MathDB

2013 ToT Spring Junior A p7 school ping pong tournament for m +n members

Source:

3/4/2020

Two teams

A

and

B

play a school ping pong tournament. The team

A

consists of

m

students, and the team

B

consists of

n

students where

m \ne n

. There is only one ping pong table to play and the tournament is organized as follows: Two students from different teams start to play while other players form a line waiting for their turn to play. After each game the first player in the line replaces the member of the same team at the table and plays with the remaining player. The replaced player then goes to the end of the line. Prove that every two players from the opposite teams will eventually play against each other.

combinatoricsgame

2013 ToT Spring Senior A p7 wizards, numbered hats, strategy

Source:

3/5/2020

The King decided to reduce his Council consisting of thousand wizards. He placed them in a line and placed hats with numbers from

1

to

1001

on their heads not necessarily in this order (one hat was hidden). Each wizard can see the numbers on the hats of all those before him but not on himself or on anyone who stayed behind him. By King's command, starting from the end of the line each wizard calls one integer from

1

to

1001

so that every wizard in the line can hear it. No number can be repeated twice. In the end each wizard who fails to call the number on his hat is removed from the Council. The wizards knew the conditions of testing and could work out their strategy prior to it. (a) Can the wizards work out a strategy which guarantees that more than

500

of them remain in the Council? (b) Can the wizards work out a strategy which guarantees that at least

999

of them remain in the Council?

combinatoricsgame strategy

2013 ToT Fall Junior A p7 game with 11 piles of ten stones each

Source:

3/22/2020

On a table, there are

11

piles of ten stones each. Pete and Basil play the following game. In turns they take

1, 2

or

3

stones at a time: Pete takes stones from any single pile while Basil takes stones from different piles but no more than one from each. Pete moves first. The player who cannot move, loses. Which of the players, Pete or Basil, has a winning strategy?

game strategygameHi

2013 ToT Fall Senior O p7 every point of self-intersection bisects links

Source:

3/22/2020

A closed broken self-intersecting line is drawn in the plane. Each of the links of this line is intersected exactly once and no three links intersect at the same point. Further, there are no self-intersections at the vertices and no two links have a common segment. Can it happen that every point of self-intersection divides both links in halves?

combinatoricsbroken line

7

Problems(4)