MathDB

p1. Vicente wrote

2005

digits on a single line with the following characteristic, each pair of adjacent digits on the line, taken in the order in which they are written, form a divisible number always or by

17

, or by

23

. If

8

is the last of the

2005

digits, which is the first?
p2. Peoples

A

B

and

C

are located in a valley (of the Pencahue commune, in the Seventh region). In

A

live

10

children of school age, while in

B

live

20

, and in

C

there are

30

children with the same characteristic. The decision was made to build a school for these children. To find a suitable place for the construction, it was decided or to opt for that place that meant the shortest journey total made by all students during a school day (children go to and from their school to foot). Now, when analyzing the geographical position of these towns on the map, it was discovered that the points

A

B

and

C

form a triangle. Help regional authorities find a point in the map that satisfies the given condition.
p3. You have a total of

80

coins, supposedly all gold, however exactly one of them is not and she has less weight than the remaining

79

that have the same weight. Indicate how to identify the smallest currency using a rudimentary scale that only allows you to compare weights. It is allowed to use the balance only four times.
p4. A figure

T

is an arrangement of squares as shown in the figure: Prove that a square

10\times 10

cannot be divided into

25

figures

T

of the same size. https://cdn.artofproblemsolving.com/attachments/9/8/3e2540b091bd2f13b30dfe4fa7e2c681597f70.png
p5. In the plane there are two squares of

20\times 20

^2

. They don't match but their four diagonals they intersect at the same point. Prove that the area with one of these squares is always greater than

310

^2

:
p6. Rita claims that she can break any rectangle into

100

pieces in such a way that one can build a square using all of these

100

pieces. Do you agree with this statement?

Problem Statement