MathDB

1

D(A) = \underset{d}{max} \underset{i}{min} \delta (P_i, d)

C

be a unit circle and

n \ge 1

be a fixed integer. For any set

A

of

n

points

P_1,..., P_n

on

C

define

D(A) = \underset{d}{max}\, \underset{i}{min}\delta (P_i, d)

, where

d

goes over all diameters of

C

and

\delta (P, \ell)

denotes the distance from point

P

to line

\ell

. Let

F_n

be the family of all such sets

A

. Determine

D_n = \underset{A\in F_n}{min} D(A)

and describe all sets

A

with

D(A) = D_n

.

7

1

1/3 p(n)= s{n)-1, product and sum of digits related

For any natural number

n= \overline{a_k...a_1a_0}

(a_k \ne 0)

in decimal system write

p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k

,

s(n)=a_0+ a_1+ ... + a_k

,

n^*= \overline{a_0a_1...a_k}

. Consider

P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}

and let

Q

be the set of numbers in

P

with all digits greater than

1

. (a) Show that

P

is infinite. (b) Show that

Q

is finite. (c) Write down all the elements of

Q

.

3

1

f (x + 1) = f (x) + 1, x_{n+1} = f (x_n), x_0 = a

A function

f: R \to R

satisfies

f (x + 1) = f (x) + 1

for all

x

. Given

a \in R

, define the sequence

(x_n)

recursively by

x_0 = a

and

x_{n+1} = f (x_n)

for

n \ge 0

. Suppose that, for some positive integer m, the difference

x_m - x_0 = k

is an integer. Prove that the limit

\lim_{n\to \infty}\frac{x_n}{n}

exists and determine its value.

9

1

subset of lattice points , 1 <= x <= 12 and 1 <= y <= 13

Let

M

be the set of all points

(x,y)

in the cartesian plane, with integer coordinates satisfying

1 \le x \le 12

and

1 \le y \le 13

. (a) Prove that every

49

-element subset of

M

contains four vertices of a rectangle with sides parallel to the coordinate axes. (b) Give an example of a

48

-element subset of

M

without this property.

8

1

length of selfintersecting curve C is an integer

A circle of perimeter

1

has been dissected into four equal arcs

B_1, B_2, B_3, B_4

. A closed smooth non-selfintersecting curve

C

has been composed of translates of these arcs (each

B_j

possibly occurring several times). Prove that the length of

C

is an integer.

5

1

partition 3d space into 3 subsets, mutual distance

The Euclidian three-dimensional space has been partitioned into three nonempty sets

A_1,A_2,A_3

. Show that one of these sets contains, for each

d > 0

, a pair of points at mutual distance

d

.

2

1

P(x) = x^n - 1987x, rational x,y with P(x) = P(y) , then x = y

Let

n

be the square of an integer whose each prime divisor has an even number of decimal digits. Consider

P(x) = x^n - 1987x

. Show that if

x,y

are rational numbers with

P(x) = P(y)

, then

x = y

.

1

sum of squares of 3 pairwise orthogonal chords of sphere independent APMC 87

Three pairwise orthogonal chords of a sphere

S

are drawn through a given point

P

inside

S

. Prove that the sum of the squares of their lengths does not depend on their directions.

4

1

problem from Austrian - Polish Math Competition

Does the set

\{1,2,3,...,3000\}

contain a subset

A

consisting of 2000 numbers that

x\in A

implies

2x \notin A

?!! :?:

1987 Austrian-Polish Competition

Subcontests

D(A) = \underset{d}{max} \underset{i}{min} \delta (P_i, d)

1/3 p(n)= s{n)-1, product and sum of digits related

f (x + 1) = f (x) + 1, x_{n+1} = f (x_n), x_0 = a

subset of lattice points , 1 <= x <= 12 and 1 <= y <= 13

length of selfintersecting curve C is an integer

partition 3d space into 3 subsets, mutual distance

P(x) = x^n - 1987x, rational x,y with P(x) = P(y) , then x = y

sum of squares of 3 pairwise orthogonal chords of sphere independent APMC 87

problem from Austrian - Polish Math Competition

1987 Austrian-Polish Competition

Subcontests

D(A) = \underset{d}{max} \underset{i}{min} \delta (P_i, d)

1/3 p(n)= s{n)-1, product and sum of digits related

f (x + 1) = f (x) + 1, x_{n+1} = f (x_n), x_0 = a

subset of lattice points , 1 &lt;= x &lt;= 12 and 1 &lt;= y &lt;= 13

length of selfintersecting curve C is an integer

partition 3d space into 3 subsets, mutual distance

P(x) = x^n - 1987x, rational x,y with P(x) = P(y) , then x = y

sum of squares of 3 pairwise orthogonal chords of sphere independent APMC 87

problem from Austrian - Polish Math Competition

subset of lattice points , 1 <= x <= 12 and 1 <= y <= 13