MathDB

p1. One hundred elements can be chosen from

\{1,2,...,200\}

, so that no one element chosen is divisor of another chosen element. Give an example of this choice, and prove that this is impossible for

101

elements.
p2. Prove that the equation

x + 1990y = z^2

has an infinite number of natural solutions.
p3. Find the quantity of natural numbers, such that none of their digits is equal to

1

, and that the product of its digits is equal to

48

.
p4. Given a sheet of material flexible, as in the first figure, and gluing the edges that have the same letters as indicated by the cast, we get a bull. What is obtained by the same procedure, starting from the second figure? https://cdn.artofproblemsolving.com/attachments/d/f/66d4242403a1d917352c58dafe4d95794bd52f.png
p5. The sequence

(C_n)

n> 0

of integers is defined by the relations:

\bullet

C_0 = 0

\bullet

C_{2n} = C_n

\bullet

C_{2n + 1} = 1-C_n

Determine

C_{1991}

.
p6. Find a finite sequence

C_0, C_1,..., C_{1991}

of naturals, check the following condition for everything

a\in \{0,1,..., 1991\}

C_a

is the number of numbers

a

in the sequence.
p7. In a

\vartriangle ABC

, with orthocenter

H

, let

A'

B'

C'

be the midpoints of the sides, and

A''

B''

C''

be the midpoints of the segments that join the orthocenter with the vertices. Prove that points

A'

B'

C'

A''

B''

C''

are concyclic.

Problem Statement