2012 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

FInd all f such that f(n!)=f(n)! and m-n divides f(m)-f(n)

\mathbb{Z}^+

be the set of positive integers. Find all functions

f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+

such that the following conditions both hold: (i)

f(n!)=f(n)!

for every positive integer

n

m-n

f(m)-f(n)

m

n

are different positive integers.

1

Approximating with sums of subsets of a geometric series

n

be a positive integer. Let

P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.

For each subset

X

P_n

S_X

for the sum of all elements of

X

, with the convention that

S_{\emptyset}=0

\emptyset

is the empty set. Suppose that

y

is a real number with

0 \leq y \leq 3^{n+1}-2^{n+1}.

Prove that there is a subset

Y

P_n

0 \leq y-S_Y < 2^n

1

Inequality involving x, y and z

\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),

for all positive real numbers

x,y

z

1

Two circles are tangent at F

A

B

C

be points lying on a circle

\Gamma

O

\angle ABC > 90

D

be the point of intersection of the line

AB

with the line perpendicular to

AC

C

l

be the line through

D

which is perpendicular to

AO

E

be the point of intersection of

l

AC

F

be the point of intersection of

\Gamma

l

that lies between

D

E

. Prove that the circumcircles of triangles

BFE

CFD

F