MathDB

1

sum 1/ h_i^2 = sum 1/ d_i^2 in tetrahedron

h_1,h_2,h_3,h_4

are the altitudes of a tetrahedron and

d_1,d_2,d_3

the distances between the pairs of opposite edges of the tetrahedron, then

\frac{1}{h_1^2} +\frac{1}{h_2^2} +\frac{1}{h_3^2} +\frac{1}{h_4^2} =\frac{1}{d_1^2} +\frac{1}{d_2^2} +\frac{1}{d_3^2}.

5

1

a_{n+1}=1/2 (a_n - 1/ a_n)

For a given real number

a

, define the sequence

(a_n)

by

a_1 = a

and

a_{n+1} =\begin{cases} \dfrac12 \left(a_n -\dfrac{1}{a_n}\right) \,\,\, if \,\,\, a_n \ne 0, \\ 0 \,\,\, if \,\,\, a_n = 0 \end{cases}

Prove that the sequence

(a_n)

contains infinitely many nonpositive terms.

4

1

sum of numbers of elements of intersections of 2 subsets of set of n elements

Let

X

be a set of

n

elements. Prove that the sum of the numbers of elements of sets

A\cap B

, where

A

and

B

run over all subsets of

X

, is equal to

n4^{n-1}

.

3

1

x^{2m}P(x,y)+y^{2m}Q(x,y) = (x+y)^{2m}R(x,y)

Prove that if

m

is a natural number and

P,Q,R

polynomials of degrees less than

m

satisfying

x^{2m}P(x,y)+y^{2m}Q(x,y) = (x+y)^{2m}R(x,y),

then each of the polynomials is zero.

2

1

partition lattice points into pairs / infinite chessboard

In a coordinate plane, consider the set of points with integer cooedinates at least one of which is not divisible by

4

. Prove that these points cannot be partitioned into pairs such that the distance between points in each pair equals

1

. In other words, an infinite chessboard, whose cells with both coordinates divisible by

4

are cut out, cannot be tiled by dominoes.

1

ray of light reflects from the rays of a given angle

A ray of light reflects from the rays of a given angle. A ray that enters the vertex of the angle is absorbed. Prove that there is a natural number

n

such that any ray can reflect at most

n

times

1978 Polish MO Finals

Subcontests

sum 1/ h_i^2 = sum 1/ d_i^2 in tetrahedron

a_{n+1}=1/2 (a_n - 1/ a_n)

sum of numbers of elements of intersections of 2 subsets of set of n elements

x^{2m}P(x,y)+y^{2m}Q(x,y) = (x+y)^{2m}R(x,y)

partition lattice points into pairs / infinite chessboard

ray of light reflects from the rays of a given angle