MathDB

1

Circles through A, tangent to DE

ABC

with

AB<AC

with altitudes

BD

and

CE

. Let the tangents to the circumcircle at

B

and

C

meet at

Y

. Let

\omega_1

be the circle through

A

tangent to

DE

at

E

; define

\omega_2

X

. Prove that

A, X, Y

are colinear.

<span class='latex-italic'>Proposed by Nikola Velov</span>

Problem 2

1

Bound of $\frac{a_{n-1}^{2}}{a_{n-2}}$

Let

n \geq 2

be a fixed positive integer and let

a_{0},a_{1},...,a_{n-1}

be real numbers. Assume that all of the roots of the polynomial

P(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0}

are strictly positive real numbers. Determine the smallest possible value of

\frac{a_{n-1}^{2}}{a_{n-2}}

over all such polynomials.
Proposed by Nikola Velov

Problem 1

1

Lamps to be turned on in at most $k$ moves

Let

n

be a fixed positive integer. There are

n \geq 1

lamps in a row, some of them are on and some are off. In a single move, we choose a positive integer

i

(

1 \leq i \leq n

) and switch the state of the first

i

lamps from the left. Determine the smallest number

k

with the property that we can make all of the lamps be switched on using at most

k

moves, no matter what the initial configuration was.
Proposed by Viktor Simjanoski and Nikola Velov

Problem 6

1

Adding and multiplying numbers on the board

The numbers 1, 2 and 3 are written on a board. Two friends are playing the following game. A player writes a number that doesn't exceed 2022 and isn't already on the board and is a sum or a product of two numbers that are written on the board. They take turns writing numbers and the winner is the player who writes 2022 on the board. Which player has a winning strategy and why?
Proposed by Ilija Jovcheski

Problem 3

1

integer valued functions with at most two values for f(2022)

We consider all functions

f: \mathbb{N} \rightarrow \mathbb{N}

such that

f(f(n)+n)=n

and

f(a+b-1) \leq f(a)+f(b)

for all positive integers

a, b, n

. Prove that there are at most two values for

f(2022)

.

\textit {Proposed by Ilija Jovcheski}

Problem 5

1

different parities of number of prime divisors of adjacent members of an AP

Given is an arithmetic progression {

a_n

} of positive integers. Prove that there exist infinitely many

k

, such that

\omega (a_k)

is even and

\omega (a_{k+1})

is odd (

\omega (n)

is the number of distinct prime factors of

n

).

\textit {Proposed by Viktor Simjanoski and Nikola Velov}

2022 Macedonian Team Selection Test

Subcontests

Circles through A, tangent to DE

Bound of $\frac{a_{n-1}^{2}}{a_{n-2}}$

Lamps to be turned on in at most $k$ moves

Adding and multiplying numbers on the board

integer valued functions with at most two values for f(2022)

different parities of number of prime divisors of adjacent members of an AP