MathDB

p1. For a positive integer, let's call

<a>

the number obtained by multiplying each figure of a by

2

and writing the numbers thus obtained. For example

<126> = 2412

and

<809> = 16018

. Prove that it is not possible find two different positive integers

a

and

b

such that

<a> = <b>

.
p2. In how many ways is it possible to cut a graph paper of

6\times 6

starting at the bottom of the paper and working to the top if it can only be cut on the lines of the grid, the two pieces in which it is divided must be equal and it can't be cut down (see figure)? https://cdn.artofproblemsolving.com/attachments/2/1/c9d6a5f0cf7230aff23cde16fd5308e08eac7e.png
Note: Two pieces are considered the same if you can place one over the other and they fit perfectly.
p3. In an equilateral triangle

ABC

with side

2

, side

AB

is extended to a point

D

so that

B

is the midpoint of

AD

. Let

E

be the point on

AC

such that

\angle ADE = 15^o

and take a point

F

AB

so that

|EF| = |EC|

. Determine the area of the triangle

AFE

.
p4. For each positive integer

n

we consider

S (n)

as the sum of its digits. For example

S (1234) = 1 + 2 + 3 + 4 = 10

. Calculate

S (1)- S (2) + S (3)- S (4) +...- S (2012) + S (2013)- S (2014)

p5. Given

102

points on a circle, next to one of them is written a

1

and next to each other a

0

. The allowed operation consists of choosing a point that has a

1

and change the number of that point, and also the number of its two neighbors, the one on the left and on the right (where there is a

1

, write

0

and where there is a

0

is written

1

). Show that it is impossible, with permitted operations, to achieve that all the points have a

0

.
p6. Consider a convex figure

P

in the plane. For a point

Z

of the plane outside the figure we denote by

|Z, P|

the smallest length of the segments that join

Z

with some point of

P

. Consider a line

L

that does not intersect the figure

P

, two points

X, Y

over

L

, and

M

the midpoint between

X

and

Y

. Prove that

\frac{|X, P| + |Y, P|}{2} \ge |M, P|

.
PS. Seniors P3, P4 were also proposed as [url=https://artofproblemsolving.com/community/c4h2690804p23355868]Juniors P3, easier P4.

Problem Statement