MathDB

1

Miklós Schweitzer 1953- Problem 3

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)

6

1

Miklós Schweitzer 1953- Problem 6

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)

7

1

Miklós Schweitzer 1953- Problem 7

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)

8

1

Miklós Schweitzer 1953- Problem 8

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)

9

1

Miklós Schweitzer 1953- Problem 9

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)

10

1

Miklós Schweitzer 1953- Problem 10

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)

1

Miklós Schweitzer 1953- Problem 1

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)

2

1

Miklós Schweitzer 1953- Problem 2

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)

4

1

Miklós Schweitzer 1953- Problem 4

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)

5

1

Miklós Schweitzer 1953- Problem 5

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)

1953 Miklós Schweitzer

Subcontests

Miklós Schweitzer 1953- Problem 3

Miklós Schweitzer 1953- Problem 6

Miklós Schweitzer 1953- Problem 7

Miklós Schweitzer 1953- Problem 8

Miklós Schweitzer 1953- Problem 9

Miklós Schweitzer 1953- Problem 10

Miklós Schweitzer 1953- Problem 1

Miklós Schweitzer 1953- Problem 2

Miklós Schweitzer 1953- Problem 4

Miklós Schweitzer 1953- Problem 5

1953 Mikl&oacute;s Schweitzer

Subcontests

Mikl&oacute;s Schweitzer 1953- Problem 3

Mikl&oacute;s Schweitzer 1953- Problem 6

Mikl&oacute;s Schweitzer 1953- Problem 7

Mikl&oacute;s Schweitzer 1953- Problem 8

Mikl&oacute;s Schweitzer 1953- Problem 9

Mikl&oacute;s Schweitzer 1953- Problem 10

Mikl&oacute;s Schweitzer 1953- Problem 1

Mikl&oacute;s Schweitzer 1953- Problem 2

Mikl&oacute;s Schweitzer 1953- Problem 4

Mikl&oacute;s Schweitzer 1953- Problem 5

1953 Miklós Schweitzer

Miklós Schweitzer 1953- Problem 3

Miklós Schweitzer 1953- Problem 6

Miklós Schweitzer 1953- Problem 7

Miklós Schweitzer 1953- Problem 8

Miklós Schweitzer 1953- Problem 9

Miklós Schweitzer 1953- Problem 10

Miklós Schweitzer 1953- Problem 1

Miklós Schweitzer 1953- Problem 2

Miklós Schweitzer 1953- Problem 4

Miklós Schweitzer 1953- Problem 5