MathDB

2

circum- and excircle intersections

BD

and

CE

be the bisectors of the interior angles

\angle B

and

\angle C

, respectively (

D\in AC

,

E\in AB

). Consider the circumcircle of

ABC

with center

O

and the excircle corresponding to the side

BC

with center

I_a

. These two circles intersect at points

P

and

Q

.
(a) Prove that

PQ

is parallel to

DE

. (b) Prove that

I_aO

is perpendicular to

DE

.

linear recurrence, finite difference divisible by 1999

A sequence

a_n

is defined by

a_0=0,\qquad a_1=3;

a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.

Find the least positive integer

h

such that

a_{n+h}-a_n

is divisible by

1999

for all

n\ge0

.

Problem 2

2

polynomials

If

a,b,c,d

are Distinct Real no. such that

a = \sqrt{4+\sqrt{5+a}}

b = \sqrt{4-\sqrt{5+b}}

c = \sqrt{4+\sqrt{5-c}}

d = \sqrt{4-\sqrt{5-d}}

Then

abcd =

geometric inequality with side ratios

In a triangle

ABC

, the bisector of the angle at

A

of a triangle

ABC

intersects the segment

BC

and the circumcircle of

ABC

at points

A_1

and

A_2

, respectively. Points

B_1,B_2,C_1,C_2

are analogously defined. Prove that

\frac{A_1A_2}{BA_2+CA_2}+\frac{B_1B_2}{CB_2+AB_2}+\frac{C_1C_2}{AC_2+BC_2}\ge\frac34.

Problem 5

1

2^n - 1 divides m^2 + 9

(a) If

m, n

are positive integers such that

2^n-1

divides

m^2 + 9

, prove that

n

is a power of

2

; (b) If

n

is a power of

2

, prove that there exists a positive integer

m

such that

2^n-1

divides

m^2 + 9

.

Problem 4

2

functions f: Q+ -> Z

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

coloring a polyhedron

Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge.
(a) Describe one polyhedron with the above property. (b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.

Problem 1

2

existence of a set of distinct powers

Find all positive integers n with the following property: There exists a positive integer

k

and mutually distinct integers

x_1,x_2,\ldots,x_n

such that the set

\{x_i+x_j\mid1\le i<j\le n\}

is a set of distinct powers of

k

.

number of distinct prime divisors of n

For a positive integer n, let

w(n)

denote the number of distinct prime divisors of n. Determine the least positive integer k such that

2^{w(n)} \leq k \sqrt[4]{n}

for all positive integers n.

1999 Brazil Team Selection Test

Subcontests

circum- and excircle intersections

linear recurrence, finite difference divisible by 1999

polynomials

geometric inequality with side ratios

2^n - 1 divides m^2 + 9

functions f: Q+ -> Z

coloring a polyhedron

existence of a set of distinct powers

number of distinct prime divisors of n

1999 Brazil Team Selection Test

Subcontests

circum- and excircle intersections

linear recurrence, finite difference divisible by 1999

polynomials

geometric inequality with side ratios

2^n - 1 divides m^2 + 9

functions f: Q+ -&gt; Z

coloring a polyhedron

existence of a set of distinct powers

number of distinct prime divisors of n

functions f: Q+ -> Z