2010 Benelux

Part of Benelux

Subcontests

(4)

1

Quadruples (a,b,p,n) - BxMO 2010

Find all quadruples

(a, b, p, n)

of positive integers, such that

p

is a prime and a^3 + b^3 = p^n\mbox{.}
(2nd Benelux Mathematical Olympiad 2010, Problem 4)

1

Collinear points - BxMO 2010

l

there are three different points

A

B

P

in that order. Let

a

be the line through

A

perpendicular to

l

b

be the line through

B

perpendicular to

l

. A line through

P

, not coinciding with

l

a

Q

b

R

. The line through

A

perpendicular to

BQ

BQ

L

BR

T

. The line through

B

perpendicular to

AR

AR

K

AQ

S

. (a) Prove that

P

T

S

are collinear. (b) Prove that

P

K

L

are collinear.
(2nd Benelux Mathematical Olympiad 2010, Problem 3)

1

Polynomial equation - BxMO 2010

Find all polynomials

p(x)

with real coeffcients such that

p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)

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

.
(2nd Benelux Mathematical Olympiad 2010, Problem 2)

1

Partition of {502,..., 2009} - BxMO 2010

A finite set of integers is called bad if its elements add up to

2010

. A finite set of integers is a Benelux-set if none of its subsets is bad. Determine the smallest positive integer

n

such that the set

\{502, 503, 504, . . . , 2009\}

can be partitioned into

n

Benelux-sets. (A partition of a set

S

n

subsets is a collection of

n

pairwise disjoint subsets of

S

, the union of which equals

S

.)
(2nd Benelux Mathematical Olympiad 2010, Problem 1)