# 1998COMC加拿大数学公开赛真题免费下载

### 1998 COMC真题免费下载

Part A共8题，每题5分

Part B共4题，每题10分

#### 部分真题预览：

Part A :

3)In the figure, each region T represents an equilateral triangle and each region S a semicircle. The complete figure is a semicircle of radius 6 with its centre O. The three smaller semicircles touch the large semicircle at points A, B and C. What is the radius of a semicircle S?

7)There are ten prizes, five A’s, three B’s and two C’s, placed in identical sealed envelopes for the top ten contestants in a mathematics contest. The prizes are awarded by allowing winners to select an envelope at random from those remaining. When the eight contestant goes to select a prize, what is the probability that the remaining three prizes are one A, one B and one C?

8)Nine spheres are placed in a closed cubical box of side lenth 32 cm. Four small spheres of radius r are first placed in the bottom corners of the box so that they touch adjacent sides of the box but not each other. A large sphere of radius 15 cm is then placed in the box so that it touches each of the four smaller spheres but not the bottom. Four spheres of radius r are then added in the upper corners and the box closed so that the lid just touches the four smaller spheres. Calculate r.

Part B :

3)Alphonse and Beryl play a game by alternately moving a disk on a circular board. The game starts with the disk already on the board as shown. A player may move either clockwise
one position or one position toward the centre but cannot move to a position that has been previously occupied. The last person who is able to move wins the game.

1. If Alphonse moves first, is there a strategy which guarantees that he will always win?
2. Is there a winning strategy for either of the players if the board is changed to five concentric circles with nine regions in each ring and Alphonse moves first? (The rules for playing this new game remain the same)