MathDB

2005 Chile Classification NMO Seniors

Part of Chile Classification NMO

Subcontests

(1)
1

2005 Chile Classification / Qualifying NMO Seniors XVII

p1. Vicente wrote 20052005 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 1717, or by 2323. If 8 8 is the last of the 20052005 digits, which is the first?
p2. Peoples A A, B B and CC are located in a valley (of the Pencahue commune, in the Seventh region). In A A live 1010 children of school age, while in B B live 2020, and in CC there are 3030 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 A, B B and CC form a triangle. Help regional authorities find a point in the map that satisfies the given condition.
p3. You have a total of 8080 coins, supposedly all gold, however exactly one of them is not and she has less weight than the remaining 7979 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 TT is an arrangement of squares as shown in the figure: Prove that a square 10×1010\times 10 cannot be divided into 2525 figures TT of the same size. https://cdn.artofproblemsolving.com/attachments/9/8/3e2540b091bd2f13b30dfe4fa7e2c681597f70.png
p5. In the plane there are two squares of 20×2020\times 20 cm2^2. 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 310310 cm2^2:
p6. Rita claims that she can break any rectangle into 100100 pieces in such a way that one can build a square using all of these 100100 pieces. Do you agree with this statement?