2013 Junior Balkan MO

Part of Junior Balkan MO

Subcontests

(4)

1

JBMO 2013 Problem 4

n

be a positive integer. Two players, Alice and Bob, are playing the following game: - Alice chooses

n

real numbers; not necessarily distinct. - Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are

\frac{n(n-1)}{2}

such sums; not necessarily distinct.) - Bob wins if he finds correctly the initial

n

numbers chosen by Alice with only one guess. Can Bob be sure to win for the following cases?
a.

n=5

n=6

n=8

Justify your answer(s).
[For example, when

n=4

, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]

1

JBMO 2013 Problem 3

\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16

for all positive real numbers

a

b

ab\geq 1

1

JBMO 2013 Problem 2

ABC

be an acute-angled triangle with

AB<AC

O

be the centre of its circumcircle

\omega

D

be a point on the line segment

BC

\angle BAD = \angle CAO

E

be the second point of intersection of

\omega

AD

M

N

P

are the midpoints of the line segments

BE

OD

AC

, respectively, show that the points

M

N

P

1

JBMO 2013 Problem 1

Find all ordered pairs

(a,b)

of positive integers for which the numbers

\dfrac{a^3b-1}{a+1}

\dfrac{b^3a+1}{b-1}

are both positive integers.