MathDB

Miklos Schweitzer 1952_8

Source:

10/12/2008

For which values of

z

does the series \sum_{n\equal{}1}^{\infty}c_1c_2\cdots c_n z^n converge, provided that

c_k>0

and \sum_{k\equal{}1}^{\infty} \frac{c_k}{k}<\infty ?

real analysisreal analysis unsolved

Miklos Schweitzer 1952_7

Source:

10/12/2008

A point

P

is performing a random walk on the

X

-axis. At the instant t\equal{}0,

P

is at a point

x_0

(

|x_0|\le N

, where

x_0

and

N

denote integers,

N>0

). If at an instant

t

(

t

being a nonnegative integer),

P

is at a point of

x

integer abscissa and

|x|<N

, then by the instant t\plus{}1 it reaches either the point x\plus{}1 or the point x\minus{}1, each with probability

\frac12

. If at the instant

t

,

P

is at the point x\equal{}N [ x\equal{}\minus{}N], then by the instant t\plus{}1 it is certain to reach the point N\minus{}1 [ \minus{}N\plus{}1]. Denote by

P_k(t)

the probability of

P

being at x\equal{}k at instant

t

(

k

is an integer). Find

\lim_{t\to \infty}P_{k}(2t)

and \lim_{t\to \infty}P_k(2t\plus{}1) for every fixed

k

.

probabilitylimitprobability and stats

Miklos Schweitzer 1952_9

Source:

10/12/2008

Let

C

denote the set of functions

f(x)

, integrable (according to either Riemann or Lebesgue) on

(a,b)

, with

0\le f(x)\le1

. An element

\phi(x)\in C

is said to be an "extreme point" of

C

if it can not be represented as the arithmetical mean of two different elements of

C

. Find the extreme points of

C

and the functions

f(x)\in C

which can be obtained as "weak limits" of extreme points

\phi_n(x)

of

C

. (The latter means that \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx holds for every integrable function

h(x)

.)

functionreal analysislimitintegrationreal analysis unsolved

Miklos Schweitzer 1952_10

Source:

10/12/2008

Let

n

be a positive integer. Prove that, for 0

n

is odd and negative if

n

is even.

trigonometryfunctionintegrationreal analysisreal analysis unsolved

Miklós Schweitzer 1956- Problem 4

Source:

10/9/2015

4. Denoting by

a(n)

the greatest prime factor of the positive integer

n

, show that

S= \sum_{n=1}^{\infty } \frac{1}{na(n)}

is convergente. (N. 13)

college contests

Miklós Schweitzer 1956- Problem 1

Source:

10/9/2015

1. Solve without use of determinants the following system of linear equations:

\sum_{j=0}{k} \binom{k+\alpha}{j} x_{k-j} =b_k

(

k= 0,1, \dots , n

),
where

\alpha

is a fixed real number. (A. 7)

college contests

Miklós Schweitzer 1956- Problem 2

Source:

10/9/2015

2. Find the minimum of

max ( |1+z|, |1+z^{2}|)

if

z

runs over all complex numbers. (F. 2)

complex numberscollege contests

Miklós Schweitzer 1956- Problem 3

Source:

10/9/2015

3. A triangulation of a convex closed polygon is the division into triangles of this poilygon by diagonals not intersecting in the interior of the polygon. Find the number of all triangulations fo a conves n-gon and also the number of those triangulations in which every triangle has at least one side in common with the given n-gon. (C. 4)

college contests

Miklós Schweitzer 1956- Problem 7

Source:

10/11/2015

7. Let

(a_n)_{n=0}^{\infty}

be a sequence of real numbers such that, with some positive number

C

,

\sum_{k=1}^{n}k\mid a_k \mid<n C

(

n=1,2, \dots

)
Putting

s_n= a_0 +a_1+\dots+a_n

, suppose that

\lim_{n \to \infty }(\frac{s_{0}+s_{1}+\dots+s_n}{n+1})= s

exists. Prove that

\lim_{n \to \infty }(\frac{s_{0}^2+s_{1}^2+\dots+s_n^2}{n+1})= s^2

(S. 7)

college contests

Miklós Schweitzer 1956- Problem 5

Source:

10/9/2015

5. On a circle consider

n

points among which there acts a repulsive force inversely proportional to the square of their distance. Prove that the point system is in stable equilibrium if and only if the points form a regular

n

-gon; in other words, considering the sum of the reciprocal distances of the

\binom{n}{2}

pairs of points which can be chosen from among the

n

given points, this sum is minimal if and only if the points lie at the vertices of a regular

n

-gon. (G. 2)

college contests

Mikl&oacute;s Schweitzer 1956- Problem 9

Source:

10/11/2015

9. Show that if the trigonometric polynomial

f(\theta)= \sum_{v=1}^{n} a_v \cos v\theta

monotonically decreases over the closed interval

[0,\pi]

, then the trigonometric polynomial

g(\theta)=\sum_{v=1}^{n}a_v \sin v\theta

is non negative in the same interval. (S. 26)

college contests

Miklós Schweitzer 1956- Problem 8

Source:

10/11/2015

8. Let

(a_n)_{n=1}^{\infty}

be a sequence of positive numbers and suppose that

\sum_{n=1}^{\infty} a_n^2

is divergent. Let further

0<\epsilon<\frac{1}{2}

. Show that there exists a sequence

(b_n)_{n=1}^{\infty}

of positive numbers such that

\sum_{n=1}^{\infty}b_n^2

is convergent and

\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}

for every positive integer

N

. (S. 8)

college contestsFunctional Analysisreal analysis

Miklós Schweitzer 1956- Problem 6

Source:

10/11/2015

6. Show that the number of the faces of a convex polyhedron is even if every face is centrally simmetric. (G. 12)

college contests

Miklós Schweitzer 1953- Problem 5

Source: Miklós Schweitzer 1953- Problem 5

8/1/2015

Show that any positive integer has at least as many positive divisors of the form

3k+1

as of the form

3k-1

. (N. 7)

number theory

Miklós Schweitzer 1956- Problem 10

Source:

10/11/2015

10. In an urn there are balls of

N

different colours,

n

balls of each colour. Balls are drawn and not replaced until one of the colours turns up twice; denote by

V_{N,n}

the number of the balls drawn and by

M_{N,n}

the expectation of the random variable

v_{N,n}

. Find the limit distribution of the random variable

\frac{V_{N,n}}{M_{N,n}}

if

N \to \infty

and

n

is a fixed number. (P. 8)

college contests

Miklós Schweitzer 1953- Problem 2

Source: Miklós Schweitzer 1953- Problem 2

8/1/2015

2. Place 32 white and 32 black chessmen on the chessboard. Two chessmen of different colours will be said to form a "related pair" if they are placed either in the same row or in the same column. Determine the maximum and minimum number of related pairs (over all possible arrangements of the 64 chessmen considered. (C. 2)

combinatorics

Miklós Schweitzer 1953- Problem 1

Source: Miklós Schweitzer 1953- Problem 1

8/1/2015

1. Let

a_{v}

and

b_{v}

,

{v= 1,2,\dots,n}

, be real numbers such that

a_{1}\geq a_{2} \geq a_{3}\geq\dots\geq a_{n}> 0

and

b_{1}\geq a_{1}, b_{1}b_{2}\geq a_{1}a_{2},\dots,b_{1}b_{2}\dots b_{n}\geq a_{1}a_{2}\dots a_{n}

Show that

b_{1}+b_{2}+\dots+b_{n}\geq a_{1}+a_{2}+\dots+a_{n}

(S. 4)

Sequences

Miklós Schweitzer 1953- Problem 3

Source: Miklós Schweitzer 1953- Problem 3

8/3/2015

3. Denoting by

E

the class of trigonometric polynomials of the form

f(x)=c_{0}+c_{1}cos(x)+\dots +c_{n} cos(nx)

, where

c_{0} \geq c_{1} \geq \dots \geq c_{n}>0

, prove that

(1-\frac{2}{\pi})\frac{1}{n+1}\leq min_{{f\epsilon E}}( \frac{max_{\frac{\pi}{2}\leq x\leq \pi} \left | f(x) \right |}{max_{0\leq x\leq 2\pi} \left | f(x) \right |})\leq (\frac{1}{2}+\frac{1}{\sqrt{2}})\frac{1}{n+1}

.
(S. 24)

Sequencespolynomialcollege contests

Miklós Schweitzer 1953- Problem 4

Source: Miklós Schweitzer 1953- Problem 4

8/1/2015

4. Show that every closed curve c of length less than

2\pi

on the surface of the unit sphere lies entirely on the surface of some hemisphere of the unit sphere. (G. 8)

geometrycollege contestsreal analysisMiklos Schweitzer

Miklós Schweitzer 1953- Problem 6

Source: Miklós Schweitzer 1953- Problem 6

8/3/2015

6. Let

H_{n}(x)

be the nth Hermite polynomial. Find

\lim_{n \to \infty } (\frac{y}{2n})^{n} H_{n}(\frac{n}{y})

For an arbitrary real y. (S.5)

H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{{-x^2}}\right)

Sequenceslimitcollege contests

Miklós Schweitzer 1953- Problem 7

Source: Miklós Schweitzer 1953- Problem 7

8/2/2015

7. Consider four real numbers

t_{1},t_{2},t_{3},t_{4}

such that each is less than the sum of the others. Show that there exists a tetrahedron whose faces have areas

t_{1},t_{2}, t_{3}

and

t_{4},

respectively. (G. 9)

3D geometrycollege contests

Miklós Schweitzer 1953- Problem 8

Source: Miklós Schweitzer 1953- Problem 8

8/2/2015

8. Does there exist a Euclidean ring which is properly contained in the field

V

of real numbers, and whose quotient field is

V

? (A.21)

abstract algebracollege contests

Miklós Schweitzer 1953- Problem 10

Source: Miklós Schweitzer 1953- Problem 10

8/2/2015

10. Consider a point performing a random walk on a planar triangular lattice and suppose that it moves away with equal probability from any lattice point along any one of the six lattice lines issuing from it. Prove that if the walk is continued indefinitely, then the point will return to its starting point with probability 1. (P. 5)

probabilitycollege contests

Miklós Schweitzer 1954- Problem 3

Source:

9/29/2015

3. Is there a real-valued function

Af

, defined on the space of the functions, continuous on

[0,1]

, such that

f(x)\leq g(x)

and

f(x)\not\equiv g(x)

inply

Af< Ag

? Is this also true if the functions

f(x)

are required to be monotonically increasing (rather than continuous) on

[0,1]

? (R.4)

functionreal analysiscollege contests

Miklós Schweitzer 1953- Problem 9

Source: Miklós Schweitzer 1953- Problem 9

8/2/2015

9. Let

w=f(x)

be regular in

\left | z \right |\leq 1

. For

0\leq r \leq 1

, denote by c, the image by

f(z)

of the circle

\left | z \right | = r

. Show that if the maximal length of the chords of

c_{1}

is

1

, then for every

r

such that

0\leq r \leq 1

, the maximal length of the chords of c, is not greater than

r

. (F. 1)

functioncollege contestscomplex analysis