MathDB

1

Miklós Schweitzer 1955- Problem 10

10. Show that if a convex polyhedron has vertices of regular distribution and congruent faces, then it is regular. (A system of points is said to be of regular distribution if every point of the system can be transformed into any other point by congruent transformations mapping the system onto itself.) (G. 11)

9

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Miklós Schweitzer 1955- Problem 9

9. Show that to any elliptic paraboloid

\varphi_1

there may be found an elliptic paraboloid

\varphi_2

(other than

\varphi_1

) and an affinity

\phi

which maps

\varphi_1

onto

\varphi_2

and has the following property: If

P

is any point of

\varphi_1

such that

\phi(P) \neq P

, then the straight line connecting

P

and

\phi(P)

is a common tangent of the two paraboloids. (G. 18)

8

1

Miklós Schweitzer 1955- Problem 8

8. Show that on any tetrahedron there can be found three acute bihedral angles such that the faces including these angles count among them all faces of tetrahedron. (G. 10)

7

1

Miklós Schweitzer 1955- Problem 7

7. Prove that for any odd prime number

p

, the polynomial

2(1+x^{ \frac{p+1}{2} }+(1-x)^{\frac {p+1}{2}})

is congruent mod

p

to the square of a polynomial with integer coefficients. (N. 21)
*This problem was proposed by P. Erdõs in the American Mathematical Monthly 53 (1946), p. 594

6

1

Miklós Schweitzer 1955- Problem 6

6. For a prime factorisation of a positive integer

N

let us call the exponent of a prime

p

the integer

k

for which

p^{k} \mid N

but

p^{k+1} \nmid N

; let, further, the power

p^{k}

be called the "contribution" of

p

to

N

. Show that for any positive integer

n

and for any primes

p

and

q

the contibution of

p

to

n!

is greater than the contribution of

q

if and only if the exponent of

p

is greater than that of

q

.

5

1

Miklós Schweitzer 1955- Problem 5

5. Show that a ring

R

is commutative if for every

x \in R

the element

x^{2}-x

belongs to the centre of

R

. (A. 18)

4

1

Miklós Schweitzer 1955- Problem 4

4. Find all positive integers

\alpha , \beta (\alpha >1)

and all prime numbers

p, q, r

which satisfy the equation

p^{\alpha}= q^{\beta}+r^{\alpha}

(

\alpha , \beta , p, q, r

need not necessarily be different). (N. 12)

3

1

Miklós Schweitzer 1955- Problem 3

3. Let the density function

f(x)

of the random variable

\xi

bean even function; let further

f(x)

be monotonically non-increasing for

x>0

. Suppose that

D^{2}= \int_{-\infty }^{\infty }x^{2}f(x) dx

exists. Prove that for every non negative

\lambda

P(\left |\xi \right |\geq \lambda D)\leq \frac{1}{1+\lambda ^{2}}

. (P. 7)

2

1

Miklós Schweitzer 1955- Problem 2

2. Let

f_{1}(x), \dots , f_{n}(x)

be Lebesgue integrable functions on

[0,1]

, with

\int_{0}^{1}f_{1}(x) dx= 0

(i=1,\dots ,n)

. Show that, for every

\alpha \in (0,1)

, there existis a subset

E

of

[0,1]

with measure

\alpha

, such that

\int_{E}f_{i}(x)dx=0

. (R. 17)

1

Miklós Schweitzer 1955- Problem 1

1. Let

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

and

b_{1}, b_{2}, \dots , b_{m}

be

n+m

unit vectors in the

r

-dimensional Euclidean space

E_{r} (n,m \leq r)

; let

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

as well as

b_{1}, b_{2}, \dots , b_{m}

be mutually orthogonal. For any vector

x \in E_{r}

, consider

Tx= \sum_{i=1}^{n}\sum_{k=1}^{m}(x,a_{i})(a_{i},b_{k})b_{k}

(

(a,b)

denotes the scalar product of

a

and

b

). Show that the sequence

(T^{k}x)^{\infty}_{ k =0}

, where

T^{0} x= x

and

T^{k} x = T(T^{k-1}x)

, is convergent and give a geometrical characterization of how the limit depends on

x

. (S. 14)

1955 Miklós Schweitzer

Subcontests

Miklós Schweitzer 1955- Problem 10

Miklós Schweitzer 1955- Problem 9

Miklós Schweitzer 1955- Problem 8

Miklós Schweitzer 1955- Problem 7

Miklós Schweitzer 1955- Problem 6

Miklós Schweitzer 1955- Problem 5

Miklós Schweitzer 1955- Problem 4

Miklós Schweitzer 1955- Problem 3

Miklós Schweitzer 1955- Problem 2

Miklós Schweitzer 1955- Problem 1

1955 Mikl&oacute;s Schweitzer

Subcontests

Mikl&oacute;s Schweitzer 1955- Problem 10

Mikl&oacute;s Schweitzer 1955- Problem 9

Mikl&oacute;s Schweitzer 1955- Problem 8

Mikl&oacute;s Schweitzer 1955- Problem 7

Mikl&oacute;s Schweitzer 1955- Problem 6

Mikl&oacute;s Schweitzer 1955- Problem 5

Mikl&oacute;s Schweitzer 1955- Problem 4

Mikl&oacute;s Schweitzer 1955- Problem 3

Mikl&oacute;s Schweitzer 1955- Problem 2

Mikl&oacute;s Schweitzer 1955- Problem 1

1955 Miklós Schweitzer

Miklós Schweitzer 1955- Problem 10

Miklós Schweitzer 1955- Problem 9

Miklós Schweitzer 1955- Problem 8

Miklós Schweitzer 1955- Problem 7

Miklós Schweitzer 1955- Problem 6

Miklós Schweitzer 1955- Problem 5

Miklós Schweitzer 1955- Problem 4

Miklós Schweitzer 1955- Problem 3

Miklós Schweitzer 1955- Problem 2

Miklós Schweitzer 1955- Problem 1