2018 Junior Balkan MO

Part of Junior Balkan MO

Subcontests

(4)

1

JBMO 2018, P3

k>1

be a positive integer and

n>2018

an odd positive integer. The non-zero rational numbers

x_1,x_2,\ldots,x_n

are not all equal and:

x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}

Find the minimum value of

k

, such that the above relations hold.

1

JBMO 2018 P2

Find max number

n

of numbers of three digits such that : 1. Each has digit sum

9

2. No one contains digit

0

2

have different unit digits 4. Each

2

have different decimal digits 5. Each

2

have different hundreds digits

1

JBMO 2018 P4

\triangle ABC

A'

B'

C'

the symmetrics of vertex over opposite sides.The intersection of the circumcircles of

\triangle ABB'

\triangle ACC'

A_1

B_1

C_1

are defined similarly.Prove that lines

AA_1

BB_1

CC_1

1

JBMO 2018 P1

Find all integers

m

n

such that the fifth power of

m

minus the fifth power of

n

16mn