1992 Vietnam Team Selection Test

Part of Vietnam Team Selection Test

Subcontests

(3)

2

Find the value of angle A'BC

ABC

a triangle be given with

BC = a

CA = b

AB = c

a \neq b \neq c \neq a

ABC

) take the points

A'

B'

C'

such that: I. The pairs of points

A

A'

B

B'

C

C'

either all lie in one side either all lie in different sides under the lines

BC

CA

AB

respectively; II. Triangles

A'BC

B'CA

C'AB

are similar isosceles triangles. Find the value of angle

A'BC

a, b, c

such that lengths

AA', BB', CC'

are not sides of an triangle. (The word "triangle" must be understood in its ordinary meaning: its vertices are not collinear.)

scientific conference participants speak 2n languages

In a scientific conference, all participants can speak in total

2 \cdot n

n \geq 2

). Each participant can speak exactly two languages and each pair of two participants can have at most one common language. It is known that for every integer

k

1 \leq k \leq n-1

there are at most

k-1

languages such that each of these languages is spoken by at most

k

participants. Show that we can choose a group from

2 \cdot n

participants which in total can speak

2 \cdot n

languages and each language is spoken by exactly two participants from this group.

2

|f(P(x))| <= c

Let a polynomial

f(x)

be given with real coefficients and has degree greater or equal than 1. Show that for every real number

c > 0

, there exists a positive integer

n_0

satisfying the following condition: if polynomial

P(x)

of degree greater or equal than

n_0

with real coefficients and has leading coefficient equal to 1 then the number of integers

x

|f(P(x))| \leq c

is not greater than degree of

P(x)

x^2 + y^2 - 5xy + 5 = 0

Find all pair of positive integers

(x, y)

satisfying the equation

x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.

2

difference is not divided by n

Let two natural number

n > 1

m

be given. Find the least positive integer

k

which has the following property: Among

k

arbitrary integers

a_1, a_2, \ldots, a_k

satisfying the condition

a_i - a_j

1 \leq i < j \leq k

) is not divided by

n

, there exist two numbers

a_p, a_s

p \neq s

m + a_p - a_s

n

every two circles touch each other

In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.