Tuesday, February 24, 2015
(in North America and South America)
Wednesday, February 25, 2015
(outside of North American and South America)
©2014 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 average (mean) of the five numbers 8, 9, 10, 11, 12 is
The value of \(\dfrac{2 \times 3 + 4}{2 + 3}\) is
Six points \(P, Q, R, S, T, U\) are equally spaced along a straight path.
Emily walks from \(P\) to \(U\) and then back to \(P\). At which point has she completed 70% of her walk?
If \(x=-3\), then \((x-3)^2\) equals
The points \(P(3,-2)\), \(Q(3,1)\), \(R(7,1)\), and \(S\) form a rectangle. What are the coordinates of \(S\)?
In the diagram, \(MNPQ\) is a rectangle with points \(M\), \(N\), \(P\), and \(Q\) on the sides of \(\triangle XYZ\), as shown.
If \(\angle{ZNM} = 68^\circ\) and \(\angle{XYZ} = 55^\circ\), what is the measure of \(\angle{YXZ}\)?
Violet has one-half of the money she needs to buy her mother a necklace. After her sister gives her $30, she has three-quarters of the amount she needs. Violet’s father agrees to give her the rest. The amount that Violet’s father will give her is
If \(x\) and \(y\) are positive integers with \(3^x 5^y = 225\), then \(x + y\) equals
At Barker High School, a total of 36 students are on either the baseball team, the hockey team, or both. If there are 25 students on the baseball team and 19 students on the hockey team, how many students play both sports?
Anca and Bruce left Mathville at the same time. They drove along a straight highway towards Staton.
Bruce drove at 50 km/h. Anca drove at 60 km/h, but stopped along the way to rest. They both arrived at Staton at the same time. For how long did Anca stop to rest?
Three-digit positive integers such as 789 and 998 use no digits other than 7, 8 and 9. In total, how many three-digit positive integers use no digits other than 7, 8 and 9?
If \(\cos 60^\circ = \cos 45^\circ \cos\theta\) with \(0^\circ \leq \theta \leq 90^\circ\), then \(\theta\) equals
At the end of the year 2000, Steve had $100 and Wayne had $10 000. At the end of each following year, Steve had twice as much money as he did at the end of the previous year and Wayne had half as much money as he did at the end of the previous year. At the end of which year did Steve have more money than Wayne for the first time?
In the diagram, \(PQRS\) is a square and \(M\) is the midpoint of \(PS\).
The ratio of the area of \(\triangle{QMS}\) to the area of square \(PQRS\) is
A music test included 50 multiple choice questions. Zoltan’s score was calculated by
Zoltan answered 45 of the 50 questions and his score was 135 points. The number of questions that Zoltan answered incorrectly is
In the diagram, the line segment with endpoints \(P(-4,0)\) and \(Q(16,0)\) is the diameter of a semi-circle.
If the point \(R(0, t)\) is on the circle with \(t>0\), then \(t\) is
If \(a\) and \(b\) are two distinct numbers with \(\dfrac{a + b}{a - b} = 3\), then \(\dfrac{a}{b}\) equals
There are two values of \(k\) for which the equation \(x^2+2kx+7k-10=0\) has two equal real roots (that is, has exactly one solution for \(x\)). The sum of these values of \(k\) is
The \(y\)-intercepts of three parallel lines are 2, 3 and 4. The sum of the \(x\)-intercepts of the three lines is 36. What is the slope of these parallel lines?
For how many integers \(a\) with \(1 \leq a \leq 10\) is \(a^{2014}+a^{2015}\) divisible by 5?
Amina and Bert alternate turns tossing a fair coin. Amina goes first and each player takes three turns. The first player to toss a tail wins. If neither Amina nor Bert tosses a tail, then neither wins. What is the probability that Amina wins?
Three distinct integers \(a\), \(b\) and \(c\) satisfy the following three conditions:
What is the value of \(a+b+c\)?
(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, \(3, 5, 7\) is an arithmetic sequence with three terms.
A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant. For example, 3, 6, 12 is a geometric sequence with three terms.)
How many pairs \((x,y)\) of non-negative integers with \(0 \leq x \leq y\) satisfy the equation \(5x^2 - 4xy + 2x + y^2 = 624\)?
In the diagram, two circles and a square lie between a pair of parallel lines that are a distance of 400 apart.
The square has a side length of 279 and one of its sides lies along the lower line. The circles are tangent to each other, and each circle is tangent to one of the lines. Each circle also touches the square at only one point – the lower circle touches a side of the square and the upper circle touches a vertex of the square. If the upper circle has a radius of 65, then the radius of the lower circle is closest to
There are \(F\) fractions \(\dfrac{m}{n}\) with the properties:
We define \(G=F+p\), where the integer \(F\) has \(p\) digits. What is the sum of the squares of the digits of \(G\)?
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