Section A
Multiple Choice Questions — 15 Questions
(+4)
1
arithmetic
easy
Brenda multiplies a number by 3, adds 3, divides by 3, adds 6, and subtracts 3 from the result to get 9. What is six times the original number?
The source PDF prints no A/B/C/D/E letters for these options; labels follow the exact on-page top-to-bottom order.
Answer: C
Work backwards to find the original number.
9 + 3 = 12, 12 − 6 = 6, 6 × 3 = 18, 18 − 3 = 15, 15 ÷ 3 = 5.
Six times the original number is 6 × 5 = 30.
2
number-theory
medium
What is the least number of 2-digit numbers that must be added to get the sum of 2021?
Answer: B
The greatest 2-digit number is 99. Since 2021 ÷ 99 = 20.41, the least number cannot be 20.
Also, 99 × 20 + 41 = 2021, so 20 numbers is not enough to reach 2021.
Thus, the least number is 21.
3
geometry
medium
Which of the following is true?
Answer: C
Option A: When the radius and the given distance are equal, the regular octagon would have a smaller area, as it would fit inside the circle fully, leaving gaps.
Option B: An octagon has 8 sides and a trapezium has 4 sides. 8 − 4 = 4, which is not a prime number.
Option C: A regular quadrilateral, by definition, is a closed figure with 4 equal sides — this statement is true.
Option D: A decagon has 10 sides, so 4 + 10 = 14. When two regular heptagons (7 sides each) are stuck to each other, sharing one side, the resulting figure has 7 × 2 − 2 = 12 sides, not 14.
Option E: This is the area formula for all triangles, not just right-angled triangles.
4
arithmetic
easy
What is the average of the sum of the first 4 counting numbers and the sum of the first 5 even numbers?
Answer: B
The first four counting numbers are: 1, 2, 3 and 4. Their sum = 1 + 2 + 3 + 4 = 10.
The first five even numbers are: 2, 4, 6, 8 and 10. Their sum = 2 + 4 + 6 + 8 + 10 = 30.
Average = (10 + 30) ÷ 2 = 40 ÷ 2 = 20.
5
fractions
easy
One-fifth of all apples in a basket are red. Three-fourths are yellow. The remaining are green. If there are 200 apples in the basket, how many of them are green?
Answer: A
1/5 of 200 = 200 ÷ 5 = 40.
3/4 of 200 = 200 ÷ 4 × 3 = 150.
Total number of red and yellow apples = 40 + 150 = 190.
Number of green apples = 200 − 190 = 10.
6
algebra
hard
Amanda's score was twice Brian's score. Cassie scored 5 points less than Brian. Dora scored 10 points more than Amanda. Dora's score was 6 times as much as Cassie's. Whose score was 10?
Answer: D
Let Cassie's score be 1 unit. Then Brian = 1 unit + 5, Amanda = 2 units + 10, and Dora = 2 units + 20.
Since Dora's score was 6 times as much as Cassie's, 2 units + 20 = 6 units, so 4 units = 20, giving 1 unit = 5.
Therefore, Brian scored 10 points.
7
number-theory
easy
The largest 4-digit number A is formed by using all the digits 4, 5, 0 and 3. The smallest 4-digit number B is formed by using all the digits 4, 5, 0 and 3. Find the value of A − B.
Answer: A
The largest 4-digit number A = 5430.
The smallest 4-digit number B = 3045.
A − B = 2385.
8
3d-geometry
medium
The 2 figures below are made up of 3 cm cubes. What is the total volume of both figures?
Answer: C
The first shape has 4 × 3 = 12 cubes.
The second shape has 1 + 2 + 3 + 2 + 1 = 9 cubes.
The total number of cubes is 12 + 9 = 21.
Volume of each cube = 3 × 3 × 3 = 27 cm³.
Total volume = 21 × 27 = 567 cm³.
9
counting
easy
Each digit of a 4-digit PIN code can be any digit from 0 to 9. How many possible 4-digit PIN codes are there?
Answer: A
All the possible PIN codes are from 0000 to 9999. Thus, there are 10 × 10 × 10 × 10 = 10,000 possibilities.
Another way to solve this is to count the number of numbers from 0000 to 9999. The count is (last number − first number + 1) = 9999 − 0 + 1 = 10,000.
10
number-theory
easy
What is the largest 3-digit number which is divisible by 32?
Answer: D
The largest 3-digit number is 999.
However, 999 ÷ 32 = 31 R7, so 999 is not divisible by 32.
Hence the largest 3-digit multiple of 32 is 32 × 31 = 992.
11
geometry
medium
The figure below is made of three regular pentagons with a length of 3 cm. What is the perimeter of the figure?
Answer: D
Counting only the exposed outer sides of the three joined pentagons (the shared internal sides do not contribute to the perimeter) gives 11 sides of length 3 cm each.
Perimeter = 3 × 11 = 33 cm.
12
patterns
medium
What is the 2021st number in the sequence below?
7, 9, 11, 13, ...
Answer: C
The common difference between any two consecutive terms is 2.
The 1st term can be written as 2 × 1 + 5. Similarly, the second and third terms can be written as 2 × 2 + 5 and 2 × 3 + 5.
Thus, the 2021st term is 2 × 2021 + 5 = 4047.
13
number-theory
medium
Which of the following numbers has the most number of even prime factors?
Answer: C
The number 2 is the only even prime number. Count the number of factor 2s for each option.
128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 has the most number of factor 2s.
14
counting-figures
medium
The following picture shows a 4 × 4 square. How many squares of any size are there comprised of the one marked with X?
Answer: D
Counting all squares of every size that contain the cell marked X:
1×1 squares: 1
2×2 squares: 4
3×3 squares: 4
4×4 squares: 1
Total = 1 + 4 + 4 + 1 = 10.
15
algebra
easy
A number p is 1.5 times another number q. If p is 18 bigger than q, then what is p + q?
Answer: E
Let q = 2 units, then p = 3 units. Then p − q = 1 unit = 18.
p + q = 5 units = 5 × 18 = 90.
Section B
Open-Ended (Integer) Questions — 10 Questions
(+5)
16
3d-geometry
medium
A 5 × 5 × 5 cube is painted red on all its faces. Find the number of smaller 1 × 1 × 1 cubes with at most two faces painted.
Answer: 117
There are 5 × 5 × 5 = 125 small cubes in total.
Only corner cubes have 3 faces painted. There are 8 small corner cubes.
The number of smaller cubes with at most 2 sides painted = 125 − 8 = 117.
17
number-theory
easy
The sum of 5 different non-zero whole numbers is 45. What is the largest possible value of the largest number among these 5 numbers?
Answer: 35
For the largest number to be as large as possible, the sum of all other 4 numbers must be as small as possible. The smallest possible sum of the 4 numbers is 1 + 2 + 3 + 4 = 10.
Therefore, the largest possible value of the fifth number is 45 − 10 = 35.
18
counting-digits
medium
Tom read a book with 2022 pages. How many digits were used to print the page numbers on the book?
Answer: 6981
Page 1 to 9: 9 digits.
Page 10 to 99: 90 × 2 = 180 digits (99 − 10 + 1 = 90 2-digit numbers).
Page 100 to 999: 900 × 3 = 2700 digits (999 − 100 + 1 = 900 3-digit numbers).
Page 1000 to 2022: 1023 × 4 = 4092 digits.
There are 9 + 180 + 2700 + 4092 = 6981 digits.
19
divisibility
easy
The 5-digit number 653A2 is a multiple of 3. How many possible digits A are there?
Answer: 3
Any number whose sum of digits is a multiple of 3 is divisible by 3.
Thus, 6 + 5 + 3 + A + 2 = 16 + A must be a multiple of 3.
16 + A = 18 or 21 or 24.
Therefore, there are 3 possible values of A: 2, 5 or 8.
20
number-theory
hard
A five-digit number is formed such that it satisfies the following conditions:
• It is a multiple of 3 and 5.
• The third digit is half of the first digit and one less than the second digit.
• The sum of the first three digits is 13 and the sum of the last three digits is 8.
• The fourth digit is the second-largest digit of that number.
Find the sum of digits of the 5-digit number.
Answer: 18
Since the third digit is half of the first digit and one less than the second digit, the possible (first, second, third) digit triples are (2,2,1), (4,3,2) or (6,4,3), with digit sums 5, 9 and 13 respectively.
As the sum of the first three digits is 13, the first three digits must be 6, 4 and 3.
Since the sum of the last three digits is 8 and the third digit is 3, the last two digits must sum to 8 − 3 = 5.
6 must be the largest digit of the number, so the fourth digit (the second-largest digit) is 5, which makes the last digit 5 − 5 = 0.
The number required is 64350, and the sum of its digits is 6 + 4 + 3 + 5 + 0 = 18.
21
geometry
medium
A rectangle with an area of 60 cm² is divided into 5 identical smaller rectangles. The length of each smaller rectangle is 3 times its width. Find the perimeter, in cm, of one smaller rectangle.
Answer: 16
Area of each smaller rectangle = 60 cm² ÷ 5 = 12 cm².
The length of the smaller rectangle is 3 times its width. Testing possible whole-number length-and-width pairs with a product of 12 cm² — (12,1), (6,2), (4,3), (1,12) — only the pair (6,2) has a length that is 3 times its width.
Thus, each small rectangle has a width of 2 cm and a length of 6 cm.
The perimeter of each small rectangle is (2 + 6) × 2 = 16 cm.
22
rate-problems
medium
A train travelling at 54 km/h passes a platform. A man is standing on the platform, and the train passes the man in 20 seconds. Find the length of the train, in metres.
Answer: 300
When the train passes the man in 20 seconds, it travels the distance equivalent to its length.
Convert seconds to hours: 20 seconds = 1/3 minute = 1/3 × 1/60 of an hour = 1/180 hour.
Distance = length of the train = (1/180) × 54 km = 3/10 km = (3/10) × 1000 = 300 metres.
23
patterns
medium
Find the sum of the numbers in the 101st group of the following sequence:
(1, 4, 6), (2, 6, 10), (3, 8, 14), ...
Answer: 711
The sum of the numbers in each group follows a pattern:
Group 1: 1 + 4 + 6 = 11 = 11 + 0 × 7
Group 2: 2 + 6 + 10 = 18 = 11 + 1 × 7
Group 3: 3 + 8 + 14 = 25 = 11 + 2 × 7
Following this pattern, Group 101's sum = 11 + 100 × 7 = 711.
24
number-theory
hard
A 4-digit number AABB is a perfect square. What is the square root of AABB?
Answer: 88
AABB = AA × 100 + BB = A × 11 × 100 + B × 11 = 11 × (A × 100 + B).
Hence AABB is a multiple of 11. Since it is a perfect square, it must be divisible by 11 × 11 = 121, so A × 100 + B = A0B must also be a multiple of 11.
From the divisibility rule of 11, A + B = 11.
Out of all possible values of A and B, only A = 7 and B = 4 satisfy the conditions, giving AABB = 7744.
√7744 = 88.
25
number-theory
medium
The sum of two whole numbers A and B is 1085. The highest common factor of these 2 numbers is 35. Find the value of (A + B) / 35.
Answer: 31
Let A = 35 × C and B = 35 × D. Then C and D do not have any common factors other than 1.
A + B = 35 × (C + D) = 1085, so C + D = 31.
(A + B) / 35 = 35 × (C + D) / 35 = C + D = 31.