MathDB

1

a_{n+1}=2a_n/(1+a_n^2), c_{n+1}=c_n^2-2c_n+2, c_n=2c_0c_1...c_{n-1}/c_n

(a_n)

and (c_n) are given by

a_0 =\frac12

,

c_0=4

, and for

n \ge 0

,

a_{n+1}=\frac{2a_n}{1+a_n^2}

,

c_{n+1}=c_n^2-2c_n+2

Prove that for all

n\ge 1

,

a_n=\frac{2c_0c_1...c_{n-1}}{c_n}

6

1

(x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 , system of equations

Let

n > 1

be an odd positive integer. Assume that positive integers

x_1, x_2,..., x_n \ge 0

satisfy:

\begin{cases} (x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 \\ (x_3 -x_2)^2 + 2(x_3 +x_2) + 1 = n^2 \\ ...\\ (x_1 - x_n)^2 + 2(x_1 + x_n)+ 1 = n^2 \end {cases}

Show that there exists

j, 1 \le j \le n

, such that

x_j = x_{j+1}

. Here

x_{n+1} = x_1

.

8

1

f(x,y) = af (x,z) + bf(y,z)

Given real numbers

a, b

, find all functions

f: R \to R

satisfying

f(x,y) = af (x,z) + bf(y,z)

for all

x,y,z \in R

.

7

1

x^a - x^b is divisible by n, for all integers x

Determine all two-digit positive integers

n =\overline{ab}

(in the decimal system) with the property that for all integers

x

the difference

x^a - x^b

is divisible by

n

.

1

if f(x +19)>=f (x) + 19 and f (x + 94)<= f (x) + 94 then f (x + 1) = f (x) + 1

A function

f: R \to R

satisfies the conditions:

f (x + 19) \le f (x) + 19

and

f (x + 94) \ge f (x) + 94

for all

x \in R

. Prove that

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

for all

x \in R

.

4

1

assigned number =|i - j | when endpoints of side are P_i,P_j

The vertices of a regular

n + 1

-gon are denoted by

P_0,P_1,...,P_n

in some order (

n \ge 2

). Each side of the polygon is assigned a natural number as follows: if the endpoints of the side are

P_i

and

P_j

, then the assigned number equals

|i - j |

. Let S be the sum of all

n + 1

assigned numbers. (a) Given

n

, what is the smallest possible value of

S

? (b) If

P_0

is fixed, how many different assignments are there for which

S

attains the smallest value?

3

1

30 square rooms situated like the cells of a 2x15 board in a rectangle bulding

A rectangular building consists of

30

square rooms situated like the cells of a

2 \times 15

board. In each room there are three doors, each of which leads to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the building?

9

1

construct point P such <APB= <BPC= <CPD iff (a+b)(b+c) < 4ac

On the plane are given four distinct points

A,B,C,D

on a line

g

in this order, at the mutual distances

AB = a, BC = b, CD = c

. (a) Construct (if possible) a point

P

outside line

g

such that

\angle APB =\angle BPC =\angle CPD

. (b) Prove that such a point

P

exists if and only if

(a+b)(b+c) < 4ac

5

1

Integer equation: ... = 1 - xyz

Solve in integers the following equation

\frac{1}{2}(x + y)(y + z)(z + x) + (x + y + z)^3 = 1 - xyz

.

1994 Austrian-Polish Competition

Subcontests

a_{n+1}=2a_n/(1+a_n^2), c_{n+1}=c_n^2-2c_n+2, c_n=2c_0c_1...c_{n-1}/c_n

(x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 , system of equations

f(x,y) = af (x,z) + bf(y,z)

x^a - x^b is divisible by n, for all integers x

if f(x +19)>=f (x) + 19 and f (x + 94)<= f (x) + 94 then f (x + 1) = f (x) + 1

assigned number =|i - j | when endpoints of side are P_i,P_j

30 square rooms situated like the cells of a 2x15 board in a rectangle bulding

construct point P such <APB= <BPC= <CPD iff (a+b)(b+c) < 4ac

Integer equation: ... = 1 - xyz

1994 Austrian-Polish Competition

Subcontests

a_{n+1}=2a_n/(1+a_n^2), c_{n+1}=c_n^2-2c_n+2, c_n=2c_0c_1...c_{n-1}/c_n

(x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 , system of equations

f(x,y) = af (x,z) + bf(y,z)

x^a - x^b is divisible by n, for all integers x

if f(x +19)&gt;=f (x) + 19 and f (x + 94)&lt;= f (x) + 94 then f (x + 1) = f (x) + 1

assigned number =|i - j | when endpoints of side are P_i,P_j

30 square rooms situated like the cells of a 2x15 board in a rectangle bulding

construct point P such &lt;APB= &lt;BPC= &lt;CPD iff (a+b)(b+c) &lt; 4ac

Integer equation: ... = 1 - xyz

if f(x +19)>=f (x) + 19 and f (x + 94)<= f (x) + 94 then f (x + 1) = f (x) + 1

construct point P such <APB= <BPC= <CPD iff (a+b)(b+c) < 4ac