Number Sense
Logic Puzzles
Geometry
Statistics & Probability
Math in Everyday Life
100
How do you check if any integer x is divisible by 12 without a calculator?
Check if the sum of its digits is divisible by 3, then check whether the digits in the last two places are divisible by 4.
100
Which is the bigger number?
99102,100101,101100,10299
Biggest number = 99102
100
How many triangles are there in the figure?
Number of triangles = 13
100
A frog jumps from the starting lilypad* and it cannot travel backwards. How many combinations can the frog travel towards land by lilypads?
* “frog” denotes starting point
Number of combinations = 6
100
Before a train enters a tunnel, it is important that it sounds its horn. The train speed is 80 km/h and the speed of sound is taken to be 340 m/s. The echo of the horn signal is reflected off the surface of the cliff through which the tunnel runs. The echo is heard 2.0 s after the driver sounds the horn. What is the distance of the train from the cliff at the time when the echo is heard? Leave your answer to 3 significant figures.
Distance = 318 m
200
State the general quadratic formula.
200
A rectangular sheet of paper ABCD is folded as shown in the figure. EF is the crease. The images of points A and B after folding are G and H, respectively. If ∠1=α, express ∠2 in terms of α.
Expression = 0.5a
200
A variation of reversi is that there is no limit to where you can play and the board is 8 x 8 big. (i.e. You can choose to not flip your opponent in any given move). Find the number of possible combinations of the game. Number of combinations = *x*= (x)(x-1)(x-2)...(1). Find x.
x = 64
Solution:
*x* = 64!
200
The construction of the Pyramid of Giza required an active team of 25000 workers, over 20 years. Two-thirds of the workers left the job half a decade after construction began. Assuming all workers work at the same rate, how long would it take in total to build the Pyramid from the start? Leave your answer to the nearest year.
Years taken = 50 years
Solution: Inverse and direct proportion between time, work rate, and number of workers
300
The number 229 is a nine digit number with nine distinct digits. Which digit is missing?
Digit missing = 4
300
You are somewhere on the Earth. If you move 1km south, 1km west and 1km north, you still end up on the same point. Where on Earth are you?
Location = North or South Pole
300
Given that AE∥BD and ∠A=∠D. The angle bisector of ∠BAE intersects CD at F. Ray FN intersects AB at G. The angle bisector of ∠CFN intersects AB at H. As shown in Figure 2, if ∠ECF=120∘,∠AFH=20∘,∠CFG=110∘, find the measure of ∠E.
∠E = 50˚
300
Given that,
where (n r) denotes n choose r
How many ways are there to arrange the following letters in a row? (W, W, W, X, X, X, Y, Y, Y)
Number of ways = 1680
Solution:
(9 3) × (6 3) × 1 = 1680
300
A large spool of rope lies on the ground. End X is pulled a distance of 2x cm in the horizontal direction. Given that the spool rolls without slipping, what is the distance moved by the spool’s centre O?
Distance moved by the spool's centre O = x cm
Solution:
Distance moved by the tallest point of the spool = 2x cm
translational distance + rotational distance = 2x = x + x
translational distance = x cm
400
If the exact fraction form of 0.3 can be shown by:
0.333.. = x
3.333.. =10x
3 =9x
x = 1/3
Find the exact fractional form of 0.00028032...
Exact fractional form = 
400
Given each letter represents a unique number from 0-9 where N, S, A cannot be equal to 0. Let the number "AGAIN" = x
Find x.
x = 14105
400
A horse keeper starts at point A with a horse. Before returning to the stable at point B, the horse must first stop at the river to drink water. Point A is 20 m from the river bank. Point B, the stable, is 10 m from the river bank. The horizontal distance between A and B is 40 m. (The blue solid line is the river bank) What is the shortest possible total distance they travel?
Shortest possible distance = 50 m
400
A set of two cards is called a “blackjack!” if the combination contains special cards: an ace and either a king, queen, jack, or “10” card. Suppose there are cards {1, 2, 3, …, 10, king, queen, jack, ace} in a game set, with each number card and special cards being distinct and only appearing ONCE. What is the probability a player obtains a “blackjack!” on the first try?
Probability = 4/91
400
A 20 m pipe with an internal radius of 5 cm is filled with water to a level of 7 cm. The surface area of the pipe that is not exposed to water is 23200 cm2. Given that the pipe is placed sideways, what is the total volume of water in the pipe? Leave your answer in cm3 to 3 significant figures.
Background Information:
Total Volume of Water in Pipe = 117000 cm3
Solution:
Divide the cross-section of the pipe into a sector and a triangle to find the individual areas, then the volume of the water.
500
It is given to you that i2 = -1.
With that knowledge, conclude the factorisation of a2+b2 in terms of a, b and i.
Factorisation = (a+bi)(a-bi)
500
As shown in the figure, squares with side lengths 1,2,4,8,… are drawn successively on the Cartesian plane. A moving point starts from the origin O(0,0) and travels along the sides of the squares according to the pattern “right → up → right → up → ⋯”. It reaches the points A1(1,0), B1(1,1), A2(2,1), B2(2,3), A3(4,3), …
Following this pattern, the coordinates of B2026 are:
A. (22025, 22026+1)
B. (22025, 22026)
C. (22027, 22027−1)
D. (22025, 22026−1)
The coordinates are D.
500
A regular fair dice is thrown twice. Suppose that the first roll lands on the number a while the second roll lands on the number b. What is the probability that a2+b2=c2, for an integer c?
Probability = 1/18
500
A man stands directly in front of a mountain, and the initial angle of elevation from his position to the top of the mountain is measured to be 24.5°. After walking 100 m in a straight line towards the mountain, the angle of elevation from his position to the top of the mountain is now measured to be 26.5°. After hiking up the mountain, the man finds that the angle of depression from the mountain's tallest point to the visible horizon is 0.74°.
Assuming the ground is perfectly flat, use the given values to estimate the Earth's radius in km. Leave your answer to 3 significant figures.
Background Information:
Estimated Radius of Earth = 6360 km (3 s.f.)
Solution:
Draw a diagram to calculate the height of the mountain
Draw a diagram of the Earth to form a right-angled triangle with the radius and known height of the mountain