Tuesday, February 26, 2019
(in North America and South America)
Wednesday, February 27, 2019
(outside of North American and South America)
©2010 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.
What is the remainder when 14 is divided by 5?
Which of the following is equal to \(20(x+y)-19(y+x)\) for all values of \(x\) and \(y\)?
The value of \(8-\dfrac{6}{4-2}\) is
In the diagram, point \(P\) is on the number line at 3 and \(V\) is at 33. The number line between \(3\) and \(33\) is divided into six equal parts by the points \(Q,R,S,T,U\).
What is the sum of the lengths of \(PS\) and \(TV\)?
Mike rides his bicycle at a constant speed of 30 km/h. How many kilometres does Mike travel in 20 minutes?
In the diagram, \(PQRS\) is a rectangle. Also, \(\triangle STU\), \(\triangle UVW\) and \(\triangle WXR\) are congruent.
What fraction of the area of rectangle \(PQRS\) is shaded?
The town of Cans is north of the town of Ernie. The town of Dundee is south of Cans but north of Ernie. The town of Arva is south of the town of Blythe and is north of both Dundee and Cans. The town that is the most north is
The product \(8 \times 48 \times 81\) is divisible by \(6^k\). The largest possible integer value of \(k\) is
The average of \(\dfrac{1}{8}\) and \(\dfrac{1}{6}\) is
The digits \(2\), \(3\), \(5\), \(7\), and \(8\) can be used, each exactly once, to form many five-digit integers. Of these integers, \(N\) is the one that is as close as possible to 30 000. What is the tens digit of \(N\)?
Line \(\ell\) is perpendicular to the line with equation \(y=x-3\). Line \(\ell\) has the same \(x\)-intercept as the line with equation \(y=x-3\). The \(y\)-intercept of line \(\ell\) is
The first part of the Genius Quiz has 30 questions and the second part has 50 questions. Alberto answered exactly 70% of the 30 questions in the first part correctly. He answered exactly 40% of the 50 questions in the second part correctly. The percentage of all of the questions on the quiz that Alberto answered correctly is closest to
Tanis looked at her watch and noticed that, at that moment, it was \(8x\) minutes after 7:00 a.m. and \(7x\) minutes before 8:00 a.m. for some value of \(x\). What time was it at that moment?
The letters A, B, C, D, and E are to be placed in the grid so that each of these letters appears exactly once in each row and exactly once in each column.
\[\begin{array}{ | c | c | c | c | c | } \hline A & & & & E \\ \hline & & C & A & \\ \hline E & & B & C & \\ \hline & * & & & \\ \hline B & & & D & \\ \hline \end{array}\]
Which letter will go in the square marked with \(*\) ?
There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. How many different-looking arrangements are possible?
In the diagram, each line segment has length \(x\) or \(y\). Also, each pair of adjacent sides is perpendicular.
If the area of the figure is 252 and \(x=2y\), the perimeter of the figure is
The five sides of a regular pentagon are all equal in length. Also, all interior angles of a regular pentagon have the same measure. In the diagram, \(PQRST\) is a regular pentagon and \(\triangle PUT\) is equilateral.
The measure of obtuse \(\angle QUS\) is
How many 7-digit positive integers are made up of the digits 0 and 1 only, and are divisible by 6?
The function \(f\) has the properties that \(f(1) =6\) and \(f(2x+1)=3f(x)\) for every integer \(x\). What is the value of \(f(63)\)?
The vertices of an equilateral triangle lie on a circle with radius 2. The area of the triangle is
In the multiplication shown, each of \(P\), \(Q\), \(R\), \(S\), and \(T\) is a digit.
The value of \(P+Q+R+S+T\) is
In the diagram, two circles touch at \(P\). Also, \(QP\) and \(SU\) are perpendicular diameters of the larger circle that intersect at \(O\). Point \(V\) is on \(QP\) and \(VP\) is a diameter of the smaller circle. The smaller circle intersects \(SU\) at \(T\), as shown.
If \(QV=9\) and \(ST=5\), what is the sum of the lengths of the diameters of the two circles?
How many positive integers \(n\) with \(n \leq 100\) can be expressed as the sum of four or more consecutive positive integers?
Consider the quadratic equation \(x^2-(r+7)x+r+87=0\) where \(r\) is a real number. This equation has two distinct real solutions \(x\) which are both negative exactly when \(p < r < q\), for some real numbers \(p\) and \(q\). The value of \(p^2+q^2\) is
In \(\triangle QRS\), point \(T\) is on \(QS\) with \(\angle QRT=\angle SRT\). Suppose that \(QT=m\) and \(TS=n\) for some integers \(m\) and \(n\) with \(n>m\) and for which \(n+m\) is a multiple of \(n-m\).
Suppose also that the perimeter of \(\triangle QRS\) is \(p\) and that the number of possible integer values for \(p\) is \(m^2+2m-1\). The value of \(n-m\) is
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