历年 Canadian Open Mathematics Challenge加拿大数学公开赛
Part A :
2)The sequence 9, 18, 27, 36, 45, 54, … consists of successive multiples of 9. This sequence is then altered by multiplying every other term by –1, starting with the first term, to produce the new sequence – 9,18, – 27, 36, – 45, 54,... . If the sum of the first n terms of this new sequence is 180, determine n.
3)The symbol n! is used to represent the product n (n –1)(n – 2)L(3)(2)(1). For example, 4!= 4(3)(2)(1). Determine n such that n!= (215)(36 )(53)(72 )(11)(13).
8)In the diagram, D ABC is equilateral and the radius of its inscribed circle is 1. A larger circle is drawn through the vertices of the rectangle ABDE. What is the diameter of the larger circle?
Part B :
- Alphonse and Beryl are playing a game, starting with the geometric shape shown in Figure 1. Alphonse begins the game by cutting the original shape into two pieces along one of the lines. He then passes the piece containing the black triangle to Beryl, and discards the other piece. Beryl repeats these steps with the piece she receives; that is to say, she cuts along the length of a line, passes the piece containing the black triangle back to Alphonse, and discards the other piece. This process continues, with the winner being the player who, at the beginning of his or her turn, receives only the black triangle. 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 use the shape in Figure 2 and Beryl goes first. As in (a), cuts may only be made along the whole length of a line in the figure. Is there a strategy that Beryl can use to be guaranteed that she will win? (Provide justification for your answer.)