1990 IberoAmerican

Part of IberoAmerican

Subcontests

(6)

1

Integer function and powers of two

f

be a function defined for the non-negative integers, such that: a)

f(n)=0

n=2^{j}-1

j \geq 0

f(n+1)=f(n)-1

otherwise. i) Show that for every

n \geq 0

k \geq 0

f(n)+n=2^{k}-1

f(2^{1990})

1

Seeds within an nxn board

A

B

are two opposite vertices of an

n \times n

board. Within each small square of the board, the diagonal parallel to

AB

is drawn, so that the board is divided in

2n^{2}

equal triangles. A coin moves from

A

B

along the grid, and for every segment of the grid that it visits, a seed is put in each triangle that contains the segment as a side. The path followed by the coin is such that no segment is visited more than once, and after the coins arrives at

B

, there are exactly two seeds in each of the

2n^{2}

triangles of the board. Determine all the values of

n

for which such scenario is possible.

1

A point of tangency, a locus and orthogonal circles

\Gamma_{1}

AB

\ell

is the tangent at

B

M

\Gamma_{1}

A

\Gamma_{2}

is a circle tangent to

\ell

\Gamma_{1}

M

. a) Determine the point of tangency

P

\ell

\Gamma_{2}

and find the locus of the center of

\Gamma_{2}

M

varies. b) Show that there exists a circle that is always orthogonal to

\Gamma_{2}

, regardless of the position of

M

1

Primes and values attained by a polynomial

b

c

be integer numbers, and define

f(x)=(x+b)^2-c

p

is a prime number such that

c

is divisible by

p

p^{2}

, show that for every integer

n

f(n)

is not divisible by

p^{2}

q \neq 2

be a prime divisor of

c

q

f(n)

for some integer

n

, show that for every integer

r

there exists an integer

n'

f(n')

is divisible by

qr

1

Concyclic points

ABC

I

is the incenter, and the incircle is tangent to

BC

CA

AB

D

E

F

, respectively.

P

is the second point of intersection of

AD

and the incircle. If

M

is the midpoint of

EF

P

I

M

D

1

About the graph of a polynomial

f(x)

be a cubic polynomial with rational coefficients. If the graph of

f(x)

is tangent to the

x

axis, prove that the roots of

f(x)

are all rational.