2018

1. Given that A,B and C are 3 non zero digits and the 3 digit numbers formed by these three digits
have the following properties:
ACB is divisible by 9
BCB is a cubic number
Find AAB
2. There are n lines that are not parallel with each other on a plane. There are no 3 lines
intersecting at a point. If they intersect 105 times, find n.
3. Andy goes north for 99km. then goes east for 15km, goes south for 78km and goes west for
87km. How far is he now from the original position?
4. 1st September ,2018 is Saturday. Which day of the week is 21st September, 2078?
5. There are 15 pieces of white chopsticks, 16 pieces of yellow chopsticks and 12 pieces of brown
chopsticks mixed together. Close your eyes. If you want to get 1 pair of chopsticks that are not
white and 1 pair of chopsticks that are not yellow, at least how many pieces of chopsticks is/are
needed to be taken?

2018

6. Given A,B are 2 non zero digits and 3 digits numbers formed bu these two digits have following
properties:
BBB is divisible by 3
BBA is a square number.
Find BAB.

7. There are n people in a room. They shake hands with each other once. If they shake hands 190
times, find n.

8. Andy goes northeast for 10v/2 km, then goes east for 12km, goes west for 8km and goes
southwest for 15v/2km. How far is he now from the original position?

9. The average of some numbers is 199. If the 7 numbers, 251,288, 364, 437, 579, 620 and 660 are
added into this group, the average increases by 42. Find the sum of all original numbers.

10. At most how many numbers can be taken from the set of integers: 2,4,6,…..,76,78,80 such that
the sum of any two numbers taken out from the set is not a multiple of the difference between
these two numbers?

11. If a0aa + bb0b + cc0c + d0dd = 27636 find a+d-b-c.

2019
 12. Given that A,B,C are3 non zero digits and the 3 digit numbers formed by these 3 digits have the
following properties:
CBB is divisible by 12
CAC is divisible by 11
C > B> A
find ABC
13. There are n lines that are not parallel with each other on a plane. There are no 3 lines
intersecting at a point. Tf they intersect 253 times, Find the value of n.
14. Alice goes southwest for 29km, then goes southeast for 30km, goes southeast for 5km and goes
northwest for 37km. How far is she now from the original position?
15. There are 15 problems in a competition. The scores of each problem are allocated in the
following ways: 2 marks will be given for a correct answer, 1 mark will be deducted from a
wrong answer and 0 will be given for a blank answer. Find the minimum number of
candidates(s) to ensure that 2 candidates will have the same sores in the competition.

2019

16. 2, 6, 15, 28, 55, 78, ……

17. There are n lines on a circle which separate the circle into 232 parts, find the minimum value of
n.
18. If x,y,z are all prime and 6x^y + y^x = 3z, find the minimum value of z.
19. There are 31 problems in a competition. The scores of each problem are allocated in the
following ways: 2 marks will be given for a correct answer, 2 marks will be deducted from a
wrong answer and 1 mark will be deducted from a blank answer. Find the minimum number of
candidates to ensure that 3 candidates will have the same scores in the competition.
20. There are 17 pieces of red socks, 13 pieces of white socks and 16 pieces of blue socks in the box.
If you want to get 2 pairs of socks that is not red color and 3 pairs of socks that is not blue color,
at least how many pieces of socks are needed to be drawn?

2019

21. 30 problems in a competition.

2 marks for 1 correct answer, -2 for 1 wrong answer and -1 for the blank answer.
Find the minimum number of candidates to ensure that 2 candidates will have the same scores
in the competition.

22. Peter goes SE for 3km, then goes SE for 9km, then goes west for 9km. How far is he now from
the original position?

23. If we separate 1 to 100 into a few groups such that the numbers in each group are co-prime to
each other, at least how many groups are needed?
 24. Andy and Amy are playing a game. They have 21 candies in total. They can take 1,2 or 4 candies
one by one. The one taking the last candies will be the winner. If Amy takes 1 candy first, how
any candies does Andy need to take to ensure his victory?

25. There are 100 red, 200 white and 300 blue balls in a bag. Andy randomly picks balls from the bag
continuously. If 2 red balls can get 1 score, 3 yellow balls can get 1 score and 4 blue balls can get
1 score. At least how many ball(s) is/are needed to pick to ensure there are 100 scores?

26. Geometric progression 1,r, r^2,……, r^n and r and n are positive integers greater than 1. Amy
accidentally puts a positive integer which is smaller than a into a in the term r^a. The outcome is
340. Find the value of the correct answer.

2020

27. Now we have 2 identical cakes and cut by n times. Peter, Mary and Alice have 2,3 and 5 pieces
of cakes where the total amounts are the same. Find the minimum value of n.

28. Now putting n blue, yellow, red balls each into a bad and drawing balls from it. If the rest are all
yellow after drawing 31st ball, find the difference between the maximum value and the minimum
value of n.

29. Given that A:B = 2:5, B:C = 3:5, C:D = 4:5 and A+B+C+D = 927, Find the value of B?

30. 22 problems in a competition.

3 marks for a correct answer, -1 for a wrong answer, and no marks will be given to the blank
answer. Find the minimum number of candidates to ensure the 3 candidates will have the same
scores.

2020

31. Now putting n blue, yellow, red, green, purple balls each into a bag and drawing balls from it. If
the rest are all red after drawing the 67th ball, find the difference between the maximum value
and the minimum value of n.
32. On a circle, there are n lines which separate the circle into 154 parts, find the minimum value of
n.
33. How many integral solutions is/are there for x if -7 < 6-x^2/9 < 5?
34. There are 242 balls. Peter and Mary need to pick 6 to 12 balls by turns. The person picking up
the last ball will be the winner. Peter is picking ball in the first turn. If he can add balls before the
game start, at least how many ball(s) does he need to add to ensure his victory?
35. There are 22 pieces of red socks, 36 pieces of white socks and 25 pieces of blue socks in the box.
If you want to get 3 pairs of socks that is not white color and 1 pair of socks that is not blue
color, at least how many piece(s) of sock(s) is/are needed to be drawn?
 2020

36. Find the smallest integer that can express in the sum of two primes in two different ways.
37. Find the nearest integer of v3/1020201.
38. Find the number of x such that x! is not divisible by 2021.
39. Given that 1.2917 < v7/6 < 1.29171. Find the largest integral a satisfying a^7 <= 6^15.
111 2
1 1
40. Let z be the complex number. If z + = -1, find the value of ∑ (z ¿ ¿ r + r ) ¿ .
z r=1 z
a a
41. If 1= 10 and m+n =am + a n for all m,n are positive integers, find the value of a 2020.

2021

42. A:B = 5:2, B:C = 6:5, C:D = 8:3 and A+B+C+D = 1561, value of B?
43. 27 problems. 3 marks for a correct answer, -1 for a wrong answer and 0 will be given for the
blank answer. Find minimum number of candidate to ensure that 4 candidates will have the
same scores in the competition.
44. There are n lines that are not parallel with each other on a plane. There are no 3 lines
intersecting at a point. If they intersect 171 times, find n.
45. Now putting n red, orange, yellow and green balls each into a bag and drawing balls from it. If
the rest are all green after drawing the 37th ball, find the difference between the maximum and
minimum of n.

2021

46. There are n lines on a circle which separate the circle into 361 parts, find the minimum value of
n.
47. Now putting n blue, yellow, red, green, purple balls each into a bag and drawing balls from it. If
the rest are all red after drawing the 87th ball, find the difference between the maximum value
and the minimum value of n.
48. How many integral solutions is/are there for x if 6<5 + x^2 / 8 < 9?
49. There are 234 balls. Peter and Mary need to pick 5 to 12 balls by turns. The picking up the last
ball will be the winner. Peter is picking ball in the first turn. If he can add balls before the game
start, at least how many balls does he need to add to ensure his victory?
50. There are 30 pieces of red socks, 38 pieces of white socks and 25 pieces of blue socks in the box.
If you want to get 3 pairs of socks that are not white color and 1 pair of socks that is not blue
color, at least how many pieces of socks is/are needed to be drawn?

2021

51. Find the minimum value of integer x such that x! is divisible by 2021.
52. If a1 = 10 and am+1 = am + 2a1 for all m>=n and m,n are positive integers, find the value of
a2021.
 123 2
1
53. ∑ (z ¿ ¿ r + r ) ¿ . If z + 1/z = -1.
r=1 z
54. Given that 1.383 < √ 6
7 < 1.3831. Find the largest integral a satisfies a^6 <= 7^13.
55. Find the smallest integer that can express in the sum of three primes in three different ways.

2022

56. Amy is reading a novel. She read 1 page on the first day, 2 pages on the second day, 3 pages on
the third day, etc, until the total pages she had read are a multiple of 50. How many pages had
she read?
57. At least how many different lines are there on a plane if the lines intersect at 91 different points
in total?
58. 1st September, 2022 is Thursday. Which day of the week is 1st October, 2058?
59. BBC is divisible by 4
ABC is cubic number
Find AAB

2022

60. There are 8 pairs of white chopsticks, 9 pairs of yellow chopsticks and 10 pairs of brown
chopsticks mixed together. Close your eyes. If you want to get 3 pairs of chopsticks with
different color, at least how many pieces of chopsticks are needed to be taken.
61. There are n lines that are not parallel with each other on a plane. There are no 3 lines
intersecting at a point. If they intersect 171 times, find n.
62. 0, 7, 26, 63, 124, 215, 342,….
63. 17th January, 2022 is Monday, which nearest year’s 17th January was Monday?
64. Use decimal number system to represent heptad number 3067 .

2022

65. If a1 = 10 and am+1 = am + 3a1 for all positive integer m, find the value of a2022.
66.