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Part of 2021 May Olympiad

Problems(2)

numbers 1-13 on vertices of 13-gon

Source: 2021 May Olympiad 2021 L2 p4

At each vertex of a

13

-sided polygon we write one of the numbers

1,2,3,…, 12,13

, without repeating. Then, on each side of the polygon we write the difference of the numbers of the vertices of its ends (the largest minus the smallest). For example, if two consecutive vertices of the polygon have the numbers

2

11

9

is written on the side they determine. a) Is it possible to number the vertices of the polygon so that only the numbers

3, 4

5

are written on the sides? b) Is it possible to number the vertices of the polygon so that only the numbers

3, 4

6

are written on the sides?

equal areas by 2 straight cuts in a convex quadr. 2021 May Olympiad L1 p4

Source:

Facundo and Luca have been given a cake that is shaped like the quadrilateral in the figure. https://cdn.artofproblemsolving.com/attachments/3/2/630286edc1935e1a8dd9e704ed4c813c900381.png They are going to make two straight cuts on the cake, thus obtaining

4

portions in the shape of a quadrilateral. Then Facundo will be left with two portions that do not share any side, the other two will be for Luca. Show how they can cut the cuts so that both children get the same amount of cake. Justify why cutting in this way achieves the objective.