2010 Olympic Revenge

Part of Olympic Revenge

Subcontests

(6)

1

Problem 6, Olympic Revenge 2010

ABC

to be a triangle and

\Gamma

its circumcircle. Also, let

D, F, G

E

, in this order, on the arc

BC

which does not contain

A

\angle BAD = \angle CAE

\angle BAF = \angle CAG

D`, F`, G`

E`

to be the intersections of

AD, AF, AG

AE

BC

, respectively. Moreover,

X

is the intersection of

DF`

EG`

Y

is the intersection of

D`F

E`G

Z

is the intersection of

D`G

E`F

W

is the intersection of

EF`

DG`

X, Y

A

are collinear, such as

W, Z

A

. Moreover, prove that

\angle BAX = \angle CAZ

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Problem 5, Olympic Revenge 2010

Secco and Ramon are drunk in the real line over the integer points

a

b

, respectively. Our real line is a little bit special, though: the interval

(-\infty, 0)

is covered by a sea of lava. Being aware of this fact, and also because they are drunk, they decided to play the following game: initially they choose an integer number

k>1

using a fair dice as large as desired, and therefore they start the game. In the first round, each player writes the point

h

for which it wants to go.
After that, they throw a coin: if the result is heads, they go to the desired points; otherwise, they go to the points

2g - h

g

is the point where each of the players were in the precedent round (that is, in the first round

g = a

g = b

for Ramon). They repeat this procedure in the other rounds, and the game finishes when some of the player is over a point exactly

k

times bigger than the other (if both of the player end up in the point

0

, the game finishes as well).
Determine, in values of

k

, the initial values

a

b

such that Secco and Ramon has a winning strategy to finish the game alive.
Observation: If any of the players fall in the lave, he dies and both of them lose the game

1

Problem 4, Olympic Revenge 2010

a_n

b_n

to be two sequences defined as below:

i)

a_1 = 1

ii)

a_n + b_n = 6n - 1

iii)

a_{n+1}

is the least positive integer different of

a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n

a_{2009}

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Problem 3, Olympic Revenge 2010

Prove that there exists a set

S

of lines in the three dimensional space satisfying the following conditions:

i)

P

in the space, there exist a unique line of

S

P

.

ii)

There are no two lines of

S

which are parallel.

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Problem 2, Olympic Revenge 2010

Joaquim, José and João participate of the worship of triangle

ABC

. It is well known that

ABC

is a random triangle, nothing special. According to the dogmas of the worship, when they form a triangle which is similar to

ABC

, they will get immortal. Nevertheless, there is a condition: each person must represent a vertice of the triangle. In this case, Joaquim will represent vertice

A

, José vertice

B

and João will represent vertice

C

. Thus, they must form a triangle which is similar to

ABC

, in this order.
Suppose all three points are in the Euclidean Plane. Once they are very excited to become immortal, they act in the following way: in each instant

t

, Joaquim, for example, will move with constant velocity

v

to the point in the same semi-plan determined by the line which connects the other two points, and which would create a triangle similar to

ABC

in the desired order. The other participants act in the same way. If the velocity of all of them is same, and if they initially have a finite, but sufficiently large life, determine if they can get immortal.
Observation: Initially, Joaquim, José and João do not represent three collinear points in the plane

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Problem 1, Olympic Revenge 2010

Prove that the number of ordered triples

(x, y, z)

(x+y+z)^2 \equiv axyz \mod{p}

gcd(a, p) = 1

p

p^2 + 1