MathDB

p1. Prove that the numbers

49

4489

444889

...

, obtained by placing the number

48

in the middle of the previous one, are squares of integer numbers.
p2. Let's consider a cube with a side

18

cm. In the center of three different and non-opposite faces, we make a rectangular hole with a side

6

cm , which crosses to the opposite face, obtaining the figure on the right. Determine the area of this body. https://cdn.artofproblemsolving.com/attachments/9/e/f4ae672c26ad9d8f8e3493393fabc6f5544151.png
p3. Prove that it is possible to find a right angle and an isosceles triangle, both with sides integers, equal perimeter and equal area.
p4. Consider the nine regions

R_1., R_2, ..., R_9

that are formed by the five olympic rings, as a sample in the figure. In each region a number from

1

9

is placed, without repeating, so that the sum in each ring it's the same. What is the largest and the smallest value for this sum? https://cdn.artofproblemsolving.com/attachments/0/e/f7ed45eb647f411be5ae4bffbf2aef5da17f38.png
p5. Determine the period of

0,\overline{19} + 0,\overline{199}

and of

0,\overline{19}\cdot 0,\overline{199}

.
p6. Let

ABCD

be a square and

E

a point inside it, such that the

ECD

is isosceles at

C

. Prove that there exists a point

E

inside the square for which, also, the

AEB

is isosceles at

E

.
p7. Let us consider the set

S = \{1, 2,...,n\}

. Let

M_1, M_2,..., M_{n + 1}

, be non-empty subsets of

S

. Prove that there are different indices

r, s, r + s, i_1, i_2 ,... ,i_r,j_1, j_2, ..., j_s

such that

M_{i1}\cup ... \cup M_{ir} = M_{j1} \cup ... \cup M_{js}

Problem Statement