MathDB

p1. Katia and Mariela play the following game: In each of their three turns, Katia replaces one of the stars in the expression

\star \star\star\star\star\star\star\star\star

for some a digit of between

1

2

3

4

5

6

7

8

9

that has not been used before in the game. In shifts of Mariela, she replaces two of the stars with two different digits that have not been used. Katia starts playing and plays alternately. Mariela wins if the resulting number at the end of the game is divisible by

27

. Does Mariela have any to ensure the triumph?
p2. Four line segments divide a rectangle into

8

regions, such as illustrated in the figure. For the three marked regions, the area is as follows:

3

5

and

8

. How much is the area of the gray quadrilateral? https://cdn.artofproblemsolving.com/attachments/b/5/60d70b8f12fcc57e63a1d221fbe8e5a324d090.png
p3. In the interior of a square with side

1

2016

points are marked. Prove that it is possible, regardless of the position of the points, to join them all using a path continuous whose length is not greater than

146

.
p4. Find all pairs of prime numbers

(p, q)

for which the following equality be true

7pq^2 + p = q^3 + 43p^3 + 1

PS. Senior p1 was also [url=https://artofproblemsolving.com/community/c4h2690898p23356626]Juniors p2.

Problem Statement