MathDB

6

Part of 2004 Tournament Of Towns

Problems(4)

TOT 2004 Spring - Junior A-Level p6 game with 2004! on blackboard

Source:

2/25/2020
At the beginning of a two-player game, the number 2004!2004! is written on the blackboard. The players move alternately. In each move, a positive integer smaller than the number on the blackboard and divisible by at most 2020 different prime numbers is chosen. This is subtracted from the number on the blackboard, which is erased and replaced by the difference. The winner is the player who obtains 00. Does the player who goes first or the one who goes second have a guaranteed win, and how should that be achieved?
combinatoricsgamegame strategy
TOT 2004 Spring - Senior A-Level p6 audience shuffles a deck of 36 cards

Source:

2/25/2020
The audience shuffles a deck of 3636 cards, containing 99 cards in each of the suits spades, hearts, diamonds and clubs. A magician predicts the suit of the cards, one at a time, starting with the uppermost one in the face-down deck. The design on the back of each card is an arrow. An assistant examines the deck without changing the order of the cards, and points the arrow on the back each card either towards or away from the magician, according to some system agreed upon in advance with the magician. Is there such a system which enables the magician to guarantee the correct prediction of the suit of at least (a) 1919 cards; (b) 2020 cards?
combinatorics
A piece of cheese

Source: Tournament of towns, Junior A-Level paper, Fall 2004

12/19/2004
Two persons are dividing a piece of cheese. The first person cuts it into two pieces, then the second person cuts one of these pieces into two, then again the first person cuts one of the pieces into two, and so until they have 5 pieces. After that the first person chooses one of the pieces, then the second person chooses one of remaining pieces and so on until all pieces are taken. For each of the players, what is the maximal amount of cheese he can get for certain, regardless of the other's actions?
combinatorics unsolvedcombinatorics
Admissible triangle

Source: Tournament of towns, Senior A-Level paper, Fall 2004

12/25/2004
Let n be a fixed prime number >3. A triangle is said to be admissible if the measure of each of its angles is of the form m⋅180∘n\frac{m\cdot 180^{\circ}}{n} for some positive integer m. We are given one admissible triangle. Every minute we cut one of the triangles we already have into two admissible triangles so that no two of the triangles we have after cutting are similar. After some time, it turns out that no more cuttings are possible. Prove that at this moment, the triangles we have contain all possible admissible triangles (we do not distinguish between triangles which have same sets of angles, i.e. similar triangles).
geometry unsolvedgeometry