Tuesday, February 25, 2020
(in North America and South America)
Wednesday, February 26, 2020
(outside of North American and South America)
©2020 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 points \(O(0, 0)\), \(P(0,3)\), \(Q\), and \(R(5,0)\) form a rectangle, as shown.
The coordinates of \(Q\) are
The value of \(3\times 2020 + 2\times 2020 - 4\times 2020\) is
For every real number \(x\), the expression \((x+1)^2 - x^2\) is equal to
Ewan writes out a sequence where he counts by 11s starting at 3. The resulting sequence is \(3, 14, 25, 36, \ldots\). A number that will appear in Ewan’s sequence is
The value of \(\sqrt{\dfrac{\phantom{\big(}\!\!\sqrt{81}+\sqrt{81}}{2}}\) is
Anna thinks of an integer.
It is not a multiple of three.
It is not a perfect square.
The sum of its digits is a prime number.
The integer that Anna is thinking of could be
In the diagram, \(WXY\) is a straight angle.
What is the average (mean) of \(p\), \(q\), \(r\), \(s\), and \(t\)?
If \(2^n = 8^{20}\), what is the value of \(n\)?
The figure consists of five squares and two right-angled triangles.
The areas of three of the squares are 5, 8 and 32, as shown. What is the area of the shaded square?
Positive integers \(s\) and \(t\) have the property that \(s(s - t) = 29\). What is the value of \(s+t\)?
In the \(5 \times 5\) grid shown, 15 cells contain X’s and 10 cells are empty.
X | X | X | X | |
X | X | X | X | |
X | X | |||
X | X | X | ||
X | X |
Any X may be moved to any empty cell. What is the smallest number of X’s that must be moved so that each row and each column contains exactly three X’s?
Harriet ran a 1000 m course in 380 seconds. She ran the first 720 m of the course at a constant speed of 3 m/s. She ran the remaining part of the course at a constant speed of \(v\) m/s. What is the value of \(v\)?
In the list \(2,x,y,5\), the sum of any two adjacent numbers is constant. The value of \(x-y\) is
In Rad’s garden there are exactly 30 red roses, exactly 19 yellow roses, and no other roses. How many of the yellow roses does Rad need to remove so that \(\frac{2}{7}\) of the roses in the garden are yellow?
Suppose that \(N=3x+4y+5z\), where \(x\) equals 1 or \(-1\), and \(y\) equals 1 or \(-1\), and \(z\) equals 1 or \(-1\). How many of the following statements are true?
Suppose that \(x\) and \(y\) are real numbers with \(-4 \leq x \leq -2\) and \(2 \leq y \leq 4\). The greatest possible value of \(\dfrac{x + y}{x}\) is
In the diagram, \(\triangle PQR\) is right-angled at \(Q\) and point \(S\) is on \(PR\) so that \(QS\) is perpendicular to \(PR\).
If the area of \(\triangle PQR\) is 30 and \(PQ=5\), the length of \(QS\) is
Four teams play in a tournament in which each team plays exactly one game against each of the other three teams. At the end of each game, either the two teams tie or one team wins and the other team loses. A team is awarded 3 points for a win, 0 points for a loss, and 1 point for a tie. If \(S\) is the sum of the points of the four teams after the tournament is complete, which of the following values can \(S\) not equal?
When \((3+2x+x^2)(1+mx+m^2x^2)\) is expanded and fully simplified, the coefficient of \(x^2\) is equal to 1. What is the sum of all possible values of \(m\)?
A cube has six faces. Each face has some dots on it. The numbers of dots on the six faces are 2, 3, 4, 5, 6, and 7. Harry removes one of the dots at random, with each dot equally likely to be removed. When the cube is rolled, each face is equally likely to be the top face. What is the probability that the top face has an odd number of dots on it?
In the diagram, the central circle contains the number 36.
Positive integers are to be written in the eight empty circles, one number in each circle, so that the product of the three integers along any straight line is 2592. If the nine integers in the circles must be all different, what is the largest possible sum of these nine integers?
Suppose that \(x\) and \(y\) are real numbers that satisfy the two equations: \[\begin{aligned} x^2+3xy+y^2 & =909 \\ 3x^2+xy+3y^2 & = 1287\end{aligned}\] What is a possible value for \(x+y\)?
There are real numbers \(a\) and \(b\) for which the function \(f\) has the properties that \(f(x) = ax+b\) for all real numbers \(x\), and \(f(bx+a)=x\) for all real numbers \(x\). What is the value of \(a+b\)?
In the diagram, the circle with centre \(X\) is tangent to the largest circle and passes through the centre of the largest circle. The circles with centres \(Y\) and \(Z\) are each tangent to the other three circles, as shown.
The circle with centre \(X\) has radius 1. The circles with centres \(Y\) and \(Z\) each have radius \(r\). The value of \(r\) is closest to
Three real numbers \(x\), \(y\), \(z\) are chosen randomly, and independently of each other, between 0 and 1, inclusive. What is the probability that each of \(x-y\) and \(x-z\) is greater than \(-\frac{1}{2}\) and less than \(\frac{1}{2}\) ?
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