MathDB

1

Every point on a circle is coloured in blue or yellow

Every point on a circle is coloured in blue or yellow such there is at least a point of each color. Prove that for every colouring of the circle there is always an isosceles triangle inscribed inside the circle 1) with all vertexes of the same colour. 2) with vertexes of both colours. For every colouring of the circle is there an equilateral triangle inscribed inside the circle 3) with all vertexes of the same colour? 4) with vertexes of both colours?

10

1

In a chess tournament participated $100$ players

In a chess tournament participated

100

players. Every player played one game with every other player. For a win

1

point is given, for loss

0

and for a draw both players get

0,5

points. Ion got more points than every other player. Mihai lost only one game, but got less points than every other player. Find all possible values of the difference between the points accumulated by Ion and the points accumulated by Mihai.

8

1

Prove that the angles $AMP$ and $BMN$ are equal.

Let

ABCD

be a trapezoid with bases

AB

and

CD

(AB>CD)

. Diagonals

AC

and

BD

intersect in point

N

and lines

AD

and

BC

intersect in point

M

. The circumscribed circles of

ADN

and

BCN

intersect in point

P

, different from point

N

. Prove that the angles

AMP

and

BMN

are equal.

4

1

(a, 2ad+b, ad^2+bd+c)

On the board there are three real numbers

(a,b,c)

. During a

procedure

the numbers are erased and in their place another three numbers a written, either

(c,b,a)

or every time a nonzero real number

d

is chosen and the numbers

(a, 2ad+b, ad^2+bd+c)

are written. 1) If we start with

(1,-2,-1)

written on the board, can we have the numbers

(2,0,-1)

on the board after a finite number of procedures? 2) If we start with

(1,-2,-1)

written on the board, can we have the numbers

(2,-1,-1)

on the board after a finite number of procedures?

2

1

Prove that the angle $GFC$ and $BAC$ are equal.

Let

\Omega

be the circumscribed circle of the acute triangle

ABC

and

D

a point the small arc

BC

of

\Omega

. Points

E

and

F

are on the sides

AB

and

AC

, respectively, such that the quadrilateral

CDEF

is a parallelogram. Point

G

is on the small arc

AC

such that lines

DC

and

BG

are parallel. Prove that the angles

GFC

and

BAC

are equal.

12

1

\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}

Let

a,b,c

be positive real numbers such that

a^2+b^2+c^2=3.

Prove that

\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}\leq4.

6

1

a^2(b+c)=b^2(c+a)=2023

Real numbers

a,b,c

with

a\neq b

verify

a^2(b+c)=b^2(c+a)=2023.

Find the numerical value of

E=c^2(a+b)

.

5

1

Prove that $abc$ is divisible by $60$.

The positive integers

a, b, c

are the lengths of the sides of a right triangle. Prove that

abc

is divisible by

60

.

3

1

A=2024^{n+1}-2023n-2024

Prove that the number

A=2024^{n+1}-2023n-2024

has at least

15

different positive divisors for every nonnegative integer

n

.

1

\frac{1}{n\cdot(\sqrt{n+1}+\sqrt{n})+\sqrt{n}}

The positive integer

n

verifies

\frac{1}{1\cdot(\sqrt{2}+\sqrt{1})+\sqrt{1}}+\frac{1}{2\cdot(\sqrt{3}+\sqrt{2})+\sqrt{2}}+\cdots+\frac{1}{n\cdot(\sqrt{n+1}+\sqrt{n})+\sqrt{n}}=\frac{2022}{2023}.

Find the sum of digits of number

n

.

9

1

collinear wanted , 3 altitudes related

Let

AD

,

BE

and

CF

be the altitudes of

\Delta ABC

. The points

P, \, \, Q, \, \, R

and

S

are the feet of the perpendiculars drawn from the point

D

on the segments

BA

,

BE

,

CF

and

CA

, respectively. Prove that the points

P, \, \, Q, \, \, R

and

S

are collinear.

11

1

Find all prime $x,y$ and $z,$ such that $x^5 +y^3 -(x+y)^2=3z^3$

Find all prime

x,y

and

z,

such that

x^5 +y^3 -(x+y)^2=3z^3

2023 Junior Balkan Team Selection Tests - Moldova

Subcontests

Every point on a circle is coloured in blue or yellow

In a chess tournament participated $100$ players

Prove that the angles $AMP$ and $BMN$ are equal.

(a, 2ad+b, ad^2+bd+c)

Prove that the angle $GFC$ and $BAC$ are equal.

\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}

a^2(b+c)=b^2(c+a)=2023

Prove that $abc$ is divisible by $60$.

A=2024^{n+1}-2023n-2024

\frac{1}{n\cdot(\sqrt{n+1}+\sqrt{n})+\sqrt{n}}

collinear wanted , 3 altitudes related

Find all prime $x,y$ and $z,$ such that $x^5 +y^3 -(x+y)^2=3z^3$