Wednesday, November 18, 2020
(in North America and South America)
Thursday, November 19, 2020
(outside of North American and South America)
©2020 University of Waterloo
Time: 2 hours
Calculating devices are allowed, provided that they do not have any of the following features: (i) internet access, (ii) the ability to communicate with other devices, (iii) information previously stored by students (such as formulas, programs, notes, etc.), (iv) a computer algebra system, (v) dynamic geometry software.
Do not open this booklet until instructed to do so.
There are two parts to this paper. The questions in each part are arranged roughly in order of increasing difficulty. The early problems in Part B are likely easier than the later problems in Part A.
PART A
PART B
For each question in Part A, full marks will be given for a correct answer which is placed in the box. Part marks will be awarded only if relevant work is shown in the space provided in the answer booklet.
The five numbers
In the diagram, two 8 by 10 rectangles overlap to form a 4 by 4 square. What is the total area of the shaded region?
A dish contains 100 candies. Juan removes candies from the dish each day and no candies are added to the dish. On day 1, Juan removes 1 candy. On day 2, Juan removes 2 candies. On each day that follows, Juan removes 1 more candy than he removed on the previous day. After day
In the diagram, nine identical isosceles trapezoids are connected as shown to form a closed ring. (An isosceles trapezoid is a trapezoid with the properties that one pair of sides is parallel and the other two sides are equal in length.) What is the measure of
Determine all pairs
There are 90 players in a league. Each of the 90 players plays exactly one match against each of the other 89 players. Each match ends with a win for one player and a loss for the other player, or with a tie for both players. Each player earns 1 point for a win, 0 points for a loss, and 0.5 points for a tie. After all matches have been played, the points earned by each player are added up. What is the greatest possible number of players whose total score can be at least 54 points?
For each question in Part B, your solution must be well-organized and contain words of explanation or justification. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks.
An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example,
The first four terms of an arithmetic sequence are
The numbers
The numbers
The numbers
Tank A and Tank B are rectangular prisms and are sitting on a flat table.
Tank A is
Tank B is
Initially, Tank A is full of water and Tank B is empty.
The water in Tank A drains out at a constant rate of
Tank B fills with water at a constant rate of
Tank A begins to drain at the same time that Tank B begins to fill.
Determine after how many seconds Tank B will be exactly
Determine the depth of the water left in Tank A at the instant when Tank B is full.
At one instant, the depth of the water in Tank A is equal to the depth of the water in Tank B. Determine this depth.
Tank A
Tank B
Tank C is a rectangular prism that is
Tank C sits on the flat table on one of its
Tank D is in the shape of an inverted square-based pyramid, as shown. It is supported so that its square base is parallel to the flat table and its fifth vertex touches the flat table.
The height of Tank D is 10 cm and the side length of its square base is 20 cm.
Initially, Tank C is full of water and Tank D is empty.
Tank D begins filling with water at a rate of
Two seconds after Tank D begins to fill, Tank C begins to drain at a rate of
At one instant, the volume of water in Tank C is equal to the volume of water in Tank D.
Determine the depth of the water in Tank D at that instant.
Tank C
Tank D
The integers
Determine whether or not
Determine whether or not
Determine the number of possible different values of
Determine the number of ordered quadruples