MathDB

p1. The figure shows the triangle

ABC

, right at

C

, its circle circumscribed and semicircles built on the two legs. If

AB = 5

AC = 4

, and

BC = 3

, find the sum of the areas of the two shaded regions. https://cdn.artofproblemsolving.com/attachments/b/1/bb5d58b24d2dc68efcd6aa218489fd379f709c.png
p2. Find the greatest power of

3

that divides

10^{2012} - 1

.
p3. In how many different ways can a board of

2 \times 20

squares be covered by

1\times 1

, with

2 \times 1

domino pieces, so that the pieces do not overlap or stick out of the board?
p4. In a certain game there are several piles of stones that can be modified according to the following rules: a) Two of the piles can be put together into one. b) If a pile has an even number of stones, it can be divided into two piles with the same number of stones each. At the beginning there are three piles, one of them has

5

stones, another has

49

and another has

51

. Determine if it is possible to achieve, with successive movements, and following rules a) and b), that at the end there are

105

piles, each one with a stone.
p5. Each vertex of a cube is assigned the value

+1

-1

, and each face the product of the values assigned to its vertices. What values can the sum of the

14

numbers thus obtained, have?
p6. The quadrilateral

ABCD

in the figure is a trapezoid, and we have

EF\parallel AB\parallel CD

. Furthermore,

EF

passes through point

G

, where the diagonals

AC

and

BD

intersect. The lengths of

AB

and

CD

are known to be, respectively,

12

and

4

. Find the length of

EF

. https://cdn.artofproblemsolving.com/attachments/5/7/968e8cc769780daad52f421aea8f35bbaf05c5.png
PS. Juniors P1,P3 were also posted as [url=https://artofproblemsolving.com/community/c4h2693420p23386115]Seniors P3, P4.

Problem Statement