MathDB

1

Integer values of a sequence

k

be a fixed positive integer. Consider the sequence definited by a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots where

\lfloor x\rfloor

denotes the greatest integer less than or equal to

x

. For each

k

find the set

A_{k}

containing all integer values of the sequence

(\sqrt[k]{a_{n}})_{n\geq 0}

.

4

1

Area inequality in quadrilateral

Prove that if

a,b,c,d

are lengths of the successive sides of a quadrangle (not necessarily convex) with the area equal to

S

, then the following inequality holds

S \leq \frac{1}{2}(ac+bd).

For which quadrangles does the inequality become equality?

5

1

2001 Austrian-Polish Mathematics Competition

The fields of the

8\times 8

chessboard are numbered from

1

to

64

in the following manner: For

i=1,2,\cdots,63

the field numbered by

i+1

can be reached from the field numbered by

i

by one move of the knight. Let us choose positive real numbers

x_{1},x_{2},\cdots,x_{64}

. For each white field numbered by

i

define the number

y_{i}=1+x_{i}^{2}-\sqrt[3]{x_{i-1}^{2}x_{i+1}}

and for each black field numbered by

j

define the number

y_{j}=1+x_{j}^{2}-\sqrt[3]{x_{j-1}x_{j+1}^{2}}

where

x_{0}=x_{64}

and

x_{1}=x_{65}

. Prove that

\sum_{i=1}^{64}y_{i}\geq 48

2

1

System of equations

Let

n

be a positive integer greater than

2

. Solve in nonnegative real numbers the following system of equations x_{k}+x_{k+1}=x_{k+2}^{2} , k=1,2,\cdots,n where

x_{n+1}=x_{1}

and

x_{n+2}=x_{2}

.

8

1

2001 Austrian-Polish Mathematics Competition

The prism with the regular octagonal base and with all edges of the length equal to

1

is given. The points

M_{1},M_{2},\cdots,M_{10}

are the midpoints of all the faces of the prism. For the point

P

from the inside of the prism denote by

P_{i}

the intersection point (not equal to

M_{i}

) of the line

M_{i}P

with the surface of the prism. Assume that the point

P

is so chosen that all associated with

P

points

P_{i}

do not belong to any edge of the prism and on each face lies exactly one point

P_{i}

. Prove that

\sum_{i=1}^{10}\frac{M_{i}P}{M_{i}P_{i}}=5

7

1

2001 Austrian-Polish Mathematics Competition

Consider the set

A

containing all positive integers whose decimal expansion contains no

0

, and whose sum

S(N)

of the digits divides

N

. (a) Prove that there exist infinitely many elements in

A

whose decimal expansion contains each digit the same number of times as each other digit. (b) Explain that for each positive integer

k

there exist an element in

A

having exactly

k

digits.

10

1

Maximize a sum

The sequence

a_{1},a_{2},\cdots,a_{2010}

has the following properties: (1) each sum of the 20 successive values of the sequence is nonnegative, (2)

|a_{i}a_{i+1}| \leq 1

for

i=1,2,\cdots,2009

. Determine the maximal value of the expression

\sum_{i=1}^{2010}a_{i}

.

1

Sum of powers

Determine the number of positive integers

a

, so that there exist nonnegative integers

x_0,x_1,\ldots,x_{2001}

which satisfy the equation

\displaystyle a^{x_0} = \sum_{i=1}^{2001} a^{x_i}

3

1

Nice (triangle inequality)

Let

a,b,c

be sides of a triangle. Prove that

2 < \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} - \frac{a^3+b^3+c^3}{abc}\leq 3

9

1

sets intersection

Let

A

be a set with

2n

elements, and let

A_1, A_2...,A_m

be subsets of

A

e ach one with n elements. Find the greatest possible m, such that it is possible to select these

m

subsets in such a way that the intersection of any 3 of them has at most one element.

2001 Austrian-Polish Competition

Subcontests

Integer values of a sequence

Area inequality in quadrilateral

2001 Austrian-Polish Mathematics Competition

System of equations

2001 Austrian-Polish Mathematics Competition

2001 Austrian-Polish Mathematics Competition

Maximize a sum

Sum of powers

Nice (triangle inequality)

sets intersection