7
Problems(2)
Santa Claus game, present behind 1 of 100 doors
Source: 2020 Dürer Math Competition Finals Day2 E7 https://artofproblemsolving.com/community/c1622639_2020_
1/7/2022
Santa Claus plays a guessing game with Marvin before giving him his present. He hides the present behind one of doors, numbered from to . Marvin can point at a door, and then Santa Claus will reply with one of the following words: "hot" if the present lies behind the guessed door, "warm" if the guess is not exact but the number of the guessed door differs from that of the present’s door by at most , "cold" if the numbers of the two doors differ by more than .At least how many such guesses does Marvin need, so that he can be certain about where his present is?Marvin does not necessarily need to make a "hot" guess, just to know the correct door with certainty.
combinatoricsgame
red and blue balls in an urn , 1024 in total
Source: 2020 Dürer Math Competition Finals Day2 E+7 https://artofproblemsolving.com/community/c1622639_2020_
1/7/2022
There are red and blue balls in an urn : in total. In one round, we do the following:
we draw the balls from the urn two by two. After all balls have been drawn, we put a new ball back into the urn for each pair of drawn balls: the colour of the new ball depends on that of the drawn pair. For two red balls drawn, we put back a red ball. For two blue balls, we put back a blue ball. For a red and a blue ball, we put back a black ball. For a red and a black ball, we put back a red ball. For a blue and a black ball, we put back a blue ball. Finally, for two black balls we put back a black ball.
Then the next round begins. After rounds, a single ball remains in the urn, which is red. What is the maximal number of blue balls that might have been in the urn at the very beginning?
combinatoricsgameColoring