历年 Canadian Open Mathematics Challenge加拿大数学公开赛
Part A :
3)A regular hexagon is a six-sided figure which has all of its angles equal and all of its side lengths equal. If P and Q are points on a regular hexagon which has a side length of 1, what is the maximum possible length of the line segment PQ?
5)Triangle PQR is right-angled at Q and has side lengths PQ = 14 and QR = 48. If M is the midpoint of PR, determine the cosine of ∠MQP.
7)If a can be any positive integer and
2x + a = y
a + y = x
x + y = z
determine the maximum possible value for x + y + z.
Part B :
- Alphonse and Beryl are playing a game, starting with a pack of 7 cards. Alphonse begins by discarding at least one but not more than half of the cards in the pack. He then passes the remaining cards in the pack to Beryl. Beryl continues the game by discarding at least one but not more than half of the remaining cards in the pack. The game continues in this way with the pack being passed back and forth between the two players. The loser is the player who, at the beginning of his or her turn, receives only one card. Show, with justification, that there is always a winning strategy for Beryl.
- Alphonse and Beryl now play a game with the same rules as in (a), except this time they start with a pack of 52 cards, and Alphonse goes first again. As in (a), a player on his or her turn must discard at least one and not more than half of the remaining cards from the pack. Is there a strategy that Alphonse can use to be guaranteed that he will win? (Provide justification for your answer.)