MathDB

1

15 teams in a soccer league

15

teams participate in a soccer league. Each team plays each of the remaining teams exactly once. If a team beats another team in a match they receive

3

points, while the loser receives

1

point. In the event of a tie, both teams receive

2

points. When all possible league matches are held, the following can be observed:

\bullet

No two teams have finished with the same amount of points.

\bullet

Each team finished the league with at least

21

points.
Let

W

be the team that finished the league with the highest score. Determine how many points

W

scored and show that there were at least four ties in the league.

6

1

QE // BC wanted, <ACB=60^o, AE / BD = BC / CE -1, 2 circumcircles

Let

\vartriangle ABC

be a triangle with

\angle ACB = 60º

. Let

E

be a point inside

\overline {AC}

such that

CE <BC

. Let

D

over

\overline {BC}

such that

\frac {AE} {BD} = \frac {BC} {CE} -1 .

Let us call

P

the intersection of

\overline {AD}

with

\overline {BE}

and

Q

the other point of intersection of the circumcircles of the triangles

AEP

and

BDP

. Prove that

QE \parallel BC

.

5

1

gifts among 5 superheroes and sorcerer Vicencio

The vigilantes are a group of five superheroes, such that each one has one and only one of the following powers: hypnosis, super speed, shadow manipulation, immortality and super strength (each has a different power). On an adventure to the island of Philippines, they meet the sorcerer Vicencio, an old wise man who offers them the following ritual to help them: The ritual consists of a superhero

A

acquiring the gift (s) of

B

without

B

acquiring the gift (s) of

A

.
Determine the fewest number of rituals to be performed by the sorcerer Vicencio so that each superhero controls each of the five gifts.
Clarification: At the end of the ritual, a superhero

A

will have his gifts and those of a superhero

B

, but

B

does not acquire those of

A

, but it does keep its own.

4

1

grasshopper jumps on the real line from 0 to 2010

Let

a_1 <a_2 <... <a_n

consecutive positive integers (with

n> 2

). A grasshopper jumps on the real line, starting at point

0

and jumping

n

to the right with lengths

a_1

,

a_2

, ...,

a_n

, in some order (each length occupies exactly once), ending your tour at the

2010

point. Find all the possible values

n

of jumps that the grasshopper could have made.

3

1

fatal numbers , numbers on vertices of regular 4k+1 polygon

Let

P

be a regular polygon with

4k + 1

sides (where

k

is a natural) whose vertices are

A_1, A_2, ..., A_ {4k + 1}

(in that order ). Each vertex

A_j

of

P

is assigned a natural of the set

\{1,2, ..., 4k + 1 \}

such that no two vertices are assigned the same number. On

P

the following operation is performed: Let

B_j

be the midpoint of the side

A_jA_ {j + 1}

for

j = 1,2, ..., 4k + 1

(where is consider

A_ {4k + 2} = A_1

). If

a

,

b

are the numbers assigned to

A_ {j}

and

A_ {j + 1}

, respectively, the midpoint

B_j

is written the number

7a-3b

. By doing this with each of the

4k + 1

sides, the

4k + 1

vertices initially arranged are erased.
We will say that a natural

m

is fatal if for all natural

k

, no matter how the vertices of

P

are initially arranged, it is impossible to obtain

4k + 1

equal numbers through a finite amount of operations from

m

.
a) Determine if the

2010

is fatal or not. Justify. b) Prove that there are infinite fatal numbers.
[color=#f00]PS. A help in translation of the 2nd paragraph is welcome. [hide=Original wording]Diremos que un natural

m

es fatal si no importa cómo se disponen inicialmente los vértices de

{P}

, es imposible obtener mediante una cantidad finita de operaciones

4k+1

números iguales a

m

.

2

1

equilateral wanted, FP = EH . cyclic quadrilateral, altitudes , orthocenter

In an acute angle

\vartriangle ABC

, let

AD, BE, CF

be their altitudes (with

D, E, F

lying on

BC, CA, AB

, respectively). Let's call

O, H

the circumcenter and orthocenter of

\vartriangle ABC

, respectively. Let

P = CF \cap AO

. Suppose the following two conditions are true:

\bullet

FP = EH

\bullet

There is a circle that passes through points

A, O, H, C

Prove that the

\vartriangle ABC

is equilateral.

1

(2009 ^ {2009 ^ {2009}}) wanted for NT functional , f(a/b)-|f(b)-f(a)|

Let

f (n)

be a function that fulfills the following properties:

\bullet

For each natural

n

,

f (n)

is an integer greater than or equal to

0

.

\bullet

f (n) = 2010

, if

n

ends in

7

. For example,

f (137) = 2010

.

\bullet

If

a

is a divisor of

b

, then:

f \left(\frac {b} {a} \right) = | f (b) -f (a) |

.
Find

\displaystyle f (2009 ^ {2009 ^ {2009}})

and justify your answer.

OIFMAT I 2010

Subcontests

15 teams in a soccer league

QE // BC wanted, <ACB=60^o, AE / BD = BC / CE -1, 2 circumcircles

gifts among 5 superheroes and sorcerer Vicencio

grasshopper jumps on the real line from 0 to 2010

fatal numbers , numbers on vertices of regular 4k+1 polygon

equilateral wanted, FP = EH . cyclic quadrilateral, altitudes , orthocenter

(2009 ^ {2009 ^ {2009}}) wanted for NT functional , f(a/b)-|f(b)-f(a)|

OIFMAT I 2010

Subcontests

15 teams in a soccer league

QE // BC wanted, &lt;ACB=60^o, AE / BD = BC / CE -1, 2 circumcircles

gifts among 5 superheroes and sorcerer Vicencio

grasshopper jumps on the real line from 0 to 2010

fatal numbers , numbers on vertices of regular 4k+1 polygon

equilateral wanted, FP = EH . cyclic quadrilateral, altitudes , orthocenter

(2009 ^ {2009 ^ {2009}}) wanted for NT functional , f(a/b)-|f(b)-f(a)|

QE // BC wanted, <ACB=60^o, AE / BD = BC / CE -1, 2 circumcircles