2021 Benelux

Part of Benelux

Subcontests

(4)

1

Sequence not containing a multiple of p (BxMO2021 P4)

a_1, a_2, a_3, \ldots

of positive integers satisfies

a_1 > 5

a_{n+1} = 5 + 6 + \cdots + a_n

for all positive integers

n

. Determine all prime numbers

p

such that, regardless of the value of

a_1

, this sequence must contain a multiple of

p

1

Circumcentre on perpendicular bisector (BxMO2021 P3)

A cyclic quadrilateral

ABXC

has circumcentre

O

D

be a point on line

BX

AD = BD

E

be a point on line

CX

AE = CE

. Prove that the circumcentre of triangle

\triangle DEX

lies on the perpendicular bisector of

OA

1

Distinct pebble sets (BxMO2021 P2)

Pebbles are placed on the squares of a

2021\times 2021

board in such a way that each square contains at most one pebble. The pebble set of a square of the board is the collection of all pebbles which are in the same row or column as this square. (A pebble belongs to the pebble set of the square in which it is placed.) What is the least possible number of pebbles on the board if no two squares have the same pebble set?

1

Replace maxima by minima (BxMO2021 P1)

(a) Prove that for all

a, b, c, d \in \mathbb{R}

a + b + c + d = 0

\max(a, b) + \max(a, c) + \max(a, d) + \max(b, c) + \max(b, d) + \max(c, d) \geqslant 0.

(b) Find the largest non-negative integer

k

such that it is possible to replace

k

of the six maxima in this inequality by minima in such a way that the inequality still holds for all

a, b, c, d \in \mathbb{R}

a + b + c + d = 0