# 2005COMC加拿大数学公开赛真题免费下载

### 2005 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 :

5)An office building has 50 storeys, 25 of which are painted black and the other 25 of which are painted gold. If the number of gold storeys in the top half of the building is added to the number of black storeys in the bottom half of the building, the sum is 28. How many gold storeys are there in the top half of the building?

6)In the grid shown, each row has a value assigned to it and each column has a value assigned to it. The number in each cell is the sum of its row and column values. For example, the “8” is the sum of the value assigned to the 3rd row and the value assigned to the 4th column. Determine the values of x and y.

7)In the diagram, the semi-circle has centre O and diameter AB. A ray of light leaves point P in a direction perpendicular to AB. It bounces off the semi-circle at point D in such a way that ∠PDO = ∠EDO. (In other words, the angle of incidence equals
the angle of reflection at D.) The ray DE then bounces off the circle in a similar
way at E before finally hitting the semicircle again at B. Determine \DOP.

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.

1. 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.)
2. Determine, with justification, the total of the nine numbers when the grid to the right is completed using Piotr’s Principle.
3. Determine, with justification, the values of x and y when the grid to the right is completed using Piotr’s Principle.

3)

1. 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.
2. 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.