2007 Italy TST

Part of Italy TST

Subcontests

(3)

2

Nice functional equation

f: R \longrightarrow R

f(xy+f(x))=xf(y)+f(x)

for every pair of real numbers

x,y

Strange inequality on primes and factorization...

p \geq 5

be a prime.
(a) Show that exists a prime

q \neq p

q| (p-1)^{p}+1

(b) Factoring in prime numbers

(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}

\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2

2

Cyclical 3-subsets

In a competition, there were

2n+1

teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team

{ A,B,C }

A

B

B

C

C

A

. (a) Find the minimum of cyclical 3-subset (depending on

n

); (b) Find the maximum of cyclical 3-subset (depending on

n

Triangle of circumcenters...

ABC

a acute triangle. (a) Find the locus of all the points

P

such that, calling

O_{a}, O_{b}, O_{c}

the circumcenters of

PBC

PAC

PAB

\frac{ O_{a}O_{b}}{AB}= \frac{ O_{b}O_{c}}{BC}=\frac{ O_{c}O_{a}}{CA}

(b) For all points

P

of the locus in (a), show that the lines

AO_{a}

BO_{b}

CO_{c}

are cuncurrent (in

X

); (c) Show that the power of

X

wrt the circumcircle of

ABC

-\frac{ a^{2}+b^{2}+c^{2}-5R^{2}}4

a=BC

b=AC

c=AB

2

Locus of centers of rectangles

ABC

an acute triangle. (a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of

ABC

; (b) Determine if exist some points that are centers of

3

distinct rectangles whose vertices lie on the sides of

ABC

An easy one about coloring graphs

We have a complete graph with

n

vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?