2009 Benelux

Part of Benelux

Subcontests

(4)

1

K on (ABI) if and only if K is on (CDJ)

Given trapezoid

ABCD

with parallel sides

AB

CD

E

be a point on line

BC

outside segment

BC

, such that segment

AE

intersects segment

CD

. Assume that there exists a point

F

AD

\angle EAD=\angle CBF

I

the point of intersection of

CD

EF

J

the point of intersection of

AB

EF

K

be the midpoint of segment

EF

, and assume that

K

is different from

I

J

K

belongs to the circumcircle of

\triangle ABI

K

belongs to the circumcircle of

\triangle CDJ

1

Girls and boys dancing from town X and Y

n\ge 1

be an integer. In town

X

n

n

boys, and each girl knows each boy. In town

Y

n

g_1,g_2,\ldots ,g_n

2n-1

b_1,b_2,\ldots ,b_{2n-1}

i=1,2,\ldots ,n

g_i

b_1,b_2,\ldots ,b_{2i-1}

and no other boys. Let

r

be an integer with

1\le r\le n

. In each of the towns a party will be held where

r

girls from that town and

r

boys from the same town are supposed to dance with each other in

r

dancing pairs. However, every girl only wants to dance with a boy she knows. Denote by

X(r)

the number of ways in which we can choose

r

dancing pairs from town

X

Y(r)

the number of ways in which we can choose

r

dancing pairs from town

Y

X(r)=Y(r)

r=1,2,\ldots ,n

1

Generalization of ISL 2008 problem

n

be a positive integer and let

k

be an odd positive integer. Moreover, let

a,b

c

be integers (not necessarily positive) satisfying the equations

a^n+kb=b^n+kc=c^n+ka

a=b=c

1

f(n) is a perfect square for all n

Find all functions

f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}

that satisfy the following two conditions:

\bullet\ f(n)

is a perfect square for all

n\in\mathbb{Z}_{>0}

\bullet\ f(m+n)=f(m)+f(n)+2mn

m,n\in\mathbb{Z}_{>0}