Subcontests
(6)Process on a blackboard
Let n be a positive integer. On a blackboard, Bobo writes a list of n non-negative integers. He then performs a sequence of moves, each of which is as follows:-for each i=1,...,n, he computes the number ai of integers currently on the board that are at most i,-he erases all integers on the board,-he writes on the board the numbers a1,a2,…,an.For instance, if n=5 and the numbers initially on the board are 0,7,2,6,2, after the first move the numbers on the board will be 1,3,3,3,3, after the second they will be 1,1,5,5,5, and so on.(a) Show that, whatever n and whatever the initial configuration, the numbers on the board will eventually not change any more.(b) As a function of n, determine the minimum integer k such that, whatever the initial configuration, moves from the k-th onwards will not change the numbers written on the board. Dedalo vs. The Minotaur
Dedalo buys a finite number of binary strings, each of finite length and made up of the binary digits 0 and 1. For each string, he pays (21)L drachmas, where L is the length of the string. The Minotaur is able to escape the labyrinth if he can find an infinite sequence of binary digits that does not contain any of the strings Dedalo bought. Dedalo’s aim is to trap the Minotaur.
For instance, if Dedalo buys the strings 00 and 11 for a total of half a drachma, the Minotaur is able to escape using the infinite string 01010101….
On the other hand, Dedalo can trap the Minotaur by spending 75 cents of a drachma: he could for example buy the strings 0 and 11, or the strings 00,11,01.
Determine all positive integers c such that Dedalo can trap the Minotaur with an expense of at most c cents of a drachma. A point variable on a semi-circle
Fix circle with center O, diameter AB and a point C on it, different from A,B. Let a point D, different from A,B, vary on the arc AB not containing C. Let E lie on CD such that BE⊥CD. Prove that CE⋅ED is maximal exactly when BOED is cyclic.