MathDB

1

Miklós Schweitzer 1956- Problem 10

N

different colours,

n

balls of each colour. Balls are drawn and not replaced until one of the colours turns up twice; denote by

V_{N,n}

the number of the balls drawn and by

M_{N,n}

the expectation of the random variable

v_{N,n}

. Find the limit distribution of the random variable

\frac{V_{N,n}}{M_{N,n}}

if

N \to \infty

and

n

is a fixed number. (P. 8)

9

1

Mikl&oacute;s Schweitzer 1956- Problem 9

9. Show that if the trigonometric polynomial

f(\theta)= \sum_{v=1}^{n} a_v \cos v\theta

monotonically decreases over the closed interval

[0,\pi]

, then the trigonometric polynomial

g(\theta)=\sum_{v=1}^{n}a_v \sin v\theta

is non negative in the same interval. (S. 26)

8

1

Miklós Schweitzer 1956- Problem 8

8. Let

(a_n)_{n=1}^{\infty}

be a sequence of positive numbers and suppose that

\sum_{n=1}^{\infty} a_n^2

is divergent. Let further

0<\epsilon<\frac{1}{2}

. Show that there exists a sequence

(b_n)_{n=1}^{\infty}

of positive numbers such that

\sum_{n=1}^{\infty}b_n^2

is convergent and

\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}

for every positive integer

N

. (S. 8)

7

1

Miklós Schweitzer 1956- Problem 7

7. Let

(a_n)_{n=0}^{\infty}

be a sequence of real numbers such that, with some positive number

C

,

\sum_{k=1}^{n}k\mid a_k \mid<n C

(

n=1,2, \dots

)
Putting

s_n= a_0 +a_1+\dots+a_n

, suppose that

\lim_{n \to \infty }(\frac{s_{0}+s_{1}+\dots+s_n}{n+1})= s

exists. Prove that

\lim_{n \to \infty }(\frac{s_{0}^2+s_{1}^2+\dots+s_n^2}{n+1})= s^2

(S. 7)

6

1

Miklós Schweitzer 1956- Problem 6

6. Show that the number of the faces of a convex polyhedron is even if every face is centrally simmetric. (G. 12)

5

1

Miklós Schweitzer 1956- Problem 5

5. On a circle consider

n

points among which there acts a repulsive force inversely proportional to the square of their distance. Prove that the point system is in stable equilibrium if and only if the points form a regular

n

-gon; in other words, considering the sum of the reciprocal distances of the

\binom{n}{2}

pairs of points which can be chosen from among the

n

given points, this sum is minimal if and only if the points lie at the vertices of a regular

n

-gon. (G. 2)

4

1

Miklós Schweitzer 1956- Problem 4

4. Denoting by

a(n)

the greatest prime factor of the positive integer

n

, show that

S= \sum_{n=1}^{\infty } \frac{1}{na(n)}

is convergente. (N. 13)

3

1

Miklós Schweitzer 1956- Problem 3

3. A triangulation of a convex closed polygon is the division into triangles of this poilygon by diagonals not intersecting in the interior of the polygon. Find the number of all triangulations fo a conves n-gon and also the number of those triangulations in which every triangle has at least one side in common with the given n-gon. (C. 4)

2

1

Miklós Schweitzer 1956- Problem 2

2. Find the minimum of

max ( |1+z|, |1+z^{2}|)

if

z

runs over all complex numbers. (F. 2)

1

Miklós Schweitzer 1956- Problem 1

1. Solve without use of determinants the following system of linear equations:

\sum_{j=0}{k} \binom{k+\alpha}{j} x_{k-j} =b_k

(

k= 0,1, \dots , n

),
where

\alpha

is a fixed real number. (A. 7)

1956 Miklós Schweitzer

Subcontests

Miklós Schweitzer 1956- Problem 10

Mikl&oacute;s Schweitzer 1956- Problem 9

Miklós Schweitzer 1956- Problem 8

Miklós Schweitzer 1956- Problem 7

Miklós Schweitzer 1956- Problem 6

Miklós Schweitzer 1956- Problem 5

Miklós Schweitzer 1956- Problem 4

Miklós Schweitzer 1956- Problem 3

Miklós Schweitzer 1956- Problem 2

Miklós Schweitzer 1956- Problem 1

1956 Mikl&oacute;s Schweitzer

Subcontests

Mikl&oacute;s Schweitzer 1956- Problem 10

Mikl&amp;oacute;s Schweitzer 1956- Problem 9

Mikl&oacute;s Schweitzer 1956- Problem 8

Mikl&oacute;s Schweitzer 1956- Problem 7

Mikl&oacute;s Schweitzer 1956- Problem 6

Mikl&oacute;s Schweitzer 1956- Problem 5

Mikl&oacute;s Schweitzer 1956- Problem 4

Mikl&oacute;s Schweitzer 1956- Problem 3

Mikl&oacute;s Schweitzer 1956- Problem 2

Mikl&oacute;s Schweitzer 1956- Problem 1

1956 Miklós Schweitzer

Miklós Schweitzer 1956- Problem 10

Miklós Schweitzer 1956- Problem 9

Miklós Schweitzer 1956- Problem 8

Miklós Schweitzer 1956- Problem 7

Miklós Schweitzer 1956- Problem 6

Miklós Schweitzer 1956- Problem 5

Miklós Schweitzer 1956- Problem 4

Miklós Schweitzer 1956- Problem 3

Miklós Schweitzer 1956- Problem 2

Miklós Schweitzer 1956- Problem 1