red book of mathematical problems pdf

Bibliographical Note This Dover edition, first published in 1996, is a slightly coircued republication of the work originally published ly Integer Press, Ottawa, Canada, in 1988 under th

OF MATHEMATICAL

PROBLEMS

KENNETH S WILLIAMS

KENNETH HARDY

Carleton University, Ottawa

Dover Publications, Inc.

Copyrt,ghtCopyright Ct) 1988 by Integer Press.

All tights teserved under Pan American and International

Copy-ight conventions

I'tihlished in Canada by (;cneral Publishing Ounpany, I rd 30

I rsmill Road,DortMilk, Toronto, OntariO

Bibliographical Note

This Dover edition, first published in 1996, is a slightly coircued

republication of the work originally published l)y Integer Press,

Ottawa, Canada, in 1988 under the title TheRed Book: lOO1'ra.cticeProblemsfor UndergradualeMaihematics Competitions A section

of theoriginal page 97 has been dekted and all subsequent CODY

"A slightly corrected republication of the work originally

pub-lished by Integer Press, Ottawa, Canada, in 1988 under the title:'Fire red book: 100 practi e problems for undergraduate rnathe-

matics comperitions"—'lp verso

Includes bibliographical references

l)over Publications, Inc., 31 East 2nd Street, Mineola, N.Y 11501

rhi' humanities The preface of The 100 problemsl'n undergraduate mathematics competitions hinted at connections betweenrrulrlt'tn-solving and all the traditional elements of a fairy tale mystery,

lr.discovery, and finally resolution Although TheRed ii ook may seem topolitical overtones, rest assured, dear reader, that the quotations (labellt'd

M:ir l'ushkinand liotsky, just (or ftrn) arc merely an inspiration for your

"II through the cur harried realms of marlrcmati

i/ic Red Book contains 100 problems for undergraduate students training

ft n mathematics competitions, pat ticu larly the Willia in Lowell PutnamM;imhematical Competition Along with the problems come useftil hints, andrimiplete solutions The book will also be useful to anyone interested in theposing and solving of mathematical problems at the trndezgradtrate level.Many of the problems were suggested by ideas originating in a variety ofsources, including Crux Mathematicorl4m, Mathematics Magazine and the

Mathematical Monthly, as well as various mathematics competi•

(suns Where possible, acknowledgement to known sources is given at the endirE thebook

Once again, we would be interested in your reaction to The Red Book, and

invite comments, alternate solutions, and even corrections We make no claim

that the solutions are the "best possible" solutions, but we trust that you willfind them elegant enough, and that The Red Book will be a practical tool intraining undergraduate competitors

We wish to thank our typesetter and our literary adviser at Integer Press fortheir valuable assistance in this project

Kenneth S Williams and Kenneth Hardy

Ottawa, Canada

May, 1988

'To be reprinted by Dover Publications in 1997

In x denotes the natural logarithm of x.

exp x denotes the exponential function ex

cl( is) denotesEuler's totient function defined for anynatural

IIL11I-ber n

GCD(a,b) denotesthe greatest common divisor of the integers a and b

denotes the binomial coefficient n!/k!(n—k)!, whereis and

k) k arenon-negative integers (the symbol having value zero

when is<k).

denotes Legendre's symbol which has value +1 (resp —1)

if the integer a is a quadratic residue (resp nonresidue)snodulo the odd prime p

deg (f(z)) denotes the degree of the polynomial f(x)

dct A denotes the determinant of the square matrix A

Z denotes the domain of rational integers

Q, R, C denote the fields of rational, real, complex numbers

respec-tively

will alwmp, be fotsiul 1/us! 1/u task itselfarises osilsiu/u is 1/u lena! conditionsfor its solution alreadyexistorareat leastirs I/u

ititi-of formation

Karl Marx (1818-1883)

1. Let p denote an odd prime and set ui =exp(2xi/p) Evaluate theproduct

(1.0) E(p) = (wni + + +w P1)/2)(cA.i't3 + +

where ri, , denote the (p — 1)/2 quadratic residues modulo p andfl(p—i)/2 denote the (p —1)/2quadratic nonresidues rnodulo p

2. Let k denote a positive integer Determine the number N(k) oftriples (x, y, z) of integers satisfying

k, y—zJ k, z—x(

3. Let pm 1 (mod 4) be prime It is known that there exists a unique

integeT to w(p) such that

w2m—1 (modp), 0<w<p/2.

2 PROBLEMS

(For example, w(5) 2,w(13) = 5.) Prove that there existintegers u,b,c,d

withad— be=1such that

(Forexample, when 7) — 5 wehave

F(2X+Y)2,tnt

13X2 -I- 1OXY + =(3X + V)2 + (2X 1- V)2.)

4. Let d,.(n), r 0, 1,2,3, denote the number of positive integraldivisors of n which are of the form 4k r Let m denote a positive integer.Prove that

(4.0) >j(di(n) —d3(n))=

5. Provethat the equation

has no solutions in integers x and p

6. Let f(x,y) — ax2+ 2bxy+cy2 be a positive-definite quadratic form.Prove that

(f(xi,yi)f(x2,y2))h/2f(xi 7/2)

(6.0)

(ac b2)(x1p2 x2pl)2,

for all real numbers x2,p1 , 1,2.

7 Let 11,S,T be three real numbers, not all the same Give a tion which is satisfied by one and only one of the three triples

Prove that neither

2L)(D + A1 A2 +131132)

2D(1)-f A1A2 — 11182)

isthe square of an integer

12. Let Q arid R denoto the fields of rationaland real nuiiilii,s

respectively Let K nnd L he Iicsnialkst su blields (>1 II which contain both

Q and thereal numbers

resI)cctiveiy. Prove that K =L

13. Let k andIbe positive integers such that

i= x2 + 2xy + 2y2,

has exactly two solutions in integers z and y

14. Let r and s he non-zero integers Prove that the equation

(14.0) (r2 — s2)z2 —4rsxy—(r2— s2)y2=1

has no solutions in integers z and y

15. Evaluate the integral

Does the infinite series a,, converge, arid if so, what is its sum?

19. Let a,,, be ra ( 2) real numbers Set

A,,=ai+a2+ +a,,, n=1,2, ,ni.

Prove that

20. Evaluate the sum

for all positive integers a

21. Let a and b he coprime positive integers For k a J)OSitiVe integer,let N(k) denote the number of integral solutions to the equation

23. Letxi, be n (> 1) real numbers Set

Let F be a function of the n(n — 1)/2 variables such that theinequality

F(x11,x17 x,,

k=1

holdsfor all

Prove (ha I equality ( ansot laId in 23.0) if 0.

24. Jo a1 ar,, he at (.> teal a hiklL are SU( h

0.Prove the inequality

cannot he expressed as the product of two non-constant polynomials withintegral coefficients

28. 'iwo people, Aand H, play a game in which the probability that

A wins is p, the probability that 13 wins is q, and the probability of a draw is

r At the beginning, A has rn dollars and B has a dollars At the end of eachgame the winner takes a dollar from the loser If A and B agree to play untilone of them loses all his/her money, what is the probabilty of A winning allthe 00003'?

29. Let f(s) be a monic polynomial of degree n 1 with complex efficients Let x1, , s,,denote the n complex roots of f(s).ThediscriminantD(f) of the polynomial f(s) is the complex Ilumber

Expressthe discriminant of f(s2) in terms of D(f)

30. Prove that for each positive integer n there exists a circle in thexy-plane which contains exactly n lattice points

31. Let a he a given non-negative integer Determine the numberS(n) of solutions of the equation

in non-negative integers x,y,z

32. Let n be a fixed integer 2 Determine all functions f(s), whichare bounded for 0 < x < a,and which satisfy the functional equation

33. Let 1 denote the closed interval [a, bi, a < b. Two functions1(x), g(x) aresaid to he completely different on I if 1(x) g(x) for all x in 1.Let q(x) and r(x) be functions defined on I such that the differential equation

+q(x)y+r(x)

has three solutions y2(x), y5(x) which are pairwise completely different

on I If x(x) is a fourth solution such that the pairs of functions z(x), y,(x)are completely different for i — 1,2,3, prove that there exists a constant

K 0,1) such, that

(330)

(K —l)yi+ (Y2 —

34. Let a,, n = 2,3, ,denote the number of ways the product

b1b.2 can be bracketed so that only two of the are multiplied together

at any one time For example, a2 = 1 since b1b2 can only be bracketed

as (b,b2), whereas 2 asbib2ba can be bracketed in two ways, namely,(b1(b2b3)) and ((b1b2)b3) Obtain aformulafor a,,

35. Evaluatethe limit

36. LetE he a real number with 0 < E < 1 Prove that there areinfinitely many integers n for which

37. Determineall the functions f which are everywhere differentiableand satisfy

forall real r ;uid p with zy -/ 1

38. A point X is chosen inside or on a circle ['wo perpendicularchords AC and liD of the circle are drawn through X (Iii the case when X

is on the cirde, the degenerate case, when one chord is a diameter and theother is reduced to a point, is allowed.) Find the greatest and least valueswhich the sum S = ACI + BDIcantake for all possible choices of the point

k=O

41. A.13,G,D arefour points lying on a circle such that ABCD is aconvex quadrilateral Determine a formula for the radius of the circle in terms

of a = All, b lid,c CDtandd = DAt

42. Let AtJCJ)hea convex quadrilateral Let I' be the point outsideAIJGD such that 41'l = Pill and LAPIJ — go°. The points Q,R,S arcsimilarly delined I'iove thai the lines andQSaic of equal lciigtliai,d

prove that z = k/I for some integer k not congruent to 2 (mod 1)

46. Let P be a point inside the triangle A tIC Let AP meet DC atI), lip meetCA at E, and CI' meet Allat F prove that

PA1 P11 IPBI PCI IPCI PAl

48. Let m and n be integers such that I in < n. Let i1,2, ,rn; j 1,2, , a,be inn integerswhich are not alL zero, and set

max nil.

Prove that the system of equations

(1 1iX1 + 012X2+ + a1x = 0,

0,(48.0)

(Z1X1 + Q,n2X2 + + = 0,

has a solution in integers ,x2, , not all zero, satisfying

49. Liouvi lie J)rovPd that if

isnot au elementary function

50. The sequence XO,2I, is defined by the coriditioiis

51. Prove that the only integers N 3 with the following property:

if I <k N and GCD(k,N) = 1 then is prime,

im mn ni

—+—+—

to the lengths of the sides of the triangle

54. Determine all the functions H R4 —?Rhaving the properties(i) H(1,0,0,1)=l,

(iii) H(a,b,c,d) =—H(b,a,d,c),

(iv) H(a+e,b,c-t- f,d) =H(a,b,c,d)+H(e,b,f.d),

where a,b, c, d, c, are real numbers

55. Let Zi z,, be the complex roots of the equation

z" + + + =0,

where as, , a ( 1) complex numbers Set

A = max

has integral coefficients if and only if k 0 (mod 4).

64. Let m be a positive integer Evaluate the determinant of the

m x in matrix Mmwhose (i,j)-th entry is GCD(i,j)

65. LetI and m be positive integers with 1 odd and for which therearc integers r and y with

fi

m

Provethat there do siot exist integers u and v with

(1(65.0)

5u2 + l6nv + 13v2

66. Let

l'rove that a,. converges and determine its sum

67. .4 I 0<i 6) he a sequence orseveii iutcgeis satisfying

0 — 0'C121

For 0, 1, ,6let

N, = itumberof a1 (0 <6)such that a3 =

Determineall sequencesAsuchthat

i0,1, ,6.

68. LetG be a finite group with identity e If G' contains elements gand h such that

ghg'=h2,

determine the order of h

69. Let a and b be positive integers such that

prove that every integer >2abheloiLgs to S

70. Provethat every integer can be expressed in the form x2+y2—5z2,where x,y,z areintegers.

71. Evaluate the sein of the infinite series

Prove that there exist integers a1,b3 with

a1 a (mod n), b1 b (mod n), GCD(a1,b1) 1.

74. Forn =1,2, let s(n) denote the sum of the of 2" Thus,for example, as =256we have s(8) =2+5+6=13.Determine all positiveintegers n such that

75. Evaluatethe sum of the infinite series

mn(m+n)I

76. Across-country racer runs a 10-mile race in 50 minutes Provethat somewhere along the course the racer ran 2 miles in exactly 10 minutes

77. Let AB he a line segnient with midpoint 0 Let II he a point on

AD between A and 0 Three semicircies are constructed on the same side of

AB as follows: Sj is the semicircle with centre 0 and radinsl0Al = OBI; S2

isthe semicircle with centre R and radius All, meeting RB at C; S3 is thesemicircle with centre S (the midpoint of CII) and radius ICSI= ISBI Thecommon tangent to S2 and S3 touches S2 at P and S3 at Q The perpendicular

to AD through C meets S3 at I) Prove that PCQ B is a rectangle

78. Determine the inverse of the n x a matrix

0l1 1 101 1

20 PROBLEMS

where n is a positive integer

80. Determine 2 x 2 matrices B and ('withintegral entries such that

81. Find two non-congruent similar triangles with sides of integrallength having the lengths of two sides of one triangle equal to the lengths oftwo sides of the other

82. Let a,b, cbe three real numbers with a <b < c The function

1(z)iscontinuous on (a,c] anddifferentiable on (a,c) Thederivative f'(x)

isstrictly increasing on (a,c) Prove that

(c— b)f(a) + (6 — a)f(c) > (c — a)f(b)

83. The sequence {a,7 = 1,2,.. is such that > 0yn+1 >

0 ,rn 1,2, ,and converges. Prove that

—

convergesand determine its sum

84. The continued fraction of where D isan odd aonsquareinteger> 5, has a period of length one What is the length of the period ofthe continued fraction of +

85. Let G bea group which has the following two properties:

(i) G has no element of order 2,

/ (ii) (zy)2 =(!/z)2, for all z,y C.

Prove that C is abçlian

86. Let A be an n x n real symmetric matrix whose entriessatisfy

2=1

for all i =1,2 n Prove that 0 det A 1

87. Let fl be a finite ring containing an element r which is not adivisor of zero Prove that 1? must have a multiplicative identity

88. Set ={1,2,.-., n) For each non-empty subset S of 1,, define

w(S)= maxS—rninS

s(S sES

Determine the average of w(S) over all non-empty subsets S of 1,,

89. Prove that the number of odd binomial coefficients in each row

of Pascal's triangle is a power of 2

22 PROBLEMS

a number is selected 'ihe row and column containing x1 arethen deleted

From the resulting array a number isselected, and its row and columndeleted as before The selection is continued until only onc number remains

available for selection Determine the sum 4- + 4- x,,.

91. Suppose that p X's and q 0's are placed on the (ircllmfere:ic-' of

a ciicle The nutitber of occuirences of two adjacent X's is a and the number

of occurrences of two adiaceil 0's is 5 l)eterniiiie — Sin ternis nip and q

92. In the triangular array

93. A sequence of ii realnumbers x1 r, satisfies

i n),

where c is a positive real number Determine a lower hound for the average

0fZ1, ,Xn asa function ofc only

94. Prove that the polynomial

in irreducible over Z for n 4.

95. Let a1 be ii ( 4) distinct real numbers Determine the

general solution of the system of 71 —2 equations

X1 4 4• 4- x,, =0, 'i 4- (L2X2 4 • 4- 0,

in the n unknowns x1 're

96. Evaluatethe sum

tation: a hint of a brig/il new day.

Aleksander Sergeevicb Pushkin (1799-1837)

at -- CS — 1. Prove that as + ci f ±1, and deduce

;iIL iiiteger q can be found so thatb(_ a — ag) and d (— I rg) satisfya!; + ed fw,tuE — be 1 and b2 + d2 - + 1)/p

4. Prove that

(d1(n) —d3(n))= >

d odd andthen interchange the order of summation of the sums on the right side

5. Rule out f-lie possibilities z 0 (mod 2) and z 3(mod 4) bycongruence considerations if x 1 (rnod 4), prove that there is at least oneprime p 3(mod4) dividing a2 3x + 9 Deduce that p divides

a contradiction

6. Use the identity

=(aaix2+ bx,y2 + bx2yj +cy1 + (cc— b)2(x;y2 —

togetherwith simple inequalities

7. Prove that exactly one of the triples

(a,b,c) =(R,S,T),(T,—S + 2T,R -S+ T), (B —S+ T,2R —S,R),satisfies

or ab>c,

by considering cases depending upon the relative sizes of fl, S and T.

8. Considiff the sign of the discriminant of

(al) — bA)z2 + 2(a(7 —cA)xy+ (bG — cL?)y2.

11. Assume that 2D(D + A1 A2 + sB1l12) is a square, where c ±1

If D is odd, show that

D—A1A2—B1B2 =2DV2A1132—eA2B1 =2DUVDeduce that U2 + V2 = 1. Then consider the four possibilities (U, V) =

(±1,0), (0,±1) The case D even can be treated similarly

and prove that

cs++cv_=13+, cv÷—cv-=/3-

14. Factorthe left side of (14.0).

15. Male the following argument mathematically rigorous:

jlnxlii(l—z)dx =

=

=

16. Takingn =1,2, ,6 in (16.0), we obtain

a(1) = 1/2, a(2) =—1/3, (((3) = 1/4,

a(4) = -.1/5, a(5) 1/6, a(6) —1/7

This suggests that a(n)= which can beproved by induction

Puttingthese two inequalitiestogether, deduce that

20. Usethe ideittity

21. Allintegral solutions of ax + by = kare given by

x=g+bt, y=h—at, i=O,i1,±2,

where (p, h) is a particular solution of ax + by =k.