MathDB

1

sequence starting with x=1 and followed by (x+1) and / or x/(x+1)

x

has two sons:

x+1

and

\frac{x}{x+1}

. Characterize all the descendants of number

1

.

5

1

for which n exists a convex polyhedron with n edges?

For which values of

n

does there exist a convex polyhedron with

n

edges?

4

1

4 black squares in every row and column of a 8x8 board (bw)

The squares of an

8 \times 8

board are colored black and white in such a way that every row and every column contains exactly four black squares. Prove that the number of pairs of neighboring white squares is the same as the number of pairs of neighboring black squares. (Two squares are neighboring if they have a side in common.)

3

1

every integer can be written in the form \pm 1 \pm 4 \pm 9\pm ... \pm n^2

We consider the sums of the form

\pm 1 \pm 4 \pm 9\pm ... \pm n^2

. Show that every integer can be represented in this form for some

n

. (For example,

3 = -1 + 4

and

8 = 1-4-9+16+25-36-49+64

.)

2

1

a^3+3a^2+a is never a perfect square for positive integer a

Prove that no number of the form

a^3+3a^2+a

, for a positive integer

a

, is a perfect square.

1

triangle of intersections of angle bisectors with circumcircle

For every triangle

ABC

inscribed in a circle

\Gamma

, let

A',B',C'

be the intersections of the bisectors of the angles at

A,B,C

with

\Gamma

. Consider the triangle

A'B'C'

. (a) Do triangles

A'B'C'

go over all possible triangles inscribed in

\Gamma

as

\vartriangle ABC

varies? If not, what are the constraints? (b) Prove that the angle bisectors of

\vartriangle ABC

are the altitudes of

\vartriangle A',B',C'

.

1991 ITAMO

Subcontests

sequence starting with x=1 and followed by (x+1) and / or x/(x+1)

for which n exists a convex polyhedron with n edges?

4 black squares in every row and column of a 8x8 board (bw)

every integer can be written in the form \pm 1 \pm 4 \pm 9\pm ... \pm n^2

a^3+3a^2+a is never a perfect square for positive integer a

triangle of intersections of angle bisectors with circumcircle