2015 Benelux

Part of Benelux

Subcontests

(4)

1

BxMO 2015, Problem 4

n

be a positive integer. For each partition of the set

\{1,2,\dots,3n\}

into arithmetic progressions, we consider the sum

S

of the respective common differences of these arithmetic progressions. What is the maximal value that

S

can attain?
(An arithmetic progression is a set of the form

\{a,a+d,\dots,a+kd\}

a,d,k

are positive integers, and

k\geqslant 2

; thus an arithmetic progression has at least three elements, and successive elements have difference

d

, called the common difference of the arithmetic progression.)

1

BxMO 2015, Problem 3

Does there exist a prime number whose decimal representation is of the form

3811\cdots11

(that is, consisting of the digits

3

8

in that order, followed by one or more digits

1

1

BxMO 2015, Problem 2

ABC

be an acute triangle with circumcentre

O

\mathit{\Gamma}_B

be the circle through

A

B

that is tangent to

AC

\mathit{\Gamma}_C

be the circle through

A

C

that is tangent to

AB

. An arbitrary line through

A

\mathit{\Gamma}_B

X

\mathit{\Gamma}_C

Y

|OX|=|OY|

1

BxMo 2015, Problem 1

Determine the smallest positive integer

q

with the following property: for every integer

m

1\leqslant m\leqslant 1006

, there exists an integer

n

\dfrac{m}{1007}q<n<\dfrac{m+1}{1008}q