历年 Canadian Open Mathematics Challenge加拿大数学公开赛
Part A :
3)In ΔABC, M is the midpoint of BC, as shown. If ∠ABM = 15° and ∠AMC = 30°, what is the size of ∠BCA?
6)Determine the number of integers n that satisfy all three of the conditions below:
- each digit of n is either 1 or 0,
- n is divisible by 6, and
- 0 < n < 107.
8)What is the probability that 2 or more successive heads will occur at least once in 10 tosses of a fair coin?
Part B :
1)Piotr places numbers on a 3 by 3 grid using the following rule, called “Piotr’s Principle”:
For any three adjacent numbers in a horizontal, vertical or diagonal line, the middle number is always the average (mean) of its two neighbours.
- Using Piotr’s principle, determine the missing numbers in the grid to the right. (You should fill in the missing numbers in the grid in your answer booklet.)
- Determine, with justification, the total of the nine numbers when the grid to the right is completed using Piotr’s Principle.
- Determine, with justification, the values of x and y when the grid to the right is completed using Piotr’s Principle.
- In the diagram, trapezoid ABCD has parallel sides AB and DC of lengths 10 and 20, respectively. Also, the length of AD is 6 and the length of BC is 8. Determine the area of trapezoid ABCD.
- In the diagram, PQRS is a rectangle and T is the midpoint of RS. The inscribed circles of 4PTS and 4RTQ each have radius 3. The inscribed circle of 4QPT has radius 4. Determine the dimensions of rectangle PQRS.