(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
Zeljko travelled at 30 km/h for 20 minutes and then travelled at 20 km/h for 30 minutes. How far did he travel, in kilometres?
The operation
Two fair six-sided dice are tossed and the numbers shown on the top face of each are added together. What is the probability that the resulting sum is less than 10?
A palindrome is a positive integer whose digits are the same when read
forwards or backwards. For example,
On a particular street in Waterloo, there are exactly 14 houses, each
numbered with an integer between 500 and 599, inclusive. The 14 house
numbers form an arithmetic sequence in which 7 terms are even and 7
terms are odd. One of the houses is numbered 555 and none of the
remaining 13 numbers has two equal digits. What is the smallest of the
14 house numbers?
(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 is an arithmetic sequence with four
terms.)
Point
Claudine has
If Claudine divides all of her candies equally among 7 friends, there
are 4 candies left over.
If Claudine divides all of her candies equally among 11 friends, there
is 1 candy left over.
What is the minimum possible value of
In the diagram,
If the height of the truncated pyramid is 4, what is the total shaded area?
Determine all pairs
Evaluate
Let
What is the area of a triangle with base of length
Let
In the diagram,
If
If
Let
If
Let
The
What is the value of
What is the sum of the
Let
In the diagram, point
If the area of
Let
One cylinder has a radius of
Let
Let
What is the value of
Let
Over the winter, Oscar counted the birds in his backyard. He counted three
different types of birds: sparrows, finches and cardinals. Three-fifths of
the birds that he counted were sparrows. One-quarter of the birds that he
counted were finches. If Oscar counted exactly
Let
A large theatre has 20 rows of seats. Each row after the first row
contains 4 more seats than the previous row. If there are
(45 minutes)
Calculating devices are not permitted.
Express answers as simplified exact numbers except where otherwise
indicated. For example,
What is the value of
The average (mean) of 3, 5, 6, 8, and
For any real number
A street magician has three cups labelled, in order, A, B, C that he has upside down on his table. He has a sequence of moves that he uses to scramble the three cups: he swaps the first and second, then he swaps the second and third, then he swaps the first and third. If he goes through this sequence of three moves a total of nine times, in what order will the cups be?
A parabola has equation
For some positive integers
A two-digit integer between 10 and 99, inclusive, is chosen at random. Each possible integer is equally likely to be chosen. What is the probability that its tens digit is a multiple of its units (ones) digit?
Rectangle
If
Suppose that
Clara takes 2 hours to ride her bicycle from Appsley to Bancroft. The reverse trip takes her 2 hours and 15 minutes. If she travels downhill at 24 km/h, on level road at 16 km/h and uphill at 12 km/h, what is the distance, in kilometres, between the two towns?
The first and second terms of a sequence are 4 and 5, respectively.
Each term after the second is determined by increasing the previous
term by one and dividing the result by the term before that. For
example, the third term equals
Austin and Joshua play a game. Austin chooses a random number equal to 1, 2, 3, 4, or 5. Joshua then chooses randomly from the remaining four numbers. Joshua’s first round score is equal to the product of his number and Austin’s number. Austin then chooses randomly from the remaining three numbers, and his first round score is the product of his second number and Joshua’s first number. The process is repeated until each of the five numbers has been chosen. The sum of each player’s two scores is their final score and the player with the highest final score wins. If Austin chooses 2 to start and Joshua then chooses 3 (making Joshua’s first round score 6), what is the probability that Austin will win?
A sphere and a cone have the same volume. The area of the lateral
surface of the cone is
(The volume of a sphere with radius
Determine the value of the following sum:
A lock has a combination that is a four-digit positive integer. The first digit is 4 and the four-digit combination is divisible by 45. How many different possible combinations are there?
In a regular
If
There are 21 marbles in a bag. The number of each colour of marble is shown in the following table:
Colour | Number |
---|---|
magenta | 1 |
puce | 2 |
cyan | 3 |
ecru | 4 |
aquamarine | 5 |
lavender | 6 |
For example, the bag contains 4 ecru marbles. Three marbles are
randomly drawn from the bag without replacement. The probability that
all three of these marbles are the same colour can be written as
For each real number
In the diagram,
If
A group of cows and horses are randomly divided into two equal rows. (The animals are well-trained and stand very still.) Each animal in one row is directly opposite an animal in the other row. If 75 of the animals are horses and the number of cows opposite cows is 10 more than the number of horses opposite horses, determine the total number of animals in the group.
Three circles each with a radius of 1 are placed such that each circle touches the other two circles, but none of the circles overlap. What is the exact value of the radius of the smallest circle that will enclose all three circles?
In a hospital, there are 7 patients (Doc, Grumpy, Happy, Sleepy, Bashful, Sneezy, and Dopey) who need to be assigned to 3 doctors (Huey, Dewey, and Louie). In how many ways can the patients be assigned to the doctors so that each patient is assigned to exactly one doctor and each doctor is assigned at least one patient?
Suppose that
(The infinite sum includes exactly the fractions of the form
In the diagram, rectangular prism
Determine the minimum possible
length of line segment
The sequences
There exist positive integers
(If