1989 Polish MO Finals

Part of Polish MO Finals

Subcontests

(3)

2

Labeling edges of a cube

The edges of a cube are labeled from

1

12

. Show that there must exist at least eight triples

(i, j, k)

1 \leq i < j < k \leq 12

so that the edges

i, j, k

are consecutive edges of a path. Also show that there exists labeling in which we cannot find nine such triples.

Inequality in 4 variables

Show that for positive reals

a, b, c, d

\left(\dfrac{ab + ac + ad + bc + bd + cd}{6} \right)^3 \geq \left(\dfrac{abc + abd + acd + bcd}{4}\right)^2

2

Three circles touching externally

k_1, k_2, k_3

are three circles.

k_2

k_3

touch externally at

P

k_3

k_1

touch externally at

Q

k_1

k_2

touch externally at

R

PQ

k_1

S

PR

k_1

T

RS

k_2

U

QT

k_3

V

P, U, V

Sphere and 3 circles

Three circles of radius

a

are drawn on the surface of a sphere of radius

r

. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.

2

Expected number of elements of a subset

n, k

are positive integers.

A_0

\{1, 2, ... , n\}

A_i

is a randomly chosen subset of

A_{i-1}

(with each subset having equal probability). Show that the expected number of elements of

A_k

\dfrac{n}{2^k}

A meeting and break

An even number of politicians are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Show that there must be two people with the same number of people sitting between them as before the break.. Additional problem: Solve the problem when the number of people is in a form

6k+3