2024 Benelux

Part of Benelux

Subcontests

(4)

1

2024 BxMO P4

For each positive integer

n

rad(n)

denote the product of the distinct prime factors of

n

. Show that there exists integers

a,b > 1

gcd(a,b)=1

rad(ab(a+b)) < \frac{a+b}{2024^{2024}}

rad(20)=rad(2^2\cdot 5)=2\cdot 5=10

1

2024 BxMO P3

ABC

be a triangle with incentre

I

and circumcircle

\Omega

\left|AC\right|\neq\left|BC\right|

. The internal angle bisector of

\angle CAB

intersects side

BC

D

and the external angle bisectors of

\angle ABC

\angle BCA

\Omega

E

F

respectively. Let

G

be the intersection of lines

AE

FI

\Gamma

be the circumcircle of triangle

BDI

E

\Gamma

G

\Gamma

1

2024 BxMO P1

a_0,a_1,\dots,a_{2024}

be real numbers such that

\left|a_{i+1}-a_i\right| \le 1

i=0,1,\dots,2023

. a) Find the minimum possible value of

a_0a_1+a_1a_2+\dots+a_{2023}a_{2024}

b) Does there exist a real number

C

a_0a_1-a_1a_2+a_2a_3-a_3a_4+\dots+a_{2022}a_{2023}-a_{2023}a_{2024} \ge C

for all real numbers

a_0,a_1,\dots,a_2024

\left|a_{i+1}-a_i\right| \le 1

i=0,1,\dots,2023

1

Dividing a 2n by 2n grid

n

be a positive integer. In a coordinate grid, a path from

(0,0)

(2n,2n)

4n

consecutive unit steps

(1,0)

(0,1)

. Prove that the number of paths that divide the square with vertices

(0,0),(2n,0),(2n,2n),(0,2n)

into 2 regions with even areas is

\frac{{4n \choose 2n} + {2n \choose n}}{2}