2019 Benelux

Part of Benelux

Subcontests

(3)

1

Rich and poor integers

m>1

is rich if for any positive integer

n

, there exist positive integers

x,y,z

n=mx^2-y^2-z^2

m>1

is poor if it is not rich.
[*]Find a poor integer.[/*] [*]Find a rich integer.[/*]

1

Bisector passes through circumcenter

\Gamma_1

\Gamma_2

intersect at points

A

Z

A\neq Z

B

be the centre of

\Gamma_1

C

be the centre of

\Gamma_2

. The exterior angle bisector of

\angle{BAC}

\Gamma_1

X

\Gamma_2

Y

. Prove that the interior angle bisector of

\angle{BZC}

passes through the circumcenter of

\triangle{XYZ}

P,Q,R

that lie on a line

\ell

in that order, and a point

S

\ell

, the interior angle bisector of

\angle{PQS}

is the line that divides

\angle{PQS}

into two equal angles, while the exterior angle bisector of

\angle{PQS}

is the line that divides

\angle{RQS}

into two equal angles.

2

Pawns and rooks on a chessboard

Pawns and rooks are placed on a

2019\times 2019

chessboard, with at most one piece on each of the

2019^2

squares. A rook can see another rook if they are in the same row or column and all squares between them are empty. What is the maximal number

p

p

p+2019

rooks can be placed on the chessboard in such a way that no two rooks can see each other?

Four variables ineq

a,b,c,d

be real numbers with

0\leqslant a,b,c,d\leqslant 1

ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\leqslant \frac{8}{27}.

[/*] [*]Find all quadruples

(a,b,c,d)

of real numbers with

0\leqslant a,b,c,d\leqslant 1

for which equality holds in the above inequality.