1987 Poland - Second Round

Part of Poland - Second Round

Subcontests

(6)

1

similar quads wanted, 4 circumcenters related

We assign to any quadrilateral

ABCD

the centers of the circles circumscribed in the triangles

BCD

CDA

DAB

ABC

. Prove that if the vertices of a convex quadrilateral

Q

do not lie on the circle, then
a) the four points assigned to quadrilateral Q in the above manner are the vertices of the convex quadrilateral. Let us denote this quadrilateral by

t(Q)

,
b) the vertices of the quadrilateral

t(Q)

do not lie on the circle,
c) quadrilaterals

Q

t(t(Q)

1

p^x-y^3 = 1

Determine all prime numbers

p

and natural numbers

x, y

p^x-y^3 = 1

.

1

2 diff. common roots for x^4 + 2ax^2 + 4bx + a^2 and x^3 + ax - b

Determine all pairs of real numbers

a, b

for which the polynomials

x^4 + 2ax^2 + 4bx + a^2

x^3 + ax - b

have two different common real roots.

1

1000 x 1000 chessboard

On a chessboard with dimensions 1000 by 1000 and squares colored in the usual way in white and black, there is a set A consisting of 1000 squares. Any two fields of set A can be connected by a sequence of fields of set A so that subsequent fields have a common side. Prove that there are at least 250 white fields in set A.

1

congruent faces in tetrahedron criterion

Prove that the sum of the plane angles at each of the vertices of a given tetrahedron is

180^{\circ}

if and only if all its faces are congruent.

1

numbered balls

From an urn containing one ball marked with the number 1, two balls marked with the number 2, ...,

n

balls marked with the number

n

we draw two balls without replacement. We assume that each ball is equally likely to be drawn from the urn. Calculate the probability that both balls drawn have the same number.