MathDB

1

Miklos Schweitzer 1949_8

The four sides of a skew quadrangle and the two segments joining the midpoints of the opposite sides are realized by rigid bars. The bars are linked by hinges. Prove that this apparatus is not rigid. P.S: The 1949 Miklos Schweitzer competition had only 8 problems!

7

1

Miklos Schweitzer 1949_7

Find the complex numbers

z

for which the series 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots converges and find its sum.

6

1

Miklos Schweitzer 1949_6

Let

n

and

k

be positive integers,

n\geq k

. Prove that the greatest common divisor of the numbers \binom{n}{k},\binom{n\plus{}1}{k},\ldots,\binom{n\plus{}k}{k} is

1

.

5

1

Miklos Schweitzer 1949_5

Let

f(x)

be a polynomial of second degree the roots of which are contained in the interval [\minus{}1,\plus{}1] and let there be a point x_0\in [\minus{}1.\plus{}1] such that |f(x_0)|\equal{}1. Prove that for every

\alpha \in [0,1]

, there exists a \zeta \in [\minus{}1,\plus{}1] such that |f'(\zeta)|\equal{}\alpha and that this statement is not true if

\alpha>1

.

4

1

Miklos Schweitzer 1949_4

Let

A

and

B

be two disjoint sets in the interval

(0,1)

. Denoting by

\mu

the Lebesgue measure on the real line, let

\mu(A)>0

and

\mu(B)>0

. Let further

n

be a positive integer and \lambda \equal{}\frac1n . Show that there exists a subinterval

(c,d)

of

(0,1)

for which \mu(A\cap (c,d))\equal{}\lambda \mu(A) and \mu(B\cap (c,d))\equal{}\lambda \mu(B) . Show further that this is not true if

\lambda

is not of the form

\frac1n

.

3

1

Miklos Schweitzer 1949_3

Let

p

be an odd prime number and

a_1,a_2,...,a_p

and

b_1,b_2,...,b_p

two arbitrary permutations of the numbers

1,2,...,p

. Show that the least positive residues modulo

p

of the numbers

a_1b_1, a_2b_2,...,a_pb_p

never form a permutation of the numbers

1,2,...,p

.

2

1

Miklos Schweitzer 1949_2

Compute \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx .

1

Miklos Schweitzer 1949_1

Let an infinite sequence of measurable sets be given on the interval

(0,1)

the measures of which are

\geq \alpha>0

. Show that there exists a point of

(0,1)

which belongs to infinitely many terms of the sequence.

1949 Miklós Schweitzer

Subcontests

Miklos Schweitzer 1949_8

Miklos Schweitzer 1949_7

Miklos Schweitzer 1949_6

Miklos Schweitzer 1949_5

Miklos Schweitzer 1949_4

Miklos Schweitzer 1949_3

Miklos Schweitzer 1949_2

Miklos Schweitzer 1949_1