MathDB

p1. During prehistoric times on a distant planet

S_{17}-T

30

tribes coexisted peacefully. The story began when a tribe - which was created to be the most intelligent and democratic - declared war. He defeated the war against the neighboring tribe and absorbed it, assimilating it completely (the losing tribe disappears).Some other tribes followed this example. According to the definition of the historians, a tribe becomes an empire after defeating and absorbing at least three other existing communities in his time (tribes or empires). What is the maximum number of empires that may have existed during all historical times on the planet

S_{17}-T

? (not necessarily simultaneously).
Note: An empire can continue to conquer other communities, but this does not make it a new empire.
p2. There are

9

positive integers, all of them with three digits and all different from each other. Prove that using some of these numbers, each number used at most once, another non-integer number

D

can be constructed such that

2 <D <3

. For this construction it is allowed to use the four operations of arithmetic.
[url=https://artofproblemsolving.com/community/c4h1846791p12438123]p3. Within a square,

2

different points are chosen. Then,

8

line segments are drawn connecting each of these points with the vertices of the square. Is it possible to divide this square into

9

parts with equal area?
p4. In each square of a rectangular board of

2005\times 2005

a number is inscribed (it can be be an integer or a fraction, positive or negative). It is known that the sum of any

4

numbers that can be covered by the figure

T

is always equal to

4

. Prove that each inscribed number is equal to

1

. https://cdn.artofproblemsolving.com/attachments/c/6/a6623418d5279f56a67f63764ffb02abd0998f.png
p5. The numbers

2^{2005}

and

5^{2005}

in their decimal expression are inscribed on a line one behind the other. How many digits are in this page?
p6. Show that there exists an integer of the form

200520052005...200500...00

which is divisible by

2006

.
[url=https://artofproblemsolving.com/community/c4h2917769p26063666]p7. Consider an convex quadrilateral. Using its sides as diameters,

4

circular discs are constructed. Prove that the disks completely cover the quadrilateral.

Problem Statement