历年 Canadian Open Mathematics Challenge加拿大数学公开赛
Part A :
5)The Rice Tent Company sells tents in two different sizes, large and small. Last year, the Company sold 200 tents, of which one quarter were large. The sale of the large tents produced one third of the company’s income. What was the ratio of the price of a large tent to the price of a small tent?
6)In the diagram, a square of side length 2 has semicircles drawn on each side. An “elastic band” is stretched tightly around the figure. What is the length of the elastic band in this position?
8)A rectangular sheet of paper, ABCD, has AD = 1 and AB = r, where 1 < r < 2. The paper is folded along a line through A so that the edge AD falls onto the edge AB. Without unfolding, the paper is folded again along a line through B so that the edge CB also lies on AB. The result is a triangular piece of paper. A region of this triangle is four sheets thick. In terms of r, what is the area of this region?
Part B :
3)A map shows all Beryl’s Llamaburgers restaurant locations in North America. On this map, a line segment is drawn from each restaurant to the restaurant that is closest to it. Every restaurant has a unique closest neighbour. (Note that if A and B are two of the restaurants, then A may be the closest to B without B being closest to A.)
- Prove that no three line segments on the map can form a triangle.
- Prove that no restaurant can be connected to more than five other restaurants.