(45 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) previously stored information such as formulas, programs, notes, etc., (iv) a computer algebra system, (v) dynamic geometry software.
Express answers as simplified exact numbers except where otherwise
indicated. For example,
What is the value of
In the diagram,
What is the value of
The average height of Ivan, Jackie and Ken is 4% larger than the average height of Ivan and Jackie. If Ivan and Jackie are each 175 cm tall, how tall is Ken?
A positive integer
A car and a minivan drive from Alphaville to Betatown. The car travels at a constant speed of 40 km/h and the minivan travels at a constant speed of 50 km/h. The minivan passes the car 10 minutes before the car arrives at Betatown. How many minutes pass between the time at which the minivan arrives in Betatown and the time at which the car arrives in Betatown?
Ruxandra wants to visit Singapore, Mongolia, Bhutan, Indonesia, and Japan. In how many ways can she order her trip to visit each country exactly once, with the conditions that she cannot visit Mongolia first and cannot visit Bhutan last?
Liesl has a bucket. Henri drills a hole in the bottom of the bucket. Before the hole was drilled, Tap A could fill the bucket in 16 minutes, tap B could fill the bucket in 12 minutes, and tap C could fill the bucket in 8 minutes. A full bucket will completely drain out through the hole in 6 minutes. Liesl starts with the empty bucket with the hole in the bottom and turns on all three taps at the same time. How many minutes will it take until the instant when the bucket is completely full?
In the diagram, the two regular octagons have side lengths of 1 and 2. The smaller octagon is completely contained within the larger octagon.
What is the area of the region inside the larger octagon and outside the smaller octagon?
The parabolas with equations
Determine the number of pairs of integers,
Evaluate
Let
What is the area of a triangle with base
Let
In the diagram,
If
If
Let
In the diagram, the larger square has side length
What is the area of the shaded region?
Let
Consider the three-digit positive integers of the form
When the integer
Let
The area of the triangle formed by the line
Let
Justin measures the heights of three different trees: a maple, a pine and
a spruce. The maple tree is 1 m taller than the pine tree and the pine
tree is 4 m shorter than the spruce tree. If the ratio of the height of
the maple tree to the spruce tree is
Suppose that
Let
If the graph of
Let
Suppose that
What is the value of
(45 minutes)
Calculating devices are not permitted.
Express answers as simplified exact numbers except where otherwise
indicated. For example,
In the diagram,
Evaluate
Bethany is told to create an expression from
If
If
The ratio of the width to the height of a rectangular screen is
In the diagram, wheel
If wheel
Martina has a cylinder with radius 10 cm and height 70 cm that is
closed at the bottom. Martina puts some solid spheres of radius 10 cm
inside the cylinder and then closes the top end of the cylinder. If
she puts the largest possible number of spheres in the cylinder, what
is the volume, in
Determine the value of the expression
Two vertical chords are drawn in a circle, dividing the circle into 3
distinct regions. Two horizontal chords are added in such a way that
there are now 9 regions in the circle. A fifth chord is added that
does not lie on top of one of the previous four chords. The maximum
possible number of resulting regions is
The product of the roots of the quadratic equation
The four positive integers
Determine all real values of
If
A solid triangular prism is made up of 27 identical smaller solid triangular prisms, as shown. The length of every edge of each of the smaller prisms is 1.
If the entire outer surface of the larger prism is painted, what fraction of the total surface area of all the smaller prisms is painted?
Two circles are drawn, as shown. Each is centered at the origin and the radii are 1 and 2.
Determine the value of
Camp Koeller offers exactly three water activities: canoeing, swimming and fishing. None of the campers is able to do all three of the activities. In total, 15 of the campers go canoeing, 22 go swimming, 12 go fishing, and 9 do not take part in any of these activities. Determine the smallest possible number of campers at Camp Koeller.
Determine the area inside the circle with centre
When the polynomial
Determine the number of pairs of real numbers,
Starting at midnight, Serge writes down the time after 1 minute, then 2 minutes after that, then 3 minutes after that, and so on. For example, the first four times that he writes are 12:01 a.m., 12:03 a.m., 12:06 a.m., and 12:10 a.m. He continues this process for 24 hours. What times does Serge write down that are exactly on the hour? (For example, 3:00 a.m. and 8:00 p.m. are exactly on the hour, while 11:57 p.m. is not.)
Figure 0 consists of a square with side length 18. For each integer
What is the smallest
positive integer
Brad answers 10 math problems, one after another. He answers the first problem correctly and answers the second problem incorrectly. For each of the remaining 8 problems, the probability that he answers the problem correctly equals the ratio of the number of problems that he has already answered correctly to the total number of problems that he has already answered. What is the probability that he answers exactly 5 out of the 10 problems correctly?
The lines with equations
A large piece of canvas is in the shape of an isosceles triangle with
The canvas is folded along median
What is the height of the tent (the distance from