Wednesday, February 24, 2016
(in North America and South America)
Thursday, February 25, 2016
(outside of North American and South America)
©2015 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.
If \(x=3\), \(y=2x\) and \(z=3y\), the value of \(z\) is
A cube has 12 edges, as shown.
The expression \(\dfrac{20+16\times 20}{20\times 16}\) equals
An oblong number is the number of dots in a rectangular grid with one more row than column. The first four oblong numbers are 2, 6, 12, and 20, and are represented below:
What is the 7th oblong number?
In the diagram, point \(Q\) is the midpoint of \(PR\).
The coordinates of \(R\) are
Carrie sends five text messages to her brother each Saturday and five text messages to her brother each Sunday. Carrie sends two text messages to her brother on each of the other days of the week. Over the course of four full weeks, how many text messages does Carrie send to her brother?
The value of \((-2)^3-(-3)^2\) is
If \(\sqrt{25-\sqrt{n}}=3\), the value of \(n\) is
If \(x\)% of 60 is 12, then 15% of \(x\) is
In the diagram, square \(PQRS\) has side length 2. Points \(W\), \(X\), \(Y\), and \(Z\) are the midpoints of the sides of \(PQRS\).
What is the ratio of the area of square \(WXYZ\) to the area of square \(PQRS\)?
In the diagram, \(\triangle PQR\) is right-angled at \(P\) and \(PR=12\).
How many of the positive divisors of 128 are perfect squares larger than 1?
The numbers \(4x, 2x-3, 4x-3\) are three consecutive terms in an arithmetic sequence. What is the value of \(x\)?
(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, \(3, 5, 7, 9\) are the first four terms of an arithmetic sequence.)
Suppose that \(a\) and \(b\) are integers with \(4<a<b<22\). If the average (mean) of the numbers \(4,a,b,22\) is 13, then the number of possible pairs \((a,b)\) is
Hicham runs 16 km in 1.5 hours. He runs the first 10 km at an average speed of 12 km/h. What is his average speed for the last 6 km?
If \(x=18\) is one of the solutions of the equation \(x^2+12x+c=0\), the other solution of this equation is
A total of \(n\) points are equally spaced around a circle and are labelled with the integers 1 to \(n\), in order. Two points are called diametrically opposite if the line segment joining them is a diameter of the circle. If the points labelled 7 and 35 are diametrically opposite, then \(n\) equals
Suppose that \(x\) and \(y\) satisfy \(\dfrac{x-y}{x+y}=9\) and \(\dfrac{xy}{x+y}=-60\).
The value of \((x+y)+(x-y)+xy\) is
There are \(n\) students in the math club at Scoins Secondary School. When Mrs. Fryer tries to put the \(n\) students in groups of 4, there is one group with fewer than 4 students, but all of the other groups are complete. When she tries to put the \(n\) students in groups of 3, there are 3 more complete groups than there were with groups of 4, and there is again exactly one group that is not complete. When she tries to put the \(n\) students in groups of 2, there are 5 more complete groups than there were with groups of 3, and there is again exactly one group that is not complete. The sum of the digits of the integer equal to \(n^2-n\) is
In the diagram, \(PQRS\) represents a rectangular piece of paper. The paper is folded along a line \(VW\) so that \(\angle VWQ = 125^\circ\). When the folded paper is flattened, points \(R\) and \(Q\) have moved to points \(R'\) and \(Q'\), respectively, and \(R'V\) crosses \(PW\) at \(Y\).
The measure of \(\angle PYV\) is
Box 1 contains one gold marble and one black marble. Box 2 contains one gold marble and two black marbles. Box 3 contains one gold marble and three black marbles. Whenever a marble is chosen randomly from one of the boxes, each marble in that box is equally likely to be chosen. A marble is randomly chosen from Box 1 and placed in Box 2. Then a marble is randomly chosen from Box 2 and placed in Box 3. Finally, a marble is randomly chosen from Box 3. What is the probability that the marble chosen from Box 3 is gold?
If \(x\) and \(y\) are real numbers, the minimum possible value of the expression \((x+3)^2+2(y-2)^2+4(x-7)^2+(y+4)^2\) is
Seven coins of three different sizes are placed flat on a table, arranged as shown in the diagram. Each coin, except the centre one, touches three other coins. The centre coin touches all of the other coins.
If the coins labelled \(C_3\) have a radius of 3 cm, and those labelled \(C_2\) have radius 2 cm, then the radius of the coin labelled \(X\) is closest to
For any real number \(x\), \(\lfloor x \rfloor\) denotes the largest integer less than or equal to \(x\).For example, \(\lfloor 4.2 \rfloor=4\) and \(\lfloor 0.9 \rfloor=0\).
If \(S\) is the sum of all integers \(k\) with \(1\leq k\leq 999\,999\) and for which \(k\) is divisible by \(\lfloor\sqrt{k}\rfloor\), then \(S\) equals
The set \(A = \{1,2,3,\ldots,2044,2045\}\) contains 2045 elements. A subset \(S\) of \(A\) is called triple-free if no element of \(S\) equals three times another element of \(S\). For example, \(\{1,2,4,5,10,2043\}\) is triple-free, but \(\{1,2,4,5,10,681,2043\}\) is not triple-free. The triple-free subsets of \(A\) that contain the largest number of elements contain exactly \(1535\) elements. There are \(n\) triple-free subsets of \(A\) that contain exactly \(1535\) elements. The integer \(n\) can be written in the form \(p^aq^b\), where \(p\) and \(q\) are distinct prime numbers and \(a\) and \(b\) are positive integers. If \(N=p^2+q^2+a^2+b^2\), then the last three digits of \(N\) are
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