1984 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

system of equations

a,b,c

be positive real numbers. Find all real solutions

(x,y,z)

ax+by=(x-y)^{2} \\ by+cz=(y-z)^{2} \\ cz+ax=(z-x)^{2}

1

decimal representations

Show that for any positive integer

m

, there exists a positive integer

n

so that in the decimal representations of the numbers

5^{m}

5^{n}

, the representation of

5^{n}

ends in the representation of

5^{m}

1

congruent quadrilaterals

ABCD

be a cyclic quadrilateral and let

H_{A}, H_{B}, H_{C}, H_{D}

be the orthocenters of the triangles

BCD

CDA

DAB

ABC

respectively. Show that the quadrilaterals

ABCD

H_{A}H_{B}H_{C}H_{D}

1

classic application of jensen

n \geq 2

be a positive integer and

a_{1},\ldots , a_{n}

be positive real numbers such that

a_{1}+...+a_{n}= 1

\frac{a_{1}}{1+a_{2}+\cdots +a_{n}}+\cdots +\frac{a_{n}}{1+a_{1}+a_{2}+\cdots +a_{n-1}}\geq \frac{n}{2n-1}