip this entry

67. page 50, line 8 from the bottom: "m^3-3m^3+27m-20" should read "m^3-9m^2+27m-20"

68. skip this entry

69. page 115, in the statement to problem 341, the condition should be "p\neq -1"

70. Another way to write the answer to problem 298 is "x_n = \lfloor{\frac{n}{2}}\rfloor^2 + \lceil{\frac{n}{2}}\rceil^2"

71. page 464 line 8 from the bottom, "u^n and \bar{u}^n" should read "u and \bar{u}"

72. page 464 line 4 from the bottom, S_2 should be "S_2=1-a"

73. pages 304, 305 solution to problem 29: there is a logical flaw in the solution, we use the sum of the angles of a polygon to conclude the existence of an angle less than $180^\circ$, but this is proved using the result itself. Here is how to fix this: If we can only find an angle greather than $180^\circ$, we use the same sweeping ray argument. Again we start by tracing a side, but this cannot go on forever, as we need to rotate the ray by more than $180^\circ$. So at some point an event happens, and this event can only be the encounter of a vertex. And again we are done.

74. page 152 line 6, the sign in front of $I_{n-1}$ in the display right above the sentence "The remaining integral..." should be changed to a minus.

75. Page 421. The second solution to problem 219 is wrong.

76. Page 584, last line: an exponent of $2$ is missing, one should have [f(\pi/2)]^2 on the right-hand side.

77. Page 261, last line. The problem was actually given at the IMO

78. page 543 solution to problem 471: there is an issue with $a$ possibly being negative, which was not ruled out in the statement. Thus one line 1 we should write $|a|<1$, on line 2, $|a|>1$ and the display should end with either $\pi \ln a^2$ or with $2\pi \ln |a|$.

79. page 331 line 14, the correct computation is 3=4^2+6^2-7^2. Also here is a suggested faster solution: We prove by induction that m=25^(n-1) can be written as the sum of 1,...,n positive squares. Base case: 1=1^2. Inductive step: Suppose 25^(n-1) can be expressed as the sum of 1,...,n positive squares. 25^n can be written as the sum of p positive squares, for any p in 1,...,n, by multiplying each addend in the decomposition of 25^(n-1) into p squares by 25. Now let 25^(n-1)=(a_1)^2+...+(a_n)^2. We have 25^n=(3a_1)^2+(4a_1)^2+(5a_2)^2+(5a_3)^2+...+(5a_n)^2, and we're done (for n=1, we simply have 25=9+16). (solution by Elliot Glazer)

80. page 382 solution to problem 136. The sentence on lines 13 and 14 should read: If $m=rn+s$, with $0\leq s< n$, then for this to happen $s$ of the $x_i$'s must be equal to $r+1$ and the remaining must be equal to $r$.

81. Page 126 last line, the display should start with lim_{t\rightarrow 0} \frac{t(lna)}{(ln(1+t))} = lim_{t\rightarrow 0} \frac{lna}{ln(1+t))^(1/t)}.

82. page 402 the third display is missing a summation sign $\sum_{j=1}^n$ at the very end.

83. page 55 the statement of problem 180 should have a strict (sharp) inequality.

84. page 513 the function defined in the solution to problem 405 should have the domain (0,\infty) (see line 10)

85. page 403 line 10 from the bottom should read: First, note that from Rolle's theorem applied to $\phi(x)=e^{-kx}P(x)$ it follows...

86. page 405 line 10 from the bottom should read: half-plane $\mbox{Re }z\leq 0$ and the disk $|z|<4$. Otherwise, if $\mbox{Re }z>0$ and $|z|\geq 4$, then...

87. page 14 statement of problem 47: to be entirely accurate we should either specify that no two diagonals are parallel, or make sure that the reader understands that angles of 0 degrees are allowed.

88. Page 334 line 10 "which does exist because the sum of the $n$ angles is $(n-2)\pi$"; this explanation is completely idiotic, because the only way to prove that the sum of the angles is what is claimed is by decomposing the polygon into triangles, so we have a vicious circle. To find an angle less than $\pi$, just pick a corner of the convex hull of the polygon.

89. Page 301: Statement of problem 872: The formula is for the Fibonacci number $F_{n+1}$ not $F_n$.

90. Page 755 line 6 replace $F_n$ by $F_{n+1}$

91. Page 511, in the solution to problem 400, the summations over the index k should all start at k=1.