Problems(2)
Luffy always lures magic things
Source: Rioplatense L3 2023 #4
12/6/2023
Luffy is playing with some magic boxes and a machine. Each box has a value(number) inside. Opening a box, Luffy sees the value, adds the value to his score(if the box value is negative, Luffy loses points) and destroys the box. Putting a box of value in the machine, this box vanishes and it is replaced by two new boxes of values and (it's not known which one has the respective value, but he can identify the new boxes). At the beginning, Luffy has points and has a box whose value is known(it is zero).
a) Prove that Luffy can reach at least points
b) Is it possible that Luffy reaches at least points, without have less than points in any moment?
combinatorics
Parallelogram Hater
Source: Rioplatense L2 2023 #4
12/6/2023
A set of points on the plane is antiparallelogram if any four points of the set are not vertices of a parallelogram. Prove that for any set of points on the plane, no three of them are collinears, there exists a subset of points, such that this subset is antiparallelogram.
geometrycombinatorics