MathDB

1

\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0

a_1 \le a_2 \le .. \le a_n

are natural numbers (

n \ge 2

), show that the inequality

\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0

holds for all

n

-tuples

(x_1,...,x_n) \ne (0,..., 0)

of real numbers if and only if

a_2 \ge 2

.

9

1

covering with rectangles with integer sides

For a rectangle

R

with integral side lengths, denote by

D(a, b)

the number of ways of covering

R

by congruent rectangles with integral side lengths formed by a family of cuts parallel to one side of

R

. Determine the perimeter

P

of the rectangle

R

for which

\frac{D(a,b)}{a+b}

is maximal.

8

1

1998 unit cubes, construct, axax 1 , b x b x1, and cxcx1 boards, 3floor tower

We are given

1988

unit cubes. Using some or all of these cubes, we form three quadratic boards

A, B,C

of dimensions

a \times a \times 1

,

b \times b \times 1

, and

c \times c \times 1

respectively, where

a \le b \le c

. Now we place board

B

on board

C

so that each cube of

B

is precisely above a cube of

C

and

B

does not overlap

C

. Similarly, we place

A

on

B

. This gives us a three-floor tower. What choice of

a, b

and

c

gives the maximum number of such three-floor towers?

7

1

each side of a regular octagon is colored blue or yellow

Each side of a regular octagon is colored blue or yellow. In each step, the sides are simultaneously recolored as follows: if the two neighbors of a side have different colors, the side will be recolored blue, otherwise it will be recolored yellow. Show that after a finite number of moves all sides will be colored yellow. What is the least value of the number

N

of moves that always lead to all sides being yellow?

5

1

b_k = a_k + 9 and a_{k+1} = 8b_k + 8, a_k does not have perfect squares

Two sequences

(a_k)_{k\ge 0}

and

(b_k)_{k\ge 0}

of integers are given by

b_k = a_k + 9

and

a_{k+1} = 8b_k + 8

for

k\ge 0

. Suppose that the number

1988

occurs in one of these sequences. Show that the sequence

(a_k)

does not contain any nonzero perfect square.

4

1

f (f(x) + y) = f(x + y) + f (0), stricly increasing

Determine all strictly increasing functions

f: R \to R

satisfying

f (f(x) + y) = f(x + y) + f (0)

for all

x,y \in R

.

1

Q(x) = P(x) +12, integer polynomials

Let

P(x)

be a polynomial with integer coefficients. Show that if

Q(x) = P(x) +12

has at least six distinct integer roots, then

P(x)

has no integer roots.

6

1

pairwise orthogonal rays in space with a common start

Three rays

h_1,h_2,h_3

emanating from a point

O

are given, not all in the same plane. Show that if for any three points

A_1,A_2,A_3

on

h_1,h_2,h_3

respectively, distinct from

O

, the triangle

A_1A_2A_3

is acute-angled, then the rays

h_1,h_2,h_3

are pairwise orthogonal.

3

1

prove that KLMN is rhombus

In a ABCD cyclic quadrilateral 4 points K, L ,M, N are taken on AB , BC , CD and DA , respectively such that KLMN is a parallelogram. Lines AD, BC and KM have a common point. And also lines AB, DC and NL have a common point. Prove that KLMN is rhombus.

1988 Austrian-Polish Competition

Subcontests

\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0

covering with rectangles with integer sides

1998 unit cubes, construct, axax 1 , b x b x1, and cxcx1 boards, 3floor tower

each side of a regular octagon is colored blue or yellow

b_k = a_k + 9 and a_{k+1} = 8b_k + 8, a_k does not have perfect squares

f (f(x) + y) = f(x + y) + f (0), stricly increasing

Q(x) = P(x) +12, integer polynomials

pairwise orthogonal rays in space with a common start

prove that KLMN is rhombus

1988 Austrian-Polish Competition

Subcontests

\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} &gt;0

covering with rectangles with integer sides

1998 unit cubes, construct, axax 1 , b x b x1, and cxcx1 boards, 3floor tower

each side of a regular octagon is colored blue or yellow

b_k = a_k + 9 and a_{k+1} = 8b_k + 8, a_k does not have perfect squares

f (f(x) + y) = f(x + y) + f (0), stricly increasing

Q(x) = P(x) +12, integer polynomials

pairwise orthogonal rays in space with a common start

prove that KLMN is rhombus

\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0