1999 Turkey Team Selection Test

Part of Turkey Team Selection Test

Subcontests

(3)

2

f(x-1-f(x)) = f(x)-x-1

Determine all functions

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

such that the set

\left \{ \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}

is finite, and for all

x \in \mathbb{R}

f(x-1-f(x)) = f(x) - x - 1

Covering plane with parabolas

Prove that the plane is not a union of the inner regions of finitely many parabolas. (The outer region of a parabola is the union of the lines not intersecting the parabola. The inner region of a parabola is the set of points of the plane that do not belong to the outer region of the parabola)

2

Midpoints of diagonals in cyclic quadrilateral

L

N

be the mid-points of the diagonals

[AC]

[BD]

of the cyclic quadrilateral

ABCD

, respectively. If

BD

is the bisector of the angle

ANC

, then prove that

AC

is the bisector of the angle

BLD

Gossip

A

B

C

D

E

F

knows a piece of gossip. They communicate by telephone via a central switchboard, which can connect only two of them at a time. During a conversation, each side tells the other everything he or she knows at that point. Determine the minimum number of calls for everyone to know all six pieces of gossip.

2

p-expansions of m and n

m \leq n

be positive integers and

p

be a prime. Let

p-

m

n

m = a_0 + a_1p + \dots + a_rp^r

n = b_0 + b_1p + \dots + b_sp^s

respectively, where

a_r, b_s \neq 0

i \in \{0,1,\dots,r\}

j \in \{0,1,\dots,s\}

0 \leq a_i, b_j \leq p-1

a_i \leq b_i

i \in \{0,1,\dots,r\}

m \prec_p n

p \nmid {{n}\choose{m}} \Leftrightarrow m \prec_p n

Area/(Perimeter)^2 of cyclic quadrilateral

Let the area and the perimeter of a cyclic quadrilateral

C

A_C

P_C

, respectively. If the area and the perimeter of the quadrilateral which is tangent to the circumcircle of

C

at the vertices of

C

A_T

P_T

, respectively, prove that

\frac{A_C}{A_T} \geq \left (\frac{P_C}{P_T}\right )^2