Secondary 1 Set 5
Logical Thinking
1. Betty goes northwest 53km, then goes northeast for 48km and goes southeast for 17km. How far
is she now from the original position?
2. How many integral solution(s) is / are there for π₯ if −4 ≤
(π₯−6)×3+10
4
< 3?
3. A bus from point A to point B takes 5 hours. The average speed from B to A is 60km/h, and the
average speed from B to A is two-thirds of the speed from A to B. Find the distance between A and
B in km.
4. It is known that π΄: π΅ = 1: 2, π΅: πΆ = 3: 5, πΆ: π· = 2: 11, π΄ + π΅ + πΆ + π· = 1480. Find the value of π· −
π΄.
5. There are some chickens and pigs in a cage and the number of chickens is 8 less than 5 times the
number of pigs. The number of legs of chickens is 26 less than 3 times the number of legs of pigs.
What is the total number of chickens and pigs in the cage?
Algebra
6. If 504 can be written as a sum of the squares of 4 consecutive even integers, find the smallest even
number among them.
7. Find the value of π₯ if (48 + 43π₯) − 7(11π₯ − 32) = 0.
8. If π₯ and π¦ are positive integers and 5π₯ − 3π¦ = 13, find the minimum value of 7π₯ + 5π¦.
9. Factorize 4π₯ 2 − 3π₯ − 10.
10. If π₯ and √π₯ 2 + 32 are both positive integers, find the sum of all possible values of x.
Number Theory
11. Find the smallest integer π₯ that is larger than 2023, such that when 3π₯−2023 is divided by 20, it has
a remainder of 3.
12. If π₯ ≡ 13(πππ17), find the maximum value of this 3-digit number π₯.
13. Find the remainder when 102023 is divided by 99.
14. A 3-digit number is divisible by 5, has a remainder 4 when divided by 7 and has a remainder 7
when divided by 11. What is such smallest 3-digit number?
15. Find the last digit of π΄ if π΄ = 4 + 10 + 28 + β― + (398 + 1) + (399 + 1).
Geometry
16. An iron wire is bent to form eleven identical squares. The area of each square is 256. If the wire is
now bent to form four identical circles, find the radius of the circles? (Take π =
22
7
).
17. The perimeter of the base of a square pyramid is 24cm. If the area of the base of the pyramid is 3
times the height of the pyramid, find the volume of the square pyramid.
18. A square and a right-angled triangle overlap. The base of the triangle is the same as the side length
of the square and the height of the triangle is twice the side length of the square. If the length of
the diagonal of the square is 10, find the area of the triangle.
19. In convex quadrilateral ππππ, ππ = 7, ππ = 24, ππ = 15, ππ = 20. Find the area of the convex
quadrilateral.
20. Combine 693 squares with sides 1 unit to form a rectangle, find the minimum perimeter of that
rectangle.
Combinatorics
21. Rearrange the letters of the word “ABCDEFG” such that A, E become the first and the last letter of
the word, how many different arrangements are there?
22. 8 identical red books, 3 identical blue books and 2 identical yellow books are arranged in a row
from left to right. How many different permutation(s) is / are there?
23. Find the number of 3-digit positive integers such that the product of its digits is 12 or 27.
24. Suppose 3 cards are drawn from an ordinary poker deck of 52 playing cards without replacement.
Find the probability that the 3 cards share the same suit.
25. Find the number of positive square numbers less than 4000 that are divisible by 6.
Secondary 1 Set 5
Logical Thinking
1. Betty goes northwest 53km, then goes northeast for 48km and goes southeast for 17km. How far
is she now from the original position?
Answer: 60
Solution:
√(53 − 17)2 + 482
= √362 + 482
= √1296 + 2304
= √3600
= 60
2. How many integral solution(s) is / are there for π₯ if −4 ≤
(π₯−6)×3+10
4
< 3?
Answer: 9
Solution:
−4 ≤
(π₯ − 6) × 3 + 10
<3
4
−4 × 4 ≤ 3π₯ − 18 + 10 < 3 × 4
−16 ≤ 3π₯ − 8 < 12
−16 + 8 ≤ 3π₯ < 12 + 8
−8 ≤ 3π₯ < 20
−8
20
≤π₯<
3
3
−2.66 ≤ π₯ < 6.66
π₯ = {−2, −1, 0, 1, 2, 3, … , 6}
πβπππ πππ 9 πππ‘πππππ π£πππ’ππ πππ π₯
3. A bus from point A to point B takes 5 hours. The average speed from B to A is 60km/h, and the
average speed from B to A is two-thirds of the speed from A to B. Find the distance between A and
B in km.
Answer: 450
Solution:
π΄π π π’ππ π₯ ππ π‘βπ π ππππ ππππ π΄ π‘π π΅
2
60 = π₯
3
180 = 2π₯
π₯ = 90
π·ππ π‘ππππ = πππππ × ππππ = 90 × 5 = 450
4. It is known that π΄: π΅ = 1: 2, π΅: πΆ = 3: 5, πΆ: π· = 2: 11, π΄ + π΅ + πΆ + π· = 1480. Find the value of π· −
π΄.
Answer: 1040
Solution:
π΄: π΅: πΆ: π· = 1 × 3: 2 × 3: 2 × 5: 11 × 5 = 3: 6: 10: 55
3 + 6 + 10 + 55 = 74
π
ππ‘ππ πΉπππ‘ππ =
1480
= 20
74
π· = 55 × 20 = 1100
π΄ = 3 × 20 = 60
π· − π΄ = 1100 − 60 = 1040
5. There are some chickens and pigs in a cage and the number of chickens is 8 less than 5 times the
number of pigs. The number of legs of chickens is 26 less than 3 times the number of legs of pigs.
What is the total number of chickens and pigs in the cage?
Answer: 22
Solution:
π΄π π π’ππ π₯ ππ ππ’ππππ ππ πβππππππ πππ π¦ ππ ππ’ππππ ππ ππππ
π₯ = 5π¦ − 8
2π₯ = 3(4π¦) − 26
ππ’ππ π‘ππ‘π’π‘π π₯ ππππ ππππ 1π π‘ πππ’ππ‘πππ π‘π π‘βπ 2ππ πππ’ππ‘πππ
2(5π¦ − 8) = 3(4π¦) − 26
10π¦ − 16 = 12π¦ − 26
−16 + 26 = 12π¦ − 10π¦
10 = 2π¦
π¦=5
ππ’ππ π‘ππ‘π’π‘π π¦ π‘π 1π π‘ πππ’ππ‘πππ π‘π πππ‘ π₯
π₯ = 5(5) − 8 = 17
π₯ + π¦ = 17 + 5 = 22
Algebra
6. If 504 can be written as a sum of the squares of 4 consecutive even integers, find the smallest even
number among them.
Answer: 8
Solution:
π΄π π π’ππ π‘βπ ππ£ππ ππ’πππππ πππ ππ π‘βπ π πππ π£πππ’π πππ πππ’ππ π‘π π₯
π₯ 2 + π₯ 2 + π₯ 2 + π₯ 2 = 504
4π₯ 2 = 504
π₯ 2 = 126
121 < 126 < 144
112 < 126 < 122
π₯ ≈ 11
π»ππππ, 11 ππ π‘βπ ππππππ ππ π‘βπ 4 ππππ πππ’π‘ππ£π ππ£ππ ππ’πππππ
π΅π¦ πππ πππ£ππ‘πππ, π‘βπ ππ’πππππ π€ππ’ππ ππ 8, 10, 12, 14
8 ππ π‘βπ π ππππππ π‘ πππππ π‘βππ
7. Find the value of π₯ if (48 + 43π₯) − 7(11π₯ − 32) = 0.
Answer: 8
Solution:
(48 + 43π₯) − 7(11π₯ − 32) = 0
48 + 43π₯ − 77π₯ + 224 = 0
−34π₯ + 272 = 0
−34π₯ = −272
π₯=8
8. If π₯ and π¦ are positive integers and 5π₯ − 3π¦ = 13, find the minimum value of 7π₯ + 5π¦.
Answer: 55
Solution:
ππ πππ 5π₯ − 3π¦ = 13,
πππ πππ£π π‘βππ‘ π€βππ π‘βπ π£πππ’π ππ π₯ ππ πππππππ πππ, π‘βπ π£πππ’π ππ π¦ ππ πππ π πππππππ πππ
π»ππππ, π‘βπ πππ π ππ π‘βπ π£πππ’π ππ π₯ πππ π¦, π‘βπ πππ π ππ π‘βπ π£πππ’π ππ 7π₯ + 5π¦
π΅π¦ ππ’ππ π πβππππππ ππ π‘ππππ πππ πππππ,
π‘βπ ππππ π‘ πππ π ππππ π£πππ’π ππ π₯ ππ 5 π€βπππ π¦ ππ 4
πβππ, 7 × 5 + 5 × 4 = 55
9. Factorize 4π₯ 2 − 3π₯ − 10.
Answer: (4π₯ + 5)(π₯ − 2) ππ (π₯ − 2)(4π₯ + 5)
Solution:
4π₯ 2 − 3π₯ − 10
4π₯ 2 − 8π₯ + 5π₯ − 10
4π₯(π₯ − 2) + 5(π₯ − 2)
(4π₯ + 5)(π₯ − 2)
10. If π₯ and √π₯ 2 + 32 are both positive integers, find the sum of all possible values of x.
Answer: 9
Solution:
π΄π π π’ππ √π₯ 2 + 32 ππ π£πππππππ π¦,
π¦ = √π₯ 2 + 32
π¦ 2 = π₯ 2 + 32
π¦ 2 − π₯ 2 = 32
(π¦ + π₯)(π¦ − π₯) = 32
πΉπππ‘πππ ππ 32: ( 1 πππ 32), (2 πππ 16), πππ (4 πππ 8)
πΉππ 1 πππ 32:
π¦ + π₯ = 32 πππ π¦ − π₯ = 1
π¦ = 16.5; π₯ = 15.5
π΅ππ‘β π₯ πππ π¦ πππ πππ‘ πππ‘πππππ
πΉππ 2 πππ 16
π¦ + π₯ = 16 πππ π¦ − π₯ = 2
π¦ = 9; π₯ = 7
πΉππ 4 πππ 8
π¦ + π₯ = 8 πππ π¦ − π₯ = 4
π¦ = 6; π₯ = 2
π»ππππ, π π’π ππ πππ π£πππ’ππ ππ π₯ = 7 + 2 = 9
Number Theory
11. Find the smallest integer π₯ that is larger than 2023, such that when 3π₯−2023 is divided by 20, it has
a remainder of 3.
Answer: 2024
Solution:
πΉππππππ π π£πππ’π ππ 3π π‘βππ‘ ππππ π€ππ‘β 3:
3, 243, …
31 = 3
π»ππππ, π₯ − 2023 = 1
π₯ = 2024
12. If π₯ ≡ 13(πππ17), find the maximum value of this 3-digit number π₯.
Answer: 999
Solution:
⌊
999
⌋ = ⌊58.76⌋ = 58
17
17 × 58 = 986
π₯ = 986 + 13 = 999
13. Find the remainder when 102023 is divided by 99.
Answer: 10
Solution:
101 ÷ 99 = π
ππππππππ ππ 10
102 ÷ 99 = π
ππππππππ ππ 1
103 ÷ 99 = π
ππππππππ ππ 10
104 ÷ 99 = π
ππππππππ ππ 1
π΅π¦ πππ πππ£ππ‘πππ, π€βππ π‘βπ ππ₯ππππππ‘ ππ πππ, π‘βπ πππππππππ ππ 10
π»ππππ, π‘βπ πππππππππ ππ 102023 ÷ 99 ππ 10
14. A 3-digit number is divisible by 5, has a remainder 4 when divided by 7 and has a remainder 7
when divided by 11. What is such smallest 3-digit number?
Answer: 480
Solution:
5, 10, 15, 20, 25, …
π₯ = 25 + 35π
π₯ = 60, 95, 130, 165, …
πππππππππ π€βππ πππ£ππππ ππ¦ 11 = 5, 7, 9, 0
π»ππππ, π₯ = 95 + 385π§
π₯ = 95 + 385 × 1 = 480
15. Find the last digit of π΄ if π΄ = 4 + 10 + 28 + β― + (398 + 1) + (399 + 1).
Answer: 8
Solution:
(31 + 1) + (32 + 1) + (33 + 1) + β― + (398 + 1) + (399 + 1)
(3 + 1) + (9 + 1) + (7 + 1) + (1 + 1) … + (798 + 1) + (799 + 1)
4+0+8+2+4+0+8+2+4+0+8+2+β―+4+0+8
(4 + 0 + 8 + 2) × 24 + 4 + 0 + 8
14 × 24 + 12
6+2=8
Geometry
16. An iron wire is bent to form eleven identical squares. The area of each square is 256. If the wire is
now bent to form four identical circles, find the radius of the circles? (Take π =
Answer: 28
Solution:
ππππ πππππ‘β ππ π‘βπ ππππ£ππ π ππ’ππππ = √256 = 16
22
7
).
πππ‘ππ πππππ‘β ππ ππππ π€πππ = 11 × 4 × 16 = 704
πππ‘ππ πππππ‘β ππ ππππ π€πππ = 4 × πππππ’ππππππππ ππ ππππππ
704 = 4 × 2ππ
704 = 8 ×
22
×π
7
π = 28
17. The perimeter of the base of a square pyramid is 24cm. If the area of the base of the pyramid is 3
times the height of the pyramid, find the volume of the square pyramid.
Answer: 144
Solution:
ππππ πππππ‘β ππ π ππ’πππ πππ π =
24
=6
4
π΄πππ ππ π ππ’πππ πππ π = 62 = 36
π»πππβπ‘ ππ π‘βπ ππ¦πππππ =
ππππ’ππ ππ π ππ’πππ ππ¦πππππ =
36
= 12
3
1
× 36 × 12 = 144
3
18. A square and a right-angled triangle overlap. The base of the triangle is the same as the side length
of the square and the height of the triangle is twice the side length of the square. If the length of
the diagonal of the square is 10, find the area of the triangle.
Answer: 50
Solution:
π΄π π π’ππ π₯ ππ πππππ‘β ππ π ππ’πππ
π΄πππ ππ π ππ’πππ = π₯ 2
π΄πππ ππ πππβπ‘ π‘πππππππ =
1
1
× π × β = × π₯ × 2π₯ = π₯ 2
2
2
πβπ’π , π΄πππ ππ π ππ’πππ = π΄πππ ππ πππβπ‘ π‘πππππππ
π΅π¦ ππ¦π‘βπππππππ π‘βπππππ,
π₯ 2 + π₯ 2 = 102
2π₯ 2 = 100
π₯ 2 = 50
19. In convex quadrilateral ππππ, ππ = 7, ππ = 24, ππ = 15, ππ = 20. Find the area of the convex
quadrilateral.
Answer: 234
Solution:
πππ πππ£π π‘βππ‘ 72 + 242 = 152 + 202
625 = 625
π»ππππ, π‘βπ ππ’πππππππ‘ππππ ππ ππππππ ππ¦ π‘π€π πππβπ‘ π‘ππππππππ
π΄πππ =
1
1
× 7 × 24 + × 15 × 20 = 234
2
2
20. Combine 693 squares with sides 1 unit to form a rectangle, find the minimum perimeter of that
rectangle.
Answer: 108
Solution:
π΅π¦ πππππππππ, π‘βππ‘ πππππ π‘βπ ππππ ππ π‘βπ ππππ‘πππππ ππ 693
πΌπ πππππ π‘π πππ‘πππ π‘βπ ππππππ’π πππππππ‘ππ,
π‘βπ ππππππ ππππ π βππ’ππ ππ ππππ ππ π‘ π‘π πππβ ππ‘βππ
693 = 21 × 33
π»ππππ, π‘βπ πππππππ‘ππ = (21 + 33) × 2 = 108
Combinatorics
21. Rearrange the letters of the word “ABCDEFG” such that A, E become the first and the last letter of
the word, how many different arrangements are there?
Answer: 240
Solution:
πΊππ‘ πππππ’π‘ππ‘πππ ππ π‘βπ π΄ πππ πΈ
πππ πππ π πππ‘ π‘βπ πππππ’π‘ππ‘πππ ππ π‘βπ ππ‘βππ 5 πππ‘π‘πππ
πππ‘ππ π΄ππππππππππ‘ = 2! × 5! = 240
22. 8 identical red books, 3 identical blue books and 2 identical yellow books are arranged in a row
from left to right. How many different permutation(s) is / are there?
Answer: 12870
Solution:
πβπππ πππ π π‘ππ‘ππ ππ 13 πππππ
13!
8! 3! 2!
13 × 12 × 11 × 10 × 9 × 8!
8! 3! 2!
13 × 12 × 11 × 10 × 9
=
3×2×1×2×1
= 12870
23. Find the number of 3-digit positive integers such that the product of its digits is 12 or 27.
Answer: 22
Solution:
π·ππππ‘π π€ππ‘β πππππ’ππ‘ ππ 12: (1, 3, 4), (2, 2, 3), (1, 2, 6)
πππ (1, 3, 4): 3! = 6 πππ π ππππ ππππππππ‘ππππ
3!
πππ (2, 2, 3): = 3 πππ π ππππ ππππππππ‘ππππ
2!
πππ (1, 2, 6): 3! = 6 πππ π ππππ ππππππππ‘ππππ
π·ππππ‘π π€ππ‘β πππππ’ππ‘ ππ 27: (1, 3, 9), (3, 3, 3)
πππ (1, 3, 9): 3! = 6 πππ π ππππ ππππππππ‘ππππ
πππ (3, 3, 3) β 1 πππ π ππππ ππππππππ‘πππ
πππ‘ππ = 6 + 3 + 6 + 6 + 1 = 22
24. Suppose 3 cards are drawn from an ordinary poker deck of 52 playing cards without replacement.
=
Find the probability that the 3 cards share the same suit.
22
Answer: 425
Solution:
ππππππππππ‘π¦ = 4 ×
13πΆ3
13 × 12 × 11
22
=4×
=
52πΆ3
52 × 51 × 50 425
25. Find the number of positive square numbers less than 4000 that are divisible by 6.
Answer: 10
Solution:
π΅π¦ πππππππ π‘βππππππ,
632 < 4000 < 642
3969 < 4000 < 4096
πβππ‘ πππππ π‘βπππ πππ 63 πππ ππ‘ππ£π π ππ’πππ ππ’πππππ πππ π π‘βππ 4000
πΌπ π‘βπ ππππ π‘ 63 πππ‘π’πππ ππ’πππππ , π‘βπ ππ’ππ‘πππππ ππ 6 πππ
{6, 12, 18, 24, … , 60}
π»ππππ, π‘βπππ πππ 10 ππ’ππ‘πππππ ππ 6 ππ π‘βπ ππππ π‘ 63 πππ‘π’πππ ππ’πππππ
