2017 Greece Team Selection Test

Part of Greece Team Selection Test

Subcontests

(4)

1

Numbers in a board, maximum value

Some positive integers are initially written on a board, where each

2

of them are different. Each time we can do the following moves: (1) If there are 2 numbers (written in the board) in the form

n, n+1

we can erase them and write down

n-2

(2) If there are 2 numbers (written in the board) in the form

n, n+4

we can erase them and write down

n-1

After some moves, there might appear negative numbers. Find the maximum value of the integer

c

such that: Independetly of the starting numbers, each number which appears in any move is greater or equal to

c

1

Functional equation with 2 fuctions

Find all fuctions

f,g:\mathbb{R}\rightarrow \mathbb{R}

f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}

g(1)=-8

1

Integer and factors

Prove that the number

A=\frac{(4n)!}{(2n)!n!}

is an integer and divisible by

2^{n+1}

n

is a positive integer.

1

Geometry: Concurrent lines and concyclic points.

ABC

be an acute-angled triangle inscribed in circle

c(O,R)

AB<AC<BC

c_1

be the inscribed circle of

ABC

which intersects

AB, AC, BC

F, E, D

respectivelly. Let

A', B', C'

be points which lie on

c

such that the quadrilaterals

AEFA', BDFB', CDEC'

are inscribable. (1) Prove that

DEA'B'

is inscribable. (2) Prove that

DA', EB', FC'

are concurrent.