Thursday, February 23, 2021
(in North America and South America)
Wednesday, February 24, 2021
(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.
A rectangle has width \(2\text{ cm}\) and length \(3\text{ cm}\).
The area of the rectangle is
The expression \(2+3\times 5+2\) equals
The number equal to 25% of 60 is
When \(x=2021\), the value of \(\dfrac{4x}{x+2x}\) is
Which of the following integers cannot be written as a product of two integers, each greater than 1?
A square piece of paper has a dot in its top right corner and is lying on a table. The square is folded along its diagonal, then rotated \(90^\circ\) clockwise about its centre, and then finally unfolded, as shown.
The resulting figure is
For which of the following values of \(x\) is \(x\) greater than \(x^2\) ?
The digits in a two-digit positive integer are reversed. The new two-digit integer minus the original integer equals 54. What is the positive difference between the two digits of the original integer?
The line with equation \(y=2x-6\) is translated upwards by 4 units. (That is, every point on the line is translated upwards by 4 units, forming a new line.) The \(x\)-intercept of the resulting line is
If \(3^{x}=5\), the value of \(3^{x+2}\) is
In the sum shown, \(P\), \(Q\) and \(R\) represent three different single digits.
The value of \(P+Q+R\) is
How many of the 20 perfect squares \(1^2, 2^2, 3^2, \dots, 19^2, 20^2\) are divisible by 9?
In the diagram, each of \(\triangle WXZ\) and \(\triangle XYZ\) is an isosceles right-angled triangle.
The length of \(WX\) is \(6\sqrt 2\). The perimeter of quadrilateral \(WXYZ\) is closest to
Natascha cycles 3 times as fast as she runs. She spends 4 hours cycling and 1 hour running. The ratio of the distance that she cycles to the distance that she runs is
Let \(a\) and \(b\) be positive integers for which \(45a + b = 2021\). The minimum possible value of \(a+b\) is
If \(n\) is a positive integer, the notation \(n!\) (read “\(n\) factorial”) is used to represent the product of the integers from 1 to \(n\). That is, \(n! = n(n-1)(n - 2) \cdots (3)(2)(1)\). For example, \(4! = 4(3)(2)(1) = 24\) and \(1!=1\). If \(a\) and \(b\) are positive integers with \(b>a\), the ones (units) digit of \(b!-a!\) cannot be
The set \(S\) consists of 9 distinct positive integers. The average of the two smallest integers in \(S\) is 5. The average of the two largest integers in \(S\) is 22. What is the greatest possible average of all of the integers of \(S\)?
In the diagram, \(\triangle PQR\) is an isosceles triangle with\(PQ = PR\). Semi-circles with diameters \(PQ\), \(QR\) and \(PR\) are drawn.
The sum of the areas of these three semi-circles is equal to 5 times the area of the semi-circle with diameter \(QR\). The value of \(\cos(\angle PQR)\) is
The real numbers \(x\), \(y\) and \(z\) satisfy the three equations \[\begin{aligned} x+y &= 7\\ xz&=-180\\ (x+y+z)^2&=4\end{aligned}\] If \(S\) is the sum of the two possible values of \(y\), then \(-S\) equals
In the diagram, \(PRTY\) and \(WRSU\) are squares.
Point \(Q\) is on \(PR\) and point \(X\) is on \(TY\) so that \(PQXY\) is a rectangle. Also, point \(T\) is on \(SU\), point \(W\) is on \(QX\), and point \(V\) is the point of intersection of \(UW\) and \(TY\), as shown. If the area of rectangle \(PQXY\) is 30, the length of \(ST\) is closest to
A function, \(f\), has \(f(2)=5\) and \(f(3)=7\). In addition, \(f\) has the property that \[f(m)+f(n)=f(mn)\] for all positive integers \(m\) and \(n\). (For example, \(f(9) = f(3)+f(3) = 14\).) The value of \(f(12)\) is
An unpainted cone has radius 3 cm and slant height 5 cm.
The cone is placed in a container of paint. With the cone’s circular base resting flat on the bottom of the container, the depth of the paint in the container is 2 cm. When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as \(\dfrac{p}{q}\) where \(p\) and \(q\) are positive integers with no common divisor larger than 1. What is the value of \(p+q\)?
(The lateral surface of a cone is its external surface not including the circular base. A cone with radius \(r\), height \(h\), and slant height \(s\) has lateral surface area equal to \(\pi rs\).)
In Figure 1, three unshaded dots are arranged to form an equilateral triangle, as shown.
Figure 2 is formed by arranging three copies of Figure 1 to form the outline of a larger equilateral triangle and then filling the resulting empty space with 1 shaded dot.
The smallest value of \(n\) for which Figure \(n\) includes at least 100 000 shaded dots is
A pair of real numbers \((a,b)\) with \(a^2+b^2\leq\tfrac{1}{4}\) is chosen at random. If \(p\) is the probability that the curves with equations \(y=ax^2+2bx-a\) and \(y=x^2\) intersect, then \(100p\) is closest to
Let \(N\) be the number of triples \((x,y,z)\) of positive integers such that \(x<y<z\) and \(xyz = 2^2 \cdot 3^2 \cdot 5^2 \cdot 7^2 \cdot 11^2 \cdot 13^2 \cdot 17^2 \cdot 19^2\). When \(N\) is divided by 100, the remainder is
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