Probability & Counting
A jar contains 6 crayons, of which 3 are red, 2 are blue, and 1 is green. Jakob reaches into the jar and randomly removes 2 of the crayons. What is the probability that both of these crayons are red?
1/5 (CSMC 2018 A3)
Which of the fractions 2/3, 3/4, 5/6, 5/8, and 11/12 is the smallest?
5/8 (CIMC 2016 A1)
Rectangle A has length 6 cm and area 36 cm2.
Rectangle B has length 12 cm and area 36 cm2.
Rectangle C has length 9 cm and area 36 cm2.
The rectangle with the smallest width has a width of x cm. What is the value of x?
3 (CIMC 2021 A1)
Stephanie has 1000 eggs to pack into cartons of 12 eggs. While packing the eggs, she breaks n eggs. The unbroken eggs completely fill a collection of cartons with no eggs left over. If n<12, what is the value of n?
4 (CIMC 2015 A1)
Ali plays a trivia game with 5 categories, each with 3 questions. She earns 1 point for each correct answer. If she answers all 3 questions in a category correctly, she earns 1 bonus point. Ali answers exactly 12 questions correctly and the remaining 3 questions incorrectly. What are her possible total scores?
14, 15, 16 (CIMC 2019 A4)
If 12x=4y+2, determine the value of the expression 6y−18x+7.
4 (CIMC 2015 A4)
Two 8 by 10 rectangles overlap to form a 4 by 4 square. What is the total area of the region not enclosed by the square?
128 (CIMC 2020 A2)
Starting on the 22nd floor of their apartment building, Taya goes up the stairs and Jenna goes up by elevator. Beginning when Taya starts going up, Jenna waits for 2 minutes for the elevator. Taya goes up from each floor to the next floor in 15 seconds. The elevator goes up from each floor to the next floor in 3 seconds. Taya and Jenna arrive on the nth floor at exactly the same time. What is the value of n?
32 (CIMC 2021 A3)
Abigail chooses an integer at random from the set {2,4,6,8,10}. Bill chooses an integer at random from the set {2,4,6,8,10}. Charlie chooses an integer at random from the set {2,4,6,8,10}. What is the probability that the product of their three integers is not a power of 2?
98/125 (Cayley 2018 20)
If a, b, c, and d satisfy ab=2/3 and cb=1/5 and cd=7/15, what is the value of abcd?
70/9 (CIMC 2022 A4)
ABCD is a square with side length 8 cm. Point E is on AB and point F is on DC so that △AEF is right-angled at E. If the area of △AEF is 30% of the area of ABCD, what is the length of AE?
4.8 cm (CSMC 2017 A2)
Let ⌊x⌋ denote the greatest integer which is less than or equal to x. For example, ⌊π⌋=3. S is the integer equal to the sum of the 100 terms shown:S=⌊π⌋+⌊π+1/100⌋+⌊π+2/100⌋+⌊π+3/100⌋+⋯+⌊π+99/100⌋ What is the value of S?
314 (CSMC 2020 A4)
Six players compete in a chess tournament. Each player plays exactly two games against every other player. In each game, the winning player earns 1 point and the losing player earns 0 points; if the game results in a draw (tie), each player earns 1/2 point. What is the minimum possible number of points that a player needs to earn in order to guarantee that he has more points than every other player?
9.5 (Cayley 2022 22)
Dina has a calculating machine, labelled f, that takes one number as input and calculates an output. The machine f calculates its output by multiplying its input by 2 and then subtracting 3. For example, if Dina inputs 2.16 into f, the output is 1.32. If Dina inputs a number x into f, she gets a first output which she then inputs back into f to obtain a second output, which is −35. What is the value of x?
-6.5 or -13/2 (CIMC 2016 A4)
ABCD is a rectangle, P is on BC, Q is on CD, and R is inside ABCD. Also, ∠PRQ=30∘, ∠RQD=w∘, ∠PQC=x∘, ∠CPQ=y∘, and ∠BPR=z∘. What is the value of w+x+y+z?
Alain and Louise are driving on a circular track with radius 25 km. Alain leaves the starting line first, going clockwise around the track at a speed of 80 km/h. Fifteen minutes after Alain starts, Louise leaves the same starting line, going counter-clockwise around the track at a speed of 100 km/h. For how many hours will Louise have been driving when the two of them pass each other for the fourth time? (give your answer in terms of pi)
(10pi-1)/9
A bag contains exactly 15 marbles of which 3 are red, 5 are blue, and 7 are green. The marbles are chosen at random and removed one at a time from the bag until all of the marbles are removed. One colour of marble is the first to have 0 remaining in the bag. What is the probability that this colour is red?
21/40 (CSMC 2022 A6)
Determine all pairs (x,y) of integers with x≤y for which 1/x+1/y=1/4
(-4,2),(-12,3),(5,20),(6,12),(8,8) (CIMC 2020 A5)
A cylinder has radius 12 and height 30. The top circular face of the cylinder is the base of a cone and the centre of the bottom circular base of the cylinder is the vertex of the cone. A sphere is placed inside so that it touches the cone, the base of the cylinder and the side of the cylinder as shown. What is the radius of the sphere to the nearest hundredth.
4.84 (Fermat 2017 22)
If n is a positive integer, a Leistra sequence is a sequence a1,a2,a3,…,an−1,an with n terms with the following properties:
Each term a1,a2,a3,…,an−1,an is an even positive integer.
Each term a2,a3,…,an−1,an is obtained by dividing the previous term in the sequence by an integer between 10 and 50, inclusive. (For a specific sequence, the divisors used do not all have to be the same.)
There is no integer m between 10 and 50, inclusive, for which an/m is an even integer.
How many Leistra sequences have a1=2^2×3^50?
17 (CSMC 2021 B2c)