Tuesday, February 28, 2017
(in North America and South America)
Wednesday, March 1, 2017
(outside of North American and South America)
©2017 University of Waterloo
Time: 60 minutes
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.
The value of \(6\times 2017-2017\times 4\) is
In the diagram, how many \(1\times 1\) squares are shaded in the \(8\times 8\) grid?
Three different numbers from the list 2, 3, 4, 6 have a sum of 11. What is the product of these numbers?
The graph shows the volume of water in a 300 L tank as it is being drained at a constant rate.
At what rate is the water leaving the tank, in litres per hour?
A sketch of \(y=-2x^2 + 4\) could be
Emilia writes down the numbers 5, \(x\) and 9. Valentin calculates the mean (average) of each pair of these numbers and obtains 7, 10 and 12. The value of \(x\) is
If \(x=1\) is a solution of the equation \(x^{2} + ax + 1 = 0\), then the value of \(a\) is
If \(\dfrac{1}{2n}+\dfrac{1}{4n}=\dfrac{3}{12}\), then \(n\) equals
Kamile turned her computer off at 5 p.m. Friday, at which point it had been on for exactly 100 hours. At what time had Kamile turned her computer on?
The sum of four different positive integers is 100. The largest of these four integers is \(n\). The smallest possible value of \(n\) is
Last Thursday, each of the students in M. Fermat’s class brought one piece of fruit to school. Each brought an apple, a banana, or an orange. In total, 20% of the students brought an apple and 35% brought a banana. If 9 students brought oranges, how many students were in the class?
Digits are placed in the two boxes of \(2\,\square\,\square\), with one digit in each box, to create a three-digit positive integer. In how many ways can this be done so that the three-digit positive integer is larger than 217?
In the diagram, \(P\) lies on the \(y\)-axis, \(Q\) has coordinates \((4,0)\), and \(PQ\) passes through the point \(R(2,4)\).
What is the area of \(\triangle OPQ\)?
The expression \[\left(1+\tfrac{1}{2}\right)\left(1+ \tfrac{1}{3}\right)\left(1+\tfrac{1}{4}\right)\left(1+\tfrac{1}{5}\right)\left(1+\tfrac{1}{6}\right)\left(1+\tfrac{1}{7}\right)\left(1+\tfrac{1}{8}\right)\left(1+\tfrac{1}{9}\right)\] is equal to
In the diagram, \(M\) is the midpoint of \(YZ\), \(\angle XMZ = 30^\circ\), and \(\angle XYZ = 15^\circ\).
The measure of \(\angle XZY\) is
If \(x+2y = 30\), the value of \(\dfrac{x}{5}+\dfrac{2y}{3}+ \dfrac{2y}{5}+\dfrac{x}{3}\) is
Aaron has 144 identical cubes, each with edge length 1 cm. He uses all of the cubes to construct a solid rectangular prism, which he places on a flat table. If the perimeter of the base of the prism is 20 cm, what is the sum of all possible heights of the prism?
For any positive real number \(x\), \(\lfloor x \rfloor\) denotes the largest integer less than or equal to \(x\). For example, \(\lfloor 4.2 \rfloor=4\) and \(\lfloor 3 \rfloor=3\). If \(\lfloor x\rfloor \cdot x = 36\) and \(\lfloor y\rfloor \cdot y = 71\) where \(x,y>0\), then \(x+y\) equals
A point is equidistant from the coordinate axes if the vertical distance from the point to the \(x\)-axis is equal to the horizontal distance from the point to the \(y\)-axis. The point of intersection of the vertical line \(x=a\) with the line with equation \(3x+8y=24\) is equidistant from the coordinate axes. What is the sum of all possible values of \(a\)?
If \(m\) and \(n\) are positive integers with \(n>1\) such that \(m^{n} = 2^{25} \times 3^{40}\), then \(m+n\) is
In the sum shown, each letter represents a different digit with \(T \neq 0\) and \(W \neq 0\).
How many different values of \(U\) are possible?
A cylinder has radius 12 and height 30. The top circular face of the cylinder is the base of a cone and the centre of the bottom circular base of the cylinder is the vertex of the cone. A sphere is placed inside so that it touches the cone, the base of the cylinder and the side of the cylinder as shown.
Which of the following is closest to the radius of the sphere?
Sylvia chose positive integers \(a\), \(b\) and \(c\).
Peter determined the value of \(a+\dfrac{b}{c}\) and got an answer of 101.
Paul determined the value of \(\dfrac{a}{c} + b\) and got an answer of 68.
Mary determined the value of \(\dfrac{a+b}{c}\) and got an answer of \(k\).
The value of \(k\) is
Eight teams compete in a tournament. Each pair of teams plays exactly one game against each other. There are no ties. If the two possible outcomes of each game are equally likely, what is the probability that every team loses at least one game and wins at least one game?
Let \(r = \sqrt{\frac{\sqrt{53}}{2}+\frac{3}{2}}\). There is a unique triple of positive integers \((a,b,c)\) such that \[r^{100} = 2r^{98}+14r^{96}+11r^{94} - r^{50} +ar^{46}+br^{44}+cr^{40}\] What is the value of \(a^2+b^2+c^2\)?
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