2003 Cuba MO

Part of Cuba MO

Subcontests

(9)

1

cuban concurrency , statring with acute isosceles

D

be the midpoint of the base

AB

of the isosceles and acute angle triangle

ABC

E

AB

O

circumcenter of the triangle

ACE

. Prove that the line that passes through

D

perpendicular to

DO

, the line that passes through

E

perpendicular to

BC

and the line that passes through

B

AC

, they intersect at a point.

1

f(uv) = f(u)f(v), f(au) = |a | f(u), f(u) + f(v) <=|u| + |v|, f : C -> R^+

Find all the functions

f : C \to R^+

such that they fulfill simultaneously the following conditions:

(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C

(ii) \ \ f(au) = |a | f(u) \ \ \forall a \in R, u \in C

(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C

1

S(S(S(2003^{2003})))., sum of digits

Let S(n) be the sum of the digits of the positive integer

n

S(S(S(2003^{2003}))).

1

4 incenters for a rectangle, in cyclic - 2003 Cuba MO 2.6

P_1, P_2, P_3, P_4

be four points on a circle, let

I_1

be incenter of the triangle of vertices

P_2P_3P_4

I_2

the incenter of the triangle

P_1P_3P_4

I_3

the incenter of the triangle

P_1P_2P_4

I_4

the incenter of the triangle

P_2P_3P_1

I_1I_2I_3I_4

is a rectangle.

1

a_{i-1} + a_{i+1} < 2a_i

a_1, a_2, ..., a_9

be non-negative real numbers such that

a_1 = a_9 = 0

and at least one of the remaining terms is different from

0

. a) Prove that for some

i

(i = 2, ..., 8

a_{i-1} + a_{i+1} < 2a_i.

b) Will the previous statement be true, if we change the number

2

1.9

in the inequality?

1

f(ab) =bf(a) + af(b), f(p) = 1, n/ gcd(n,f(n)) is square free

f : N \to N

f(p) = 1

for all p prime and

f(ab) =bf(a) + af(b)

a, b \in N

. Prove that if

n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k

is the canonical distribution of

n

p_i

does not divide

a_i

i = 1, 2, ..., k

\frac{n}{gcd(n,f(n))}

is square free (not divisible by a square greater than

1

2

bw in 4x4 board

4 \times 4

board has all its squares painted white. An allowed operation is to choose a rectangle that contains

3

squares and paint each of the boxes as follows: a) If the box is white then it is painted black. b) If the box is black then it is painted white. Prove that by applying the allowed operation several times, it is impossible get the entire board painted black.

(MNP) cuts triangle ABC and froms equilateral - 2003 Cuba MO 2.3

ABC

be an acute triangle and

T

be a point interior to this triangle. that

\angle ATB = \angle BTC = \angle CTA

M,N

P

be the feet of the perpendiculars from

T

BC

CA

AB

respectively. Prove that if the circle circumscribed around

\vartriangle MNP

cuts again the sides

BC

CA

AB

M_1

N_1

P_1

respectively, then the

\vartriangle M_1N_1P_1

It is equilateral.

2

1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003

Given the following list of numbers:

1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003

where the number

2003

12

times. Is it possible to write these numbers in some order so that the

100

-digit number that we get is prime?

x^2 + (3a + b)x + a^2 + 2b^2 = 0, x^2 - 2a(3a + 2b)x + 5a^2b^2 + 4b^4 = 0

The roots of the equation

x^2 + (3a + b)x + a^2 + 2b^2 = 0

x_1

x_2

x_1 \ne x_2

. Determine the values of

a

b

so that the roots of the equation

x^2 - 2a(3a + 2b)x + 5a^2b^2 + 4b^4 = 0

x^2_1

x^2_2

2

<PEQ = 2<APB , circles, tangents, cuba - romania

A

be a point outside the circle

\omega

. The tangents from

A

touch the circle at

B

C

P

be an arbitrary point on extension of

AC

C

Q

the projection of

C

PB

E

the second intersection point of the circumcircle of

ABP

with the circle

\omega

\angle PEQ = 2\angle APB

p/q = 1-1/2+1/3+...+1/1335

\frac{p}{q}=1-\frac{1}{2} + \frac{1}{3}- \frac{1}{4} + ... -\frac{1}{1334} + \frac{1}{1335}

p, q \in Z_+

p

is divisible by

2003