1988 Vietnam National Olympiad

Part of Vietnam National Olympiad

Subcontests

(3)

2

Partitioned plane

The plane is partitioned into congruent equilateral triangles such that any two of them which are not disjoint have either a common vertex or a common side. Is there a circle containing exactly 1988 points in its interior?

Three pairwise skew lines in space

a

b

c

be three pairwise skew lines in space. Prove that they have a common perpendicular if and only if

S_a \circ S_b \circ S_c

is a reflection in a line, where

S_x

denotes the reflection in line

x

2

Determine the coefficients

Suppose P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0 be a real polynomial of degree

n > 2

with a_n \equal{} 1, a_{n\minus{}1} \equal{} \minus{}n, a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2} such that all the roots of

P

are real. Determine the coefficients

a_i

tan A, tan B, tan C are roots of an equation

ABC

is an acute triangle such that

\tan A

\tan B

\tan C

are the three roots of the equation x^3 \plus{} px^2 \plus{} qx \plus{} p \equal{} 0, where

q\neq 1

. Show that p \le \minus{} 3\sqrt 3 and

q > 1

2

Birds and cages

1988

994

cages, two in each cage. Every day we change the arrangement of the birds so that no cage contains the same two birds as ever before. What is the greatest possible number of days we can keep doing so?

Prove that the sequence has a finite limit

A bounded sequence

(x_n)_{n\ge 1}

of real numbers satisfies x_n \plus{} x_{n \plus{} 1} \ge 2x_{n \plus{} 2} for all

n \ge 1

. Prove that this sequence has a finite limit.