MathDB

1

Miklos Schweitzer 1951_17

\alpha

be a projective plane and

c

a closed polygon on

\alpha

. Prove that

\alpha

will be decomposed into two regions by

c

if and only if there exists a straight line

g

in

\alpha

which has an even number of points in common with

c

.

16

1

Miklos Schweitzer 1951_16

Let

\mathcal{F}

be a surface which is simply covered by two systems of geodesics such that any two lines belonging to different systems form angles of the same opening. Prove that

\mathcal{F}

can be developed (that is, isometrically mapped) into the plane.

15

1

Miklos Schweitzer 1951_15

Let the line z\equal{}x, \, y\equal{}0 rotate at a constant speed about the

z

-axis; let at the same time the point of intersection of this line with the

z

-axis be displaced along the

z

-axis at constant speed. (a) Determine that surface of rotation upon which the resulting helical surface can be developed (i.e. isometrically mapped). (b) Find those lines of the surface of rotation into which the axis and the generators of the helical surface will be mapped by this development.

14

1

Miklos Schweitzer 1951_14

For which commutative finite groups is the product of all elements equal to the unit element?

13

1

Miklos Schweitzer 1951_13

Of how many terms does the expansion of a determinant of order

2n

consist if those and only those elements

a_{ik}

are non-zero for which i\minus{}k is divisible by

n

?

12

1

Miklos Schweitzer 1951_12

By number-theoretical functions, we will understand integer-valued functions defined on the set of all integers. Are there number-theoretical functions

f_0(x),f_1(x),f_2(x),\dots

such that every number theoretical function

F(x)

can be uniquely represented in the form F(x)\equal{}\sum_{k\equal{}0}^{\infty}a_kf_k(x),

a_0,a_1,a_2,\dots

being integers?

11

1

Miklos Schweitzer 1951_11

Prove that, for every pair

n

,

r

of positive integers, there can be found a polynomial

f(x)

of degree

n

with integer coefficients, so that every polynomial

g(x)

of degree at most

n

, for which the coefficients of the polynomial f(x)\minus{}g(x) are integers with absolute value not greater than

r

, is irreducible over the field of rational numbers.

10

1

Miklos Schweitzer 1951_10

Let

f(x)

be a polynomial with integer coefficients and let

p

be a prime. Denote by z_1,...,z_{p\minus{}1} the (p\minus{}1)th complex roots of unity. Prove that f(z_1)\cdots f(z_{p\minus{}1})\equiv f(1)\cdots f(p\minus{}1) \pmod{p}.

9

1

Miklos Schweitzer 1951_9

Let

\{m_1,m_2,\dots\}

be a (finite or infinite) set of positive integers. Consider the system of congruences (1) x\equiv 2m_i^2 \pmod{2m_i\minus{}1} ( i\equal{}1,2,... ). Give a necessary and sufficient condition for the system (1) to be solvable.

8

1

Miklos Schweitzer 1951_8

Given a positive integer

n>3

, prove that the least common multiple of the products

x_1x_2\cdots x_k

(

k\geq 1

) whose factors

x_i

are positive integers with x_1\plus{}x_2\plus{}\cdots\plus{}x_k\le n, is less than

n!

.

7

1

Miklos Schweitzer 1951_7

Let

f(x)

be a polynomial with the following properties: (i) f(0)\equal{}0; (ii) \frac{f(a)\minus{}f(b)}{a\minus{}b} is an integer for any two different integers

a

and

b

. Is there a polynomial which has these properties, although not all of its coefficients are integers?

6

1

Miklos Schweitzer 1951_6

In lawn-tennis the player who scores at least four points, while his opponent scores at least two points less, wins a game. The player who wins at least six games, while his opponent wins at least two games less, wins a set. What minimum percentage of all points does the winner have to score in a set?

5

1

Miklos Schweitzer 1951_5

In a lake there are several sorts of fish, in the following distribution:

18\%

catfish,

2\%

sturgeon and

80\%

other. Of a catch of ten fishes, let

x

denote the number of the catfish and

y

that of the sturgeons. Find the expectation of \frac {x}{y \plus{} 1}

4

1

Miklos Schweitzer 1951_4

Prove that the infinite series 1\minus{}\frac{1}{x(x\plus{}1)}\minus{}\frac{x\minus{}1}{2!x^2(2x\plus{}1)}\minus{}\frac{(x\minus{}1)(2x\minus{}1)}{3!(x^3(3x\plus{}1))}\minus{}\frac{(x\minus{}1)(2x\minus{}1)(3x\minus{}1)}{4!x^4(4x\plus{}1)}\minus{}\cdots is convergent for every positive

x

. Denoting its sum by

F(x)

, find \lim_{x\to \plus{}0}F(x) and

\lim_{x\to \infty}F(x)

.

3

1

Miklos Schweitzer 1951_3

Consider the iterated sequence (1) x_0,x_1 \equal{} f(x_0),\dots,x_{n \plus{} 1} \equal{} f(x_n),\dots, where f(x) \equal{} 4x \minus{} x^2. Determine the points

x_0

of

[0,1]

for which (1) converges and find the limit of (1).

2

1

Miklos Schweitzer 1951_2

Denote by

\mathcal{H}

a set of sequences S\equal{}\{s_n\}_{n\equal{}1}^{\infty} of real numbers having the following properties: (i) If S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}, then S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H}; (ii) If S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H} and T\equal{}\{t_n\}_{n\equal{}1}^{\infty}, then S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H} and ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}; (iii) \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}. A real valued function

f(S)

defined on

\mathcal{H}

is called a quasi-limit of

S

if it has the following properties: If S\equal{}{c,c,\dots,c,\dots}, then f(S)\equal{}c; If

s_i\geq 0

, then

f(S)\geq 0

; f(S\plus{}T)\equal{}f(S)\plus{}f(T); f(ST)\equal{}f(S)f(T), f(S')\equal{}f(S) Prove that for every

S

, the quasi-limit

f(S)

is an accumulation point of

S

.

1

Miklos Schweitzer 1951_1

Choose terms of the harmonic series so that the sum of the chosen terms be finite. Prove that the sequence of these terms is of density zero in the sequence

1,\frac12,\frac13,\dots,\frac1n,\dots

1951 Miklós Schweitzer

Subcontests

Miklos Schweitzer 1951_17

Miklos Schweitzer 1951_16

Miklos Schweitzer 1951_15

Miklos Schweitzer 1951_14

Miklos Schweitzer 1951_13

Miklos Schweitzer 1951_12

Miklos Schweitzer 1951_11

Miklos Schweitzer 1951_10

Miklos Schweitzer 1951_9

Miklos Schweitzer 1951_8

Miklos Schweitzer 1951_7

Miklos Schweitzer 1951_6

Miklos Schweitzer 1951_5

Miklos Schweitzer 1951_4

Miklos Schweitzer 1951_3

Miklos Schweitzer 1951_2

Miklos Schweitzer 1951_1