MathDB

1

Geometrical inequality

ABCDEF

be a convex hexagon. The diagonals

AC

and

BD

cross at

P,

the diagonals

AE{}

and

DF

cross at

Q,

and the line

PQ

crosses the sides

BC

and

EF

at

X

and

Y,{}

respectively. Prove that the length of the segment

XY

does not exceed the sum of the lengths of one of the diagonals through

P{}

and one of the diagonals through

Q{}

.
The Problem Selection Committee

P3

1

Divisibility with a quadratic and a factorial

Given a positive integer

a,

prove that

n!

is divisible by

n^2 + n + a

for infinitely many positive integers

n.{}

Proposed by Andrei Bâra

P4

2

The legendary "Sudoku" problem

Consider a

4\times 4

array of pairwise distinct positive integers such that on each column, respectively row, one of the numbers is equal to the sum of the other three. Determine the least possible value of the largest number such an array may contain.
The Problem Selection Committee

Subsets of points lying in disks

Fix a positive integer

n.{}

Consider an

n{}

-point set

S{}

in the plane. An eligible set is a non-empty set of the form

S\cap D,{}

where

D

is a closed disk in the plane. In terms of

n,

determine the smallest possible number of eligible subsets

S{}

may contain.
Proposed by Cristi Săvescu

P2

1

A slightly unusual geometry problem

A diagonal line of a (not necessarily convex) polygon with at least four sides is any line through two non-adjacent vertices of that polygon. Determine all polygons with at least four sides satisfying the following condition: The reflexion of each vertex in each diagonal line lies inside or on the boundary of the polygon.
The Problem Selection Committee

P1

1

Sum os several powers of two

Let

m

and

n

be positive integers, where

m < 2^n.

Determine the smallest possible number of not necessarily pairwise distinct powers of two that add up to

m\cdot(2^n- 1).

The Problem Selection Committee

2023 Romania Team Selection Test

Subcontests

Geometrical inequality

Divisibility with a quadratic and a factorial

The legendary "Sudoku" problem

Subsets of points lying in disks

A slightly unusual geometry problem

Sum os several powers of two

2023 Romania Team Selection Test

Subcontests

Geometrical inequality

Divisibility with a quadratic and a factorial

The legendary &quot;Sudoku&quot; problem

Subsets of points lying in disks

A slightly unusual geometry problem

Sum os several powers of two

The legendary "Sudoku" problem