MathDB

Miklós Schweitzer 1954- Problem 1

Source: Miklós Schweitzer 1954- Problem 1

8/3/2015

1. Given a positive integer

r>1

, prove that there exists an infinite number of infinite geometrical series, with positive terms, having the sum 1 and satisfying the following condition: for any positive real numbers

S_{1},S_{2},\dots,S_{r}

such that

S_{1}+S_{2}+\dots+S_{r}=1

, any of these infinite geometrical series can be divided into

r

infinite series(not necessarily geometrical) having the sums

S_{1},S_{2},\dots,S_{r}

, respectively. (S. 6)

Sequencescollege contestsreal analysis

Miklós Schweitzer 1954- Problem 2

Source: Miklós Schweitzer 1954- Problem 2

8/3/2015

2. Show that the series

\sum_{n=1}^{\infty}\frac{1}{n}sin(asin(\frac{2n\pi}{N}))e^{bcos(\frac{2n\pi}{N})}

is convergent for every positive integer N and any real numbers a and b. (S. 25)

seriesSequencescollege contests

Miklós Schweitzer 1954- Problem 7

Source:

9/29/2015

7. Find the finite groups having only one proper maximal subgroup. (A.12)

group theoryabstract algebracollege contests

Miklós Schweitzer 1954- Problem 4

Source:

9/29/2015

4. Find all functions of two variables defined over the entire plane that satisfy the relations

f(x+u,y+u)=f(x,y)+u

and

f(xv,yv)= f(x,y) v

for any real numbers

x,y,u,v

. (R.12)

functioncollege contests

Miklós Schweitzer 1954- Problem 5

Source:

9/29/2015

5. Let

\xi _{1},\xi _{2},\dots ,\xi _{n},...

be independent random variables of uniform distribution in

(0,1)

. Show that the distribution of the random variable

\eta _{n}= \sqrt[]{n}\prod_{k=1}^{n}(1-\frac{\xi _{k}}{k}) (n= 1,2,...)

tends to a limit distribution for

n \to \infty

. (P. 6)

probabilitycollege contests

Miklós Schweitzer 1954- Problem 6

Source:

9/29/2015

6. Prove or disprove the following two propositions: (i) If

a

and

b

are positive integers such that

a<b

, then in any set of

b

consecutive integers there are two whose product is divisible by

ab

(ii) If

a,b

and

c

are positive integers such that

a<b<c

, then in any set of

c

consecutive integers there are three whose product is divisible by

abc

. (N.8)

number theory

Miklós Schweitzer 1954- Problem 9

Source:

9/29/2015

9. Lep

p

be a connected non-closed broken line without self-intersection in the plane

\varphi

. Prove that if

v

is a non-zero vector in

\varphi

and

p

has a commom point with the broken line

p+v

, then

p

has a common point with the broken line

p+\alpha v

too, where

\alpha =\frac{1}{n}

and

n

is a positive integer. Does a similar statemente hold for other positive values of

\alpha

? (

p+v

denotes the broken line obtained from

p

through displacemente by the vector

v

.) (G. 1)

college contestsreal analysisgeometry

Miklós Schweitzer 1955- Problem 3

Source:

9/30/2015

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)

functioncollege contests

Miklós Schweitzer 1954- Problem 8

Source:

9/29/2015

8. Prove the following generalization of the well-known Chinese remainder theorem: Let

R

be a ring with unit element and let

A_{1},A_{2},\dots . A_{n} (n\geqslant 2)

be pairwise relative prime ideals of

R

. Then, for arbitrary elements

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

of

R

, there exists an element

x\in R

such that

x-c_{k} \in A_{k} (k= 1,2, \dots , n)

. (A. 17)

Ring Theorycollege contests

Miklós Schweitzer 1954- Problem 10

Source:

9/29/2015

10. Given a triangle

ABC

, construct outwards over the sides

AB, BC, CA

similiar isosceles triangles

ABC_{1}, BCA_{1}

and

CAB_{1}

. Prove that the straight lines

AA_{1}. BB_{1}

and

CC_{1}

are concurrent. Is this statemente true in elliptic and hyperbolic geometry, too? (G. 19)

geometrycollege contests

Miklós Schweitzer 1955- Problem 1

Source:

9/30/2015

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)

vectorcollege contests

Miklós Schweitzer 1955- Problem 2

Source:

9/30/2015

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)

real analysisfunctioncollege contests

Miklós Schweitzer 1955- Problem 6

Source:

10/8/2015

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

.

college contestsNumberTheory

Miklós Schweitzer 1955- Problem 4

Source:

9/30/2015

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)

number theoryprime numberscollege contests

Miklós Schweitzer 1955- Problem 5

Source:

9/30/2015

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)

college contests

Miklós Schweitzer 1955- Problem 8

Source:

10/8/2015

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)

geometry3D geometrytetrahedroncollege contests

Miklós Schweitzer 1955- Problem 9

Source:

10/8/2015

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)

college contests

Miklós Schweitzer 1955- Problem 10

Source:

10/8/2015

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)

college contests

Miklós Schweitzer 1955- Problem 7

Source:

10/8/2015

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

number theorycollege contests

Miklós Schweitzer 1958- Problem 6

Source:

10/23/2015

6. Prove that if

a_n \geq 0

and

\frac{1}{n}\sum_{k=1}^{n} a_k \geq \sum_{k=n+1}^{2n}a_k

(n=1, 2, \dots)

,
then

\sum_{k=1}^{\infty} a_k

is convergent and its sum is less than

2ea_1

. (S. 9)

college contests

Miklós Schweitzer 1958- Problem 1

Source:

10/21/2015

1. Find the groups every generating system of which contains a basis. (A basis is a set of elements of the group such that the direct product of the cyclic groups generated by them is the group itself.) (A. 14)

group theorycollege contests

Miklós Schweitzer 1958- Problem 2

Source:

10/21/2015

2. Let

A(x)

denote the number of positive integers

n

not greater than

x

and having at least one prime divisor greater than

\sqrt[3]{n}

. Prove that

\lim_{x\to \infty} \frac {A(x)}{x}

exists. (N. 15)

college contests

Miklós Schweitzer 1958- Problem 3

Source:

10/21/2015

3. Let

n

be a positive integer having at least one prime factor with expoente

\geq 2

. Show that

n

has as many factorizations into an odd number of factors as into an even number of factors. (Factorizations into the same factors arranged in different order are considered different.)(N. 10)

college contests

Miklós Schweitzer 1958- Problem 4

Source:

10/21/2015

4. Let

P_1 P_2 P_3 P_4 P_5 P_6

be a convex hexagon. Denote by

T

its area and by

t

the area of the triangle

Q_1 Q_2 Q_3

, where

Q_1,Q_2

and

Q_3

are the midpoints of

P_1P_4,P_2P_5,P_3P_6

respectively. Prove that

t<\frac{1}{4}T

. (G. 3)

geometrycollege contests

Miklós Schweitzer 1958- Problem 5

Source:

10/21/2015

5. Prove that neither the closed nor the open interval can be decomposed into finitely many mutually disjoint proper subsets which are all congruent by translation. (St. 2)

geometrygeometric transformationcollege contests