Problems published in "Wiskundige opgaven met de oplossingen"

Problems published in "Wiskundige opgaven met de

oplossingen"

Citation for published version (APA):

Bruijn, de, N. G. (1964). Problems published in "Wiskundige opgaven met de oplossingen". Technische Hogeschool Eindhoven.

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! '

_I

published in

"'NISKUNDIGE OPGAVEN' MET DE OPLOSSINGEN"

by N.G. de Bruijn Technological University, Eindhoven, Netherlands. September 1964.

(S"'i',~

(4r£')

•

The references at the end of each problem indicate the publication of the solution in

Wiskundige Opgaven; volume, issue, year,

problem number. An

additional D indicates

that the problem

was

published in Dutch language.

1.

Assume

.e>0,

a >0, b >0 (n=1,2,3, ••• ), lim b a-1

=.e.

n n n-oo n n

Rearranging the sequence a

l , a2, ••• we obtain the non-increasing sequence IXl po IX2 ? ~ ;;.. •••• Simil.arly we obtain from b

1, b2, •••

the sequence

~

... , ?

~2

?

~3

;:t •••• Show that lim

~

,,1

=.e~

n-lJO n n

If & > 0, show that

t N

,fn ..

.P.::.1 _.1

If

n (

2 ) (~+&)

.e

=1

PI.e

1'+1 - 3 l' ,1- 1'(1'+1) +

~

(1' runs through the primes, and

p/.e

:means that p runs through all

primes dividing.e).

[17 (.5) 1942, Nr. 174, D]

Let B(N) denote the number of triples (,e,m,n) with ~,m,nare.

positive integers ~N, mutually co-prime (that is (.e,m)

=

(m,n)

=

=(n,.e) = 1~) Show that'

N

L

a

B(N)

=

-2~3

1:

.e=1

~e::cp(.e)

J.L(k)

{t

d/.e J.L(d)[kd]}:

(where kE::cp(.e) means 1

~

k

~

.e,

(k,£)

=

1, and J.L

den~tes

the Mebius' function), and that, for every & >0,

If p ~ q ? 0, p+q

=

n, and

l:P A=1 then show that

n

I

a_llA

I

Ei 1, t A=p+1

I

d et (a , ) _{- IlA - -}

I

~ 2 q.

(1l=1, ••• ,n)

and that there is a matrix a~A for which the sign

=

holds.

[17 (5) 1942, Nr. 177 jointly with D. van Dantzig, D]

Show that for every integer n >

°

and for every integer x we have

xn/d

=

°

(mod n),

when Il(d) indicates Mobius' function.

[18 (1) 194} Nr. 15, D]

Assume

°

< a < 1, ~ complex. If z >

°

we define

GO _ma a~-1 F(z,~ta) = tn=1 e n ."

r

(~)' lHz,~) =

zP

(J3;l 0, -1, -2, ... ), cp(z,~)

=

_{(-1) +1} ~

(-~)lZP

log z (~= 0, -1, -2 t • • • ) . 1

Show that F(z,~, at) .-, (i cp(Zt~) can be continued anal.ytically

throughout the complex z - planeo

)

I "

Let fez) = a z

1 + a2 z2 +

B:s

z 3 + •• ~ be holomorphic for / z/ < 1, and assume that for no value of z inside that circl,e fez) takes a real

1

value < - !j: • Show that la

I

~n (n=l,2, 3, ••• ).

n

;

[18

(2B)

1946, Nr. 64, D

J

Assume d > 0, d::' 3 (mod 4), and let C({d) denote the quadratic

field obtained by adjoining

Vd

to the rationals.

If t;. is an integer of the field, and if t;. divides i(d+1), then show that the norm of; is positive.

~8 (2B)

1946, Nr. 65, DJ

Let G be a finite abelian group and let R be a subset having k elements. Assume that for every character X of G the aum t aE:R X (a) is either 0 or k. Show that R is a subgroup.

Show that for every positive integer N we have

Let s be fixed, s > 1. Let F(z,s) be defined by F(~,s) 00 -z(log n)2 = ~ .;::e _ _ _ _ _ n=1 s n 1

- Vz

e 2 (s-1)

4z

r

·s-1 2V£

if z > O. Show that F(z,s) can be continued analytically throughout the

complex z-plane.

[18(3) 1946, Nr. 109,

DJ

,

On the interval -1 ~x ~ 1 we are given n point charges (n ~ 2)

situated in ~ , ••• ,x , respectively, repulsing each other mutually

1 n with

forces Ix. - x.j-1. They are in

1. J equi]hrum. Show that x , •• ,x are the 1 n

zeros of (1_x2 ) pI 1(x), where

n-st

p 1 is the (n-1) Legendre polynomial.

n-n8(3) 1946 Nr.110,

DJ

Let the complex number fen) be defined for all integers n,m and satisfy

jf(n+m) + f(n-m) - 2f(n) - 2f(m)/ < 1.

Show that there ex1sts a complex constant ~ such that

I fen) -~ n2

j

<

t

for all n.

[19(1) 1950, Nr. 22,

D)

Let the sequences {aJ and {c

n} satisfy the conditions that

-p

<

a :5; 1 (where

P

is a constant, -1 < - P E; 1) for all n, and that

n

l:::=1 (cn+1 - ancn ) converges. Show that limn...co cn exists.

If 0 < q < 1, show that

co 2 2

J (

CO n -n n)-1

o t-ooq t dt =

in

{ 1+q+q +q +q + ••• :5 6 10 } J, •

[19 (1) 1950,Nr.24, DJ

Two players play the following game, moving alternately the

game is lost by the player who is unable to move.

On an infinite row of squares, labeled 1,2,3, ••• , we have a

finite number of counters, at most one on each square~ A move consists

of shifting a counter to a square with a lower number, which is

allowed only if the new square and all squares between the old square and the new one, are empty.

Show how to play this game in all possible situations.

D9( 1) 1950, Nr. 25, D]

Let, for x ~ 1, the function f be decreasing and positive, and

co

assume that t 1f(n) converges. Let a1 ' ••• '~k be positive, and

Show that

co co co

tn=1 f(nO':!.) + ••• + tn=1 f(nak ) =e;"

.li

f(n).

Assume 0 < ex < 1, ex -1 + ~ -1 = 1, and let

[X ]

denote the integral part of x. Show that

and that the equality sign holds if and only if ex is irrational.

[19(2) 1951, Nr. 52, D]

Show that (i):if'

p~1,

r.

oo /a

I

p < 00 then we have

o n

-x n

a e x

n p

n!

(ii) if p> 1, the equality sign holds only if a =a = •••

=

0;

o 1

(iii) for no value of p > 1 we can replace the right hand side

with C not depending on aot ai'

...

,

and 0 ~ C < 1.

[19(2) 1951, Nr. 53, D)

Consider partitions of the positive integer n into positive integral parts, and identify two partitions if and only if the

one can be transformed into the other by cyclic permutation ~f the

summands (Thus 3+2+1+1+1 and 1+1+3+2+1 are the same partitions, but

1+1+2+3+1 is different). Show that there are ~

1

-

n

Two players play the following game, moving alternately; the game is lost by the player who is unable to move. We have a number of heaps

"I ,

of matches. A move consists of removing either one match from one heap, or removing simultaneously one match from a number of heaps (one from each heap). The latter operation'is only allowe~ if all heaps from which a match is taken have the same size (although there may be heaps of the same size, from whioh no match is taken).

Show how to play this game.

[19Gn 1951, Hr. 55, D]

Show that the conic passing through the centres of the circumscribed circle and all four inscribed circles of the triangle

ABC,

touches Euler's

line. _, . [19(3) 1952, Hr. 91,

lj

Show that

,

.... / - j ; : {t (3t" +1) - 3t" (1+t"

)t }

dt a

2~"~

(r

(~»

J

\~-3.

[19(4) 1953, Hr. 131. Jj. Let ~ 0

1' f 1, ••• , fn be continuously differentiable funotions of the variables xi' •••• x_n' and assume that they are periodiO/'_=. -f,m"ctionS of each variable. with period 1. Let

" "'

'Sf, ••• , f

~.

1

xi t " 0 t xn

5

=

1 n .

denote the Jaoobian 01·f

1 ••••• fn•

Show that 1

...

/

_o J (f₁ ' •• • ,f ) dx , ••• , dx

=

0 ; n 1 n

=

I:~ _l.=o det. . a.. = 1, l.,J =o, ••• ,n l.J we have

b9(4)

1952, Nr. 132,

nJ

Consi~er the confocal system of hypersurfaces

2 X 1 a +A 1

=

1

in n-dimensional euclidean space, where a , ••• ta are d~stinct real

1 n

numberse Let P

=

(p , ••• ,p ) be a real point, and

p.#

0 (~=1,.8.,n).

1 n l.

Show that there are n-1 different re~l values for

A

such that it is

possible to draw a set of n mutually orthogonal tangents through P. ~9(4)1952,Nr. 133,~

nJ

Let fn be defined by f

=

1 f - I:n-1 1 ' n - k=1 where d k = a (k=1,3,5, ••• ), dk = b (k=2,4,6, ••• ).

Find explicit expressions for f2n and f 2n+_{1 e}

Find the asymptotic behaviour of f2n+1 and of f

2n,

if it is assumed that a > 0, b

>

0.

[19(4) 1952, Nr. 135,

D)

Let 'A be a positive constant. Define' £ by £(0)=f(1)=0, and

Snow that we have for the n-th derivative

,-1 n

{c

n 1+11. }

where C depends on A only.

,

If a~n) represents the sum of the A-th powers of the divisors of n, then show that

=

~d/n en/d) A a

A(d2 ) ..

[19 (5) 1954 Nr. 169, DJ

-A finite sequence of irreducible fractions Pk/q k (qk > 0, k=O, •••

,m)

is called a chain if

(k=1, ••• ,m).

,

'

Consider a second chain Pk/q~ (k=~, •• ,n) and assume that ~

pI

n

<qr

n

Show that the two chains have a fraction in common.

For which values of A do the roots of the equation x:5 + 3i~

=

A. form an equilateral trianGle in the complex plane?

[20(1) 1955 Nr. 13,

nJ

A sequence v ,v •••• (v ~ 0) will be called a T-sequence if it

1 2' n

has the property that the conditions (i)a

1 +a2 + ••• is C1 -summable, and

(1i) /a

I

=:; v (n=1,2, ••• ), imply that ta converges. It is known that

n n 'n

implies that {v } is a T-sequence. Show that n

a. If

{Vn}

is a T-sequence, lim vn=O, then (1) holds.

be There exists a T-sequence that does not satisfy (1).

[20 (1) 1955 Nr. 14]

It is known that the interval (0,1) is not the~ of a sequence

of nowhere dense sets, nor the~ of a sequence of sets of measure

zero. Is it possible to dissect the interval into Sl U.5₂ U ••• , such that

"-each 5

i is the

union

of a nowhere dense set and a set of measure, zero?

[20(1) 1955 Nr. 15J

If f1 ,f2 I • • • is a sequence of measurable functions defined on

_00 < x < 00, and fn (x)-O for all x, then a set S is called regular whenever

the convergence is uniform on 5. A set R" is called a residue set whenever

its complement R' is the union of a countable number of regular sets.

:Sgoroff's theorem shows that there exists a residue set of measure zero.

Construct a sequence f ,f If , ••• , satisfying the above conditions, such

1 2 3

that every residue set is an everywhere dense set of power c (the power of

the continuum). [20(1) 1955 Nr. 16]

I

. I

11 ..

Let A , ••• ,A be impenetrable rigid bodies in n-dimensional

1 m

euclidian space (n > 1). Assume that the bodies are convex and bounded.

Show that it is possible to move the bodies such that Ai tends to infinity whereas the others do not.

[20 ( 1 )

1955

Nr. 1?

J

If in the previous problem we moreover assume n=2, then there is at least one index i such that A. can be moved to infinity without

l.

moving the others at all.

[20(

1) 1955

Nr.

18J

A set S of positive integers is called convex if u

e:::S,

w E:S,

u/v, 'v/w always imply v E: S. Let f and g be functions defined on S.

stow that the validity of one of the following relations for all nE:S implies the validity of the other one for all n e::: S:

Here ~ indicates Mobius' function.

,

(20(2)

1956

Nr.

53J

(For definitions see Nr.

53).

Let S be a finite convex set, consisting

of the distinct positive integer a •••• ,a •

1 n

The matrix T = (t

ij) (i, j = 1, ••• ,n) be defined by tij= 1 if aj/ai,tij

=

0

otherwise. Determine the determinant and the inverse of T.

Let S be a finite convex set consisting of the distinct positive

integers a₁, ••• ,a

n• Let l' and g be functions related according to Hr. 53.

Let the matrix M = (m .. ) (i,j = 1, ••• ,n) be defined by m .. = f«a., a.»,

l. J . l.J l. J

where (ai,a

j ) denotes the G.C.D. of ai and aj • We define f(a) = 0 if a is

not:m S.

Show that det M = g(a ) ••• g(a ), and determine the inverse matrix of _{1 .. n} X.

[20(2)1956 Nr. 55J

Let nand k be integers, 1 E; k E; n. Evaluate

j + j2+ ••• +jk

I: (-1) 1

,

where the sum runs over all sets j 1 , •• • , jk which sa tisfy 1 E; j 1 < j 2 < ••• <

< jk E; n. [20(2)1956 Nr. 56J Show that 00 n dx -xI: L - 1- Y',

J

e 1E;n E;[XJ n! x2 -1

where Y is Euler's constant.

Consider permutations f(1), ••• ,f(n) of the elements 1, ••• ,n with the property that f(k+1)-f(k)! 1 (k=1, ••• ,n-1).

Let N(n,m) be the number of such permutations which moreover satisfy

Show that the following statement is true for any sequence

of integers> 1, if lif ... oo nk = 00.

n ,n ,n , •••

1 2 3

Let S be an infinite sett and assume that for any k(k=1,2,3, ••• ) there

.nk

is a dissection S = U j=1 Skj into non-empty disjoint subsets.

ShoVl that it is possible to select, for each k, an index m

k(1:S:: ~

:s::

nk) such that

has infinitely many elements.

[20(2) 1950 Nr • .59 jointly with P. ErdosJ

Show that the statement made in the preceding problem is false for any sequence n ,n , ••• that does not tend to infinity.

1 2

[20(2) 1956 Nr. 60 jointly with P. ErdosJ

Let a ,a t • • • be a monotonic sequence of positive numbers, and let

1 2

the sequence b ,b , ••• be a rearrangement of a--,a

,.0 ••

Assume that

1 2 1 2

b

n + 1/bn converges to a limit

,e.

Show that an + 1/an also tends to

,e'.

[20(3) 1957 Nr. 91

J

- b f ' d 'to . t and abbreviate e2ni

x/

n __ e(x).

Let n e a ~xe pos~ ~ve ~n eger,

Let the matrix (a .. (t» = ACt), for any complex number t, be defined by

~J

Show that, for all s and t, we have A(s)A(t)=A(s+t).

If A( t) is defined as in Nr. 92, then the mapping t -. A( t) is a representation of the additive group of all complex numbers.

Show that this representation is irreducible, and, if n > 1, also faithful.

[20(3)1957 Nr. 93]

If lxl < 1, show that

00

l:n=1 ( n2 x -x +x n

rP

+n) ( / 1-x n)

=

O.

If.

Ixl

< 1, show that

00 l: n=1 ~0(4)1958 Nr. 131] + .... [20(4)1958 Nr. /132]

Let Sk be the k-th. elementary symmetrical function of the variables

e2nitl, ••• ,e2nitn(0 ~ k =iii n). Show that

s: ..

0

s:

-n(n-1)

n( th-t.> • dt 1 ••• d t

=

2 n! ( .

J n

Let the real sequence a

1,a2 ,a3, ••• have the property that to any

e

> 0 there exists a number

keel

such that k(e) < n < m always implies

a -a < & 0 Show that a can be written in the form a =b +c , where b ~ 0,

nm n n n n n

Show that for sequences

{e },

with 0 <

e

< 1 (n=1,2, ••• ),

n n

the following two conditions are equivalent:

-s A

C9 n

(1) There exists a real number s such that tn=1 e converges

(>..

=

(1-

e )

-1)~

n n

00

(2) If a₁,a₂, ••• are positive and such that tn=1 a converges,

00

Eh

n

then En=1 an converges as well.

t20(4)1958 Nr. 135 jOintly with H./ Freudenthal.J

(~ ;. 0, ex ~-1, ex+2~ < y) ..

In n-dimensional euclidean space R (n~ 2) we take a two-dimensional n

plane V, which we consider as the complex plane. The unit of length

in Rn equals the unit of length in the complex plane (thus, iflzE:V, z2 €.V, the distance of

2i

and z..a is

I

zl -zJ

I).

Let d be the zero point of V, and let Alt ••• tA

n be points of Rn such

that the vectors OA , ••• ,OA form an orthonormal base for R • The

1 n n

orthogonal projections of A, ••• ,A onto V produce

1 n the complex numbf'

z

,.J.

,z •

1 ' n Show that t

n _zk2

₌

_{0, t} n _{I 1,-,2}

₌

_2.

1 1 . .lib

[21(1) 1960, Nr. 9, D]

As in the previous problem, let the complex plane V be embedded

in

R

(n ~

2).

Let

A

l, •••

,A 1

be the vertices of a regular simplex,

n n+

and let the complex numbersw

l , ••• ,wn+1 qe their orthogonal projections

on

V.

Show that

[21(1) 1960, Nr. 10,

n]

In a non-equilateral triangle ABC let 0 be the orthocenter and

G the center of gravity. Let P be a point on OG with

PiG

=

p. OG where

p is a real number. Denote the direotions (i.e. the points at infinity)

of BC,CA,AB and CP by a,b,c, and l~ (a,b;c) repectively ( lp only depends

If 1 (a,b;Q=d then 1 (a,bjd)=c.

p p

Remark. If p=oo t!1is is a theorem of P. Zeeman:

Ehch cneoftre four directions a,b,c,d is the direction of Euler's line in the triangles having sides paralell to the other three as soon as this is the case for one of them.

If a , ••• ,a are real, Z ,.o.,Z complex, then show that

1 n 1 n

L21(1)1960, Nr. 12,J

Let p be a positive constant, and let c denote the coefficient

n

of xn in the power series expansion of epxex+1)(x+2) ••• (x+n-1)(n=1,2, ••• ).

00 n

Show that the series 1:1 cnZ has a positive radius of convergence P, and

-1

thatp is the smallest positive solution of the equation w +log w = p+1.

[21(1)1960, Nr. 13,]

Consider 2n distinct points on a circle. We want to split this set into n pairs, such that ,the n chords connecting the pairs do not intersect inside the circle. In how many ways is this possible?

Let a < b, and let f be a real function in the interval a === x === b.

We assume that

Show that f(a) === f(b).

[21(3)1962, Nr. 106J

Let a ,a ,a , ••• be a sequence of positive numbers, with lim_n .... _oo a_n

=

O.

1 2 3

Let S be the set of all positive integers n with the property

n /

_{p n} p >na _n

(p runs through the primes dividing n). Leta(x) be the number of elements of S which do not exceed x. Show that a(x)/x .... 1 if x .... ~.

Let C be a closed continuously differentiable Jordan curve in ~ plane, and let Fl and F2 be points inside C.

For every point P on C we consider the angle $ between the inside normal and the bisectrix of the angle Fl PF

2 • Show that

cos fi do:: 21t t

if r

1 :: PF1 t r2 :;?~, and if do denotes the line element of C.

[21(4)1963, Hr.'131,

DJ

Show by a counter-example that the following theorem is not true: "Let II .11

1 and II il2 be different norms in a linear space M. If a sequence

x ,x , ••• converges in both norms then the limits are equal". 1 2 Evaluate [21(4)1963, Hr. 132., jointly with G.W. VeltkampJ [21(5)1964, Nr. 168~ jOintly with C.J. BouwkampJ

Let k be Cl positive integer, and let Xl' ••• t~ be real numbers

satisfying IXl

j+ ••• +

i~

1<

n. Show that

r,co 1

~1

in-1n-j sin nx sin nx ••• sin nx,.

n= 1 2 A

has the value 0 if k-j is an even integer with 0 < j < k, and has the value iXl ••• ~ if j=ko

[21(5) 1964 Nr. 175]

Let ~ be an odd prime. We want to construct an infinite sequence of integers Xl' X

2' x3, ••• such that X12

¥

1 (mod p) and such that

2x x =x 2+ 1 (mod p) for n=1,2,3, ••• Show that this construction n+1 n n

is possible if and only if p is not a Fermat primeo

Let the function cp satisfy q{X) = 0 cp(x)

=

x q(x) - q>(x_- 1)

=

xq>'(x) (X < 0) t (0:s;x~1), (x >0) #J

Show that q> ex) tends to a positive limit if x .... 0 0 .

Show that the limit mentioned in the previous problem equals e-1, where y' is Euler's constant.

178]-Let P be a polynomial of degree n. ShoVi that for all

x

( _{n - . e - k '} n + 1 )

where the summation extends over all

.e,

k with

.e?J;

0, k?J; 0 t

.e+

k ~~.

Fo"r each rational number r in the interval 0 < r < 1 we form

( ) -2 -2 /

f r

=

a b , i f a b is the irreducible fraction representing r.

Show that the sum of all these f(r)'s equals

t.

Let a(1), a(2), ••• be a sequence of complex numbers, and let A(n) =" n -1 t dl n da(d) (d runs through the divisors of n). Assume

00 00

that I: n=1 a(n) converges with sum 0 and I: n=1 a(n) log n converges

00

with sum s. Show that I: 1A(n) converges with sum -so

n=

~2

(1) 1965,

Nr. 10J

Let ill be an integer ?J;2 and let K be a finite field. Consider

the m-~imensional projective geometry over K. Show that there exists a projective transformation T with the property that both the set of

all points and the set of all (m-1)-dimensional hyperplanes are per-

.

muted cyclically by T.

[22 (1)

1965,

Nr. 11]

Let ~'~tX be bounded continuous functions of x,y,z (-00<:;':< 00, _oo<y<oo '. _00< z< 00). Show that the system

2 2 2 x(x +y +z )

=

( 2 2 2) Y X +y +z

=

_{.,2~_ •• 2 L _2\ U \ A TJ T ... I = has a solution. ~(x,y,z) ~(x,y,z)

Show that the equation

[22 (1) 1965, lir. 12] Iz 1 5 - 2z 5 + Z 4 - Re z

= "

has at least one solution, for every complex number ".

Let ::;.' be the nxn matrix with first roV! 0100 •••

.o,

second ro\'!

0010 ••• 0, ••• ~ (n-1)-th row 00 ••• 01, nnd n-th row

,

...

Determine the products Tk (k=O, .::!::. 1,

z.

2, ••• ).

..

[22 ( 1) 1 965, Nr. 14]

Find the Jordan canonical normal form of the matrix T of the

previous problem, and construct a matrix V that transforms T into

th~t normal form (by means of T~V-1TV).

Consider the complete graph of order n, i.e. the graph with vertices P1 , ••• ,P_{n connected in a11 possib1e ways} (BO there are

in(n-1) edges). A subset T of the set of all edges is ca11ed a

complete tree if it forms a tree with vertices P , ••• ,P • According

" ! n .

. n-2 (

to Cayley 'there are n complete trees. Show that for each k 1E; k :!Sin=1)

n-2 ( )n-k-1

there are (k-1) n-1 complete trees with the property that the

number of edges meeting at P₁ equals k.

[22 (1) 1965. Nr. 16)

Let p and q be real numbers. each chosen at random from the interval (O,N). We define the sequence a , a

l , ••• by a

=

1. · 0 0 a_n+

1

=

"exp(p exp (qan

».

Let PN be the probabi1ity that the sequence {an} converges. Show that P N

=

2K2 (2) N-2 +

O(

e -NN-3) as N ....

ClO.

where K stands for the modified Bessel function of the second kind.

[22 (1) 1965. Nr. 17)

Let the real function

+

be positive and increasing for 0 <h <1, and lim

4

(h)

=

0 if h '0. Construct a real function

t • .

continuous on

(_00, go), such that there does not exist an x with the property that

(f(x + h) "-" f(X»(cjI(h»-1

is bounded on O<h <1.

Let the real function f be increasing and continuously differentiable

on the interval [0,00). Assume that f - log f' is a concave function in

that interval. Show that f itself is concave.

Let f and g be non-negative measurable functions on measure

spaces X and

Y,

respectively, and assume that IXf(x)dX

=

lyg(Y)dYe

Let C

f

,e

denote the following condition: for each measurable set

E cy we have

ix

min(f,I1(E»dx 'if>

IE

g(y)dy.

C is obtained from C

f by interchanging the roles of X and Y

g,f ,g

and of f and g. Show that C

f ,g and C g,f are equivalent.

[22 (1) 196.5, 20]

constant as t - ~

..

Show the following asymptotic equivalence for x .. w:

00 2n2/( 2), X2(2)-t -1

~oo

-2 (k_J:)2

Find posi ti ve ·constants c , c1, C2 t • • • such that

o

tends to 1 if x > 0, x

_00.

From 1965 onward the "Wiskundige opgaven met de oplosBingen" ceased to exist as a separate journal.

Both the following problems and their solutions were published in the "Nieuw Archief voor Wiskunde". The references at the end of the problem

, .

indicate problemnlmber, volume, year and page of publication of the problem, and possibly, volume, year and page of publication of the solution.

A function cp is oalled. "tamell _{if it is defined}

and posi ti ve for x ~ 0 and if

Let f be positive for x ~ 0 and integrable over every finite interval 0 < x<A.

Put L (x) - x -1

J

~

f( t )dt. Assume that lim _{x_} L(Ax)JL(X) a 1

oo

for tNery A > 0, and that either f or 1

If

is tame. Show that

f(x) ~ L(x) (x .. co ).

[Nr. 45,

12

(1965), p. 47.

1.2.

(1965)

,p.

248].

As usual in number theory (m,n) denotes the g.o.d. of m and n, ~ denotes

the Mt5bius funotion, &i . the Kronecker delta. I f a, b, 0 are positive integers ,

,J put

S(a,b,c) -

~d/a,(bd,c).a ~(d)

(the summation extends over all divisors d of a whioh satisfy (bd,o) - d).

Show that

[Nr. 46,

12

(1965), p. 47,

Let the real function K(x,y) be given for 0 .; x .; 1, 0 .; Y .;; 1, and

2

assume that K, aK/ax, a K/a y, a K/axay are continuous. Moreover assume that

K is a non-negative kernel, i.e. K(x,y) - K(y,x), and

J

1

J1

K(x,y)f(x)f(Y) dxdy ;?:;

°

o 0

for all real continuous functions f. ShO'l that

K( a 2K/axay) - (aK/ax) (aK/ay) is also non-negative definite.

[Nr.

47,

12

(1965),

p.

47;

14, (1966),

p~

138-139

~

Let ,(x) be a twice differentiable real function satisfying ,(x) > 0,

,"(x) ;?:;

°

(_00 <x < 00). Let Y1 and Y2 be real solutions of the second order

differential equation

Show that if a < b, y~(a) - y~(b) - 0, then Y2 has at least one zero in the

interval a =s;; x E; b.

Show that, if t - +00.

has the asymptotic expansion

[Nr.

48,

12

(1965),

p.

47-48,

12

(1965),

p.

250].

[Nr.

49,

12,

(1965),

p.

48,

Assume a₁ - 1,0< ~< 1 (k - 2,3, ••• ), 1:~ a_k< 00. Put S1(n) _

- 1: _v~n a_v (n - 1,2, ••• ), and define S2(n), s3(n), ••• by induction:

sk+1(n)

=

1:v~n _{av Sk(V).}

Determine li~_ 00 Sk(1).

[Nr. 64,

12

(1965), p. 120,

~ (1966), p. 66 - 67].

I f b is a fixed positive number, show that

(n _ 00 ) .

[Nr. 65,12 (1965), p. 121J

14

(1966), p. 61].

Let C be the space of all continuous real functions on the interval

(-~ ~. If f € C we define its norm

Ilf II

by

Let B be the set of all f € C with finite norm. Show that B is a Banach space • •

[Nr. 66,

12

(1965) p. 121,

~ (1966), p. 61-68]'

The space S consists of all real functions f defined on _00 <x< co

with the property that f has at most finitely many discontinuities, and

that f is constant in every interval where it is continuous. In S we take as a norm

IIfll

=

max _..DO<x<oo If(x)l •

Determine all bounded linear functionals on S.

[Nr. 81,

12

(1965), p. 233, j j (1966), p. 141-149].

Let m and n be relatively prime positive integers. Let, for each i (i = O, ••• ,m-1 ), A. bea cyclicnxn matrix. Show that the partitionl;'d

~ matrix ••• A _m-1 A Ai 0 A m-1 ••• A 2

m-•

• • • • •• ••• • •••• •••• ••• ••• • •••• •••• ••• ••• • •••• A o

1

can be written in the form pcp-1, where C is an mn x mn cyclic matrix

and P is a permutation matrix.

[Nr. 82,

.1.2

(1965), p. 233,

1.4

(1966), P. 149-150]. ~_1 -n k/ ( \ IfN~1weputSk=~=oe n k! k=1,2""J.Showthat [Nr. 83,

12

(1965), p. 233;

.:!..4

(1966),150-151

J.

Derive the following with the aid of the result of the previous

-~-problem: If a ,a_{o 1}, ••• is a sequence of real numbers with limN _{_co} (N_1)EN-₀ 1 a _n

=

A

co -n ko fk. ( ) ~-1

and if b

n ..

lk=o

e n lc' _1, then we have li~_ ClO N-1 0 b_n

=

A.

[Nr. ~,

.1i

(1965), p. 234~

j j (1966), p. 151-152].

If x runs through the real. numbers, x ~ 1, show that

=1 ( -1) 2/ ( -1)

r:

n log 1 + nx

=

1t 12 + 0 x •

1E;;n<x

[Nr. 98, j j (1966), p. 49,

.:!.4

(1966), p. 281

J.

I f cp denotes Euler's indicator, show that

~ k-~

(k) log(1 + kiN) =

~

+ 0(N-1 log N)

~.1

[Nr. 119,

.1.4

(1966), p. 135;·

12.

(1967), p. 82-83].

Assume a.

>

0, 0 < A < 1. If 0 < b < a. we define Nb(t) on the i"''lterval O<t <b by

Let f be an integrable function on the interval 0 < x < a, and assume that the left-hand limit f(a - 0) exists. Show that

[Nr. 120,

.1A

(1966), p. 135,

.12.

(1967), p. 83-84].

Let p be a positive constant and let a

1,a2, ••. be a sequenoe of real

-1 ~

-numbers such that 1im.~ _~-~ N _{_} 1 a. _{K ·} exists. For each n ~ 1 we define b _n

.1. .1.

as the average of a

k, taken over all k for which n - pn

2 _{~k ~n + pn}2 •

Show that 1imN ... oo

N-1

_E~=1

b_k exists.

[Nr. 155,

.12

(1967), p. 153,

1£

(1968), p. 38J.

Assume c > 0, 0 < A < 1. Let the real function f be defined on the interval 0 < x < c. Put rex) ... X-A g(x), assume that g(x) has bounded variation on 0 ~ x < c and that g(O+)=O. Define F by F(O) = 0 and

x

F(x):::

J

£(t) (x_t)A_1 dt (0 < x

~

c).

o

Show that on 0 ~ x ~ c the function F is continuous and of bounded variation.

[Nr. 165,

.12

(1967), p. 250,

(Continuation of no. 165). Show (with the aid of no. 165 that for o < x E; c we have for the left-hand limit

f(x-O) = n-1 sin nil.

J

x (x-y)dF (y), ~

o

where the integral is an improper Stieltjes integral (to be taken in the sense of

li~

tx

J

b).

o

[Nr. -166,

.12

(1961), p.25C;

.1.§ (1968), p. 130 ~

-31-If in no. 165 the assumption is added that g is continuous on 0 < x'" c axu- continuously differentiable on 0 < x~ ct then show that F is continuously

dif.'.'erentiable (0 < x ~ 0), that F(O) = 0 and that

( ) - 1 .

f x = n s~n ni\.. (0 < Yo ... c).

[Nr. 167,

.12.

(1967 ), p. 251; .1§. (19 68 ), p. 1;OJ.

Let g(x) be absolutely continuous (0 ~ x ~ c), and g(O)

=

O. Put x-i\.g(x) =f(x). Show that Abel's integral equation

J

x I..

e (x-y)- <P (y)dy

=

f(x)

has a solution <p which is integrable on [O,cJ, and even such that for every x( 0 E; x ~ c) the function (x-y) -A <p (y) is an integrable function

of y on 0 E; Y E; x.

[Nr. 168, .1.§.(1968), p. 124;

Let B be the number of equivalence relationships that can be defined _{n _} on a set of n points. (So B1

=

1, B2

=

2, B3 - 5, B4

=

15, B5 m 52, B6== 203,

B7 "" 877, • u). Show that Bn is even if and only if n:, 2 (mOd 3).

[Nr. 179₁ with S.D. Chatterji,

/-:

o

o

o

o

-I

o

-I

o

o

o

o -}

0

o

o -}

0

-} -I -I

o

(the matrix formed by the first n-I rows and n-I columns ~s cyclic).

[Nr. 188, 16 (1968), p.12S; ll(1969)p. 75-77,D].

A conic K passes through the points Pl,···,P4' QI'" .,Q4' The pencil of conics through p

1"",P4 contains a conic touching both Q

IQ2 and Q3Q4' Show that this pencil also contains a conic touching both Q₁Q₃ and Q2Q4' as well as a conic touching both Q_IQ₄ and Q2Q3' The six po~nts of contact lie on a single straight line.

[Nr. 189, 16 (1968), p.125~_ 17 (1969) p.77-78,D].

Let n be a positive integer, and let f E L2 (R

n). Let g be its Fourier transform, defined by

where

T T

Show that

r' .

~.

(" If (x I ' . . . ,x ) I 2 dx I . . . dx .

)-00 "-00 n n

When does the equality sign hold?

[Nr.22l, 22.(1969) p.151-152; 18 (1970) p. 298-299~

Prove the following n-dimensional extension of Weyl's uncertainty relation (using the notation of problem No.221):

with -1 Do(f)Do(g) ~ n(4;r)

r

oo

fOO

2 21 12 1 ( ... (x l+ ... +x )f(xl , ••• ,x) dxl ••• dXn

)2

"-00 -00 n n Do(f) := 2

f:oo ..•

rOO

_{I f(x l ,··· ,xn)}

I

_{dx l ·· .dxn}

and a similar definition for Do(g). When does the equality s~gn hold?

[Nr.222,

.!2.

(1969) p. 152; 18 (1970), p. 301J

-3>-Let S

n be the closed unit sphere in n-space (n > I). Assume Snc Clu

u ••• U C

k, where CI, ••• ,Ck are cylinders C. are obtained from

~

measurable cylinders. These measurable any measurable set B. in any

«n-2)-dimen-~

sional) hyperplane, if we take all points x + tb., where b. is a unit

1.

reals. By "cross section" of that cylinder we denote the (n-I)-dimen-sional volume of B ..

1.

Show that the sum of the cross sections of CI"",C

k is not less than the (n-l)-dimensional volume of S l'

n-[Nr. 232, 12(I969) p.225;

19(1971), p. 79-80J.

Let X and. Ybetopo16gTcal spaces, and let X-be compact. Let Ube-a subset of the cUbe-artesiUbe-an product X x Y. We put

vex) := {y E Y

I

(x,y) E U}

W(y) := {x E X

I

(x,y) E U}

(x E X) ,

(y E Y) •

Assume that Vex) 1.S an upper semi-continuous function of x. (This means

that for every x E X and every open set Q with Vex ) c Q c Y there exists

o 0

an open set P with x E P c X such that Vex) c Q for all x E P.) Moreover o

assume that for every x E X and for every y E Y either y E Vex) or there

exist disjoint open sets Ql,Q2 with Vex) c Q1' Y E Q2'

Show that W(y) is an upper semi-continuous function of y, and that W(y) is closed for every y E Y. (The special case that Y is Hausdorff,

and that U is the graph of a continuous injection of X into Y, is a well-known theorem).

[Nr.263, ~(1970) p. 295;

11(1971) p. I67-168J

From an urn containing k black balls and k red balls, all 2k balls are drawn out, one at a time, without replacement. Each time, before drawing a ball, one guesses the colour of the ball to be drawn. We want to optimize the number of correct guesses. The best strategy is: always take as a guess the colour which has the majority in the urn, guessing at random if there are as many black balls as red balls. Determine the expectation of the number of correct guesses under this strategy.

[Nr. 273, ~(1971) p. 78; 19(1971) p. 229-232, DJ

Let p and q t.e integers, both'" 0, and assume p and q to be relatively prime. L~t C· denote the curve in 3-space, given by

p,q

x = (2 - cos pt) cos qt

!

y = (2 - cos pt) sin qt (0 :5 t :5 2'IT). z

=

sin pt

Show that Cp,q and Cq,p belong to the same isotopy type. (Note that C₃,2 and C

3,-2 are clover leaf knots).

[Nr.330, 20 (1972) p.252; ~ (1974) p.84-85].

Let I be an open interval on the real line, and let f be a real function on I. Show that the following two conditions are equivalent:

(i) For all U,V,WE I we have

where I ~ [ :5 [(u-v) (v-w) (w-u)

I,

u,v,w feu) ~ := u u,v,w f(v) v few) w 1

(ii) f is differentiable on I, and for all u,v E I we have

If' (u) - ff (v) I :5 2Iu-vl.

[Nr. 337, 21 (1973) p.97; 21 (1973) p.228-290J.

Players P aiid-Q playa game,·of.which the rules are determined by positive integers k,£,m. There is a countable set of-markers, labelled!l,2,3,; ••• P and Q move al ternatE!ly; P 'cmoves ' first. Each move0f P "consists 0f taking k markers I and

each move of;Q consists of taking £.markers. P has won as soon'as his set of markers contains a sequence of m consecutive integers. Determine all cases

(k;£,m) where P has-a winning strategy.

Let u be a con~il).·;;\oU? function on [0,00). Put p(x) :=

IX

u(t)dt o -1 We assume that x with p(x)

=

O(xA), J~ u(t)dt -+ .l(x) .

-+ 00), and that A is a real number, A < 1,

u(x)

=

O(xA- 1),

ex

-+ 00).

[Nr. 4;. 24 (1976) p. 78; 24 (1976) p. 279-280J.

Let m

1 fm2' ••• be sequences of p.lements of:IN (= {I, 2,3, ..• }) •

Show the aI' a

2 ' ••• E:IN such that.~ ",,'ery k E:N there ar e exac t 1 Y

~ values of iE:N with a

i = k and exactly ~ values d j E:IN with laj+1 - ajl = k. (The special ca that n

1

=

n2

= ... =

1 is due to P.~ Slater and W.Y. Velez, Pacific J .Math. 71 (1977) 193-196).'

..,... -.

Let u be a con~inuous function on [O,~). Put

p(x) :=

IX,

u(t)dt - xu(x).

, o·

We assume that x-I J~ u(t)dt -+ 0 (x -+ co), and that A is a real number, A < 1,

A A~

with p(x)

=

O(x), (x -+ 00). Show that u(x)

=

O(x }, (x -+ 00).

[Nr. 427, 24 (1976) p. 78; 24 (1976) p •. 279-280J.

Let n

1,n2, ..• and m11m2 , •.. be sequences of elements ofJN (= {1,2,3, ••• }). Show the existence of a

1 ,a2, ••. EJN such that for every k EJN there are exactly

~ values of i EJN with a

i

=

k and exactly ~ values of j EJN with

I

aj+1 .;: aj

I

=

k. (The special case that n

1

=

n2 = ...

=

1 is due to P.J.Slater and W.Y. Velez, Pacific J .Math. 71 (1977) 193-196).·