MathDB

p1. In the figure,

ABCD

and

EFGH

are squares with

AB = 2

cm.

E

M

and

N

are midpoints of the sides

AD

AB

and

CD

respectively. Find the area of the pentagon

AFGHD

. https://cdn.artofproblemsolving.com/attachments/6/1/800b099849b9aaf38bee0ddf492ffd248bfb6a.png
p2. For some positive integer

n

that does not contain

0

as a digit, we define

f (n)

as the number that It results from arranging the digits of

n

in reverse order. For example:

f(1735) = 5371

f(43) = 34

f (22) = 22

. It is said that

n

is salsero if

n > f (n)

and furthermore, the number

n - f (n)

results from ordering the digits of

n

in some way. Find all salsero numbers less than

2023

.
p3. Ali has

2023

cube-shaped blocks, which have a number written on each face, so that in every block there are three faces with the numbers

1

2

and

-3

, and each pair of opposite faces has equal numbers. There he prepares to build a snake using these blocks. Starts by placing a block on the ground and then repeats the following procedure for each step: pick up a block and decide whether to glue it above, in front or to the right of the last placed block, so that the glued faces contain equal numbers. The figure shows an example of how he could have glued the first

6

blocks. https://cdn.artofproblemsolving.com/attachments/8/e/6a6045312329a715b973015bc9792befd43415.png Ali continues in this way until all the blocks have been used. When the snake is finished, he lifts it off the ground and realizes that the sum of the numbers of all the visible faces is

4046

. Determine the minimum number of times two faces could have been glued together with the number

-3

in them.
p4. Let

a

and

b

be positive integers such that

b^2 > 9a

and gcd

(a, b) = 1

. Prove that if

a(a + b)

is a perfect square, then

b

is a composite number.
p5. The supreme leader of the kingdom of Camelot owns

30

gold bars and decides to give them to his two sons Arthur and Morgause. Since they do not know how to distribute the bars, the king proposes the following game: Morgause distributes the

30

bars into three boxes so that each box contains at least two bars and there are two boxes such that one has exactly

6

more bars than the other. Next Arturo (who does not know how many bars Morgause put in each box), must label each box with a number from

1

30

, not necessarily different. Let

\ell

be the number of bars inside a certain box and

n

the number written by Arturo on it. Then, if

n \le \ell

, then Arthur takes

n

bars of this box and Morgause keeps the

\ell- n

remaining bars. Otherwise, Morgause takes the

\ell

bars from the box. Arturo performs the previous procedure for each of the three boxes. Determine the maximum number of bars that Arturo can obtain, regardless of the initial distribution made by Morgause, and describe how it can reach said maximum.
PS. You should use hide for answers.

Problem Statement