MathDB

2

Poblem 4 (1st jbmo tst), moldova

n

be a positive integer,

n\geq 4

.

n

cards are arranged on a circle and the numbers

1

or

-1

are written on each of the cards. in a

question

we may find out the product of the numbers on any

3

cards. What is the minimum numbers if questions needed to find out the product of all

n

numbers?

Poblem 4 (2nd jbmo tst)

Determine all real solutions of the equation:

{ \frac{x^{2}}{x-1}+\sqrt{x-1}+\frac{\sqrt{x-1}}{x^{2}}}=\frac{x-1}{x^{2}}+\frac{1}{\sqrt{x-1}}+\frac{x^{2}}{\sqrt{x-1}} .

3

2

Problem 3 (2nd jbmo tst)

Determine all 2nd degree polynomials with integer coefficients of the form

P(X)=aX^{2}+bX+c

, that satisfy:

P(a)=b

,

P(b)=a

, with

a\neq b

.

Poblem 3 (1st jbmo tst), moldova

The convex polygon

A_{1}A_{2}\ldots A_{2006}

has opposite sides parallel

(A_{1}A_{2}||A_{1004}A_{1005}, \ldots)

. Prove that the diagonals

A_{1}A_{1004}, A_{2}A_{1005}, \ldots A_{1003}A_{2006}

are concurrent if and only if opposite sides are equal.

2

Poblem 2 (jbmo tst), moldova

Prove that there infinitely many numbers of the form

18^{m}+45^{m}+50^{m}+125^{m}

, divisible by 2006.

m\in N

Problem 2 (2nd jbmo tst)

Let

ABCD

be a rectangle and denote by

M

and

N

the midpoints of

AD

and

BC

respectively. The point

P

is on

(CD

such that

D\in (CP)

, and

PM

intersects

AC

in

Q

. Prove that

m(\angle{MNQ})=m(\angle{MNP})

.

1

2

Poblem 1 (jbmo tst), moldova

Five segments have lengths such that any three of them can be sides of a - possibly degenerate - triangle. Also, the lengths of these segments are nonzero and pairwisely different. Prove that there exists at least one acute-angled triangle among these triangles.

Problem 1 (2nd jbmo tst)

Let the number

x

. Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number

x

(for example,

x\cdot x=x^{2}

,

x^{2}\cdot x^{2}=x^{4}

,

x^{4}: x=x^{3}

, etc). Determine the minimal number of operations needed for calculating

x^{2006}

.

2006 Junior Balkan Team Selection Tests - Moldova

Subcontests

Poblem 4 (1st jbmo tst), moldova

Poblem 4 (2nd jbmo tst)

Problem 3 (2nd jbmo tst)

Poblem 3 (1st jbmo tst), moldova

Poblem 2 (jbmo tst), moldova

Problem 2 (2nd jbmo tst)

Poblem 1 (jbmo tst), moldova

Problem 1 (2nd jbmo tst)