SAMPLE TEST FOR TIMO
HEAT ROUND

TÀI LIỆU ƠN TẬP TIMO
VỊNG CHUNG KẾT QUỐC GIA

PRIMARY 6 - SECONDARY 1

KHỐI 6 - KHỐI 7

ĐỀ 1: (Đề TIMO Khối 6 Vòng Chung Kết Quốc Gia 2019 – 2020)
Logical Thinking
1.

There are 16 pieces of white chopsticks, 27 pieces of yellow chopsticks and 46
pieces of blue chopsticks mixed together. If you want to get 2 pairs of white
chopsticks and 3 pairs of chopsticks with other different colours in the dark, at least
how many pieces of chopsticks are needed to be taken?
At least: Ít nhất.

2.

There are some chickens and rabbits in a cage. The number of chickens is 4 times as
the number of rabbits. They have 240 legs in total. How many chicken(s) is / are
there?
4 times: gấp 4 lần; In total: Tổng số.

3.

It requires 16 minutes to cut a piece of wood into 9 sections. If the time required to
cut into each section is the same, how many minute(s) is / are required to cut the

wood into 11 sections?

4.

Find the average of the following 8 numbers.
2019、2065、2046、1985、2034、1922、46、1827
Average: Trung bình cộng.

5.

Leo needs 18 days to finish a project. Bruce needs 21 days to finish the same
project. If Leo, Bruce and Mary do the project together, it takes 9 days. How many
days does Mary take to finish the project alone?
Arithmetic
1

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


6.

Find the value of 1  4  7  ...  31  34 .

7.

Find the value of 1369 1722 1364 1717 .


8.

Find the value of

9.

1
1
1
1

 ... 

.
110 132
342 380
1
1
1
169  5
10  4
113 .
Find the value of 2
400
800
200

10. Using method S  (3S  S )  2 , find the value of 1  3  9  ...  59049 .
Number Theory
11. If a 9-digit number 202015 A2B is divisible by 12 and A, B  0 , find the sum of

possible value of AB .
9-digit number: Số có 9 chữ số; Divisible: Chia hết; Sum: Tổng.
12. Find the last digit of A if A  5  12  19  26  ...   5  7  99    5  7 100  .
Last digit: Chữ số tận cùng.
13. Find the numbers of all positive factors of 996.
Positive factors: Ước dương.
14. The remainder of dividing a positive integer K by 17 is 11. What is the remainder of
dividing  6K  16  by 17?
Remainder: Số dư; Divide: Chia; Positive integer: Số nguyên dương.
15. The sum of 25 consecutive numbers is 900. Find the second largest number.
Sum: Tổng; Consecutive: Liên tiếp.

2

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


Geometry
16. How many rectangle(s) is / are there in the figure below?
Rectangle: hình chữ nhật.

17. Small cubes with side length 1 are combined according to the pattern shown below.
Find the total volume of the 30th generation.
Cubes: Hình lập phương; Side length: Độ dài cạnh; Pattern: Quy luật; Total
Volume: Tổng diện tích.

1st generation


2nd generation

3rd generation

3

KỲ THI OLYMPIC TỐN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


18. The figure below is a square overlapped by two right-angled triangles and one
quarter circle. Find the area of the shaded region. (Take  

22
)
7

Figure: Hình; Right-angled triangles: Tam giác vng; One quarter circle: Một
phần tư đường trịn; Area: Diện tích.

19. How many rectangle(s) with both 2 “*” is / are there in the figure below?
Rectangle: Hình chữ nhật.

*
*

4


KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


20. The perimeter of a circle is 18.84. If the diameter of the circle is the same as the
diagonal of a square, what is the area of the square? (Take   3.14 )
Perimeter: Chu vi; Circle: Hình trịn; Diameter: Đường kính; Diagonal: Đường
chéo; Square: Hình vng; Area: Diện tích.
Combinatorics
21. Choose 3 digits, without repetition, from 1, 2, 3, 6, 7, 8 to construct 3-digit numbers.
Of these 3-digit numbers, how many of them are divisible by 4?
Digit: Chữ số ; Without repetition: Không lặp lại; Divisible: Chia hết.
22. A flight of stairs has 11 steps. Peter can go up for 1 step or 3 steps each time. The 6th
step cannot be stepped on as it is destroyed. How many way(s) is / are there for Peter
to go up the stairs?
23. What is the nearest time while the hour hand and the minute hand are perpendicular
to each other between 6pm and 7pm?
Perpendicular: Vng góc.
24. In how many possible way(s) can 10 identical balls be distributed to 3 distinct boxes
so that every box contains at least one ball?
Distinct: Phân biệt (Khác nhau)
25. 6 identical black books and 8 identical white books are put from left to right. How
many different permutation(s) is / are there?
Permutations: Hốn vị (Cách sắp xếp)

5


KỲ THI OLYMPIC TỐN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


ĐỀ 2: (Đề TIMO Khối 7 Vòng Chung Kết Quốc Gia 2019 – 2020)
Logical Thinking
1.

There are 22 pieces of white chopsticks, 18 pieces of yellow chopsticks and 39
pieces of blue chopsticks mixed together. If you want to get 2 pairs of blue
chopsticks and 2 pairs of chopsticks with other colours in the dark, at least how
many pieces of chopsticks are needed to be taken?

2.

A, B, C, D and E are passing balls. Starting from A, 5th time pass will return to B.
Within the pass, there is at least 1 time to A, how many passing methods are there?

3.

If abcd  badc  17743 , calculate a  b  c  d .
Calculate: Tính.

4.

There are some chickens and rabbits in a cage. The number of chickens is 5 times as
the number of rabbits. The total legs of chicken’s is 72 more than that of rabbit’s.
How many rabbit(s) is / are there?

In total: Tổng số; More than: Nhiều hơn.

5.

According to the pattern shown below, what is the number in the blank?
Pattern: Quy luật.
1 、 1 、 4 、 13 、 43 、 142 、 __ 、….
Algebra

6.

Factorize x2  2 x  63 .
Factorize: Phân tích thành nhân tử.

7.

How many negative integral solution(s) is / are there for x if 8x  92  16 ?
Negative integral solutions: Nghiệm nguyên âm.

6

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


8.

Given that a is a negative real number and a  1 . If a 


10
 8 , find the
a 1

minimum value of a.
Negative real number: Số thực âm; Minimum value: Giá trị nhỏ nhất.
9.

Find the value of x if (96  23x)  (11x  48)  0 .

10. Given a and b are real numbers and satisfy equations a  b  6 and ab  6 , find the

a  b .
2
2
3

value of

a b

Real numbers: Số thực; Equations: Phương trình.
Number Theory
11. It is known that x is rational, x  0 and x  3 

10
. Find the value of x.
10
3

3  ...

Rational: Hữu tỉ.
12. Find the sum of all positive factors of 786.
Sum: Tổng; Positive factors: Ước số dương.
13. Given x  0 and x 

1
1
 22 , find the value of x 2  2 .
x
x

14. Find the unit digit of A if A  13  132  133  134  ...  1324  1325 .
Unit digit: Chữ số hàng đơn vị.
15. The difference between two square numbers is 253. Find the value of the larger
square given that it is a 3-digit number.
Difference: Hiệu; Square numbers: Số chính phương; 3-digit number: Số có 3 chữ
số.
Geometry
7

KỲ THI OLYMPIC TỐN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


16. M (18, 22) is the mid-point of A( x,8) and B(10, y) . Find the value of x  y .
Mid-point: Trung điểm.

17. A interior angle of a regular n-sided polygon is 17 times of an exterior angle. Find n.
Interior angle: Góc trong; Regular n-sided polygon: Đa giác đều n cạnh; Exterior
angle: Góc ngồi
18. Small cubes with side length 1 are combined according to the pattern shown below.
Find the total volume of the 12th generation.
Cubes: Khối lập phương; Side length: Độ dài cạnh; Pattern: Quy luật; Total
volume: Tổng thể tích.

1st generation

2nd generation

3rd generation

19. A solid cuboid is formed by merging 770 cubes with side length 1cm. Find the
minimum value of the total surface area of the cuboid.
Solid cuboid: Hình hộp chữ nhật đặc; Merge: Ghép; Side length: Độ dài cạnh;
Minimim value: Giá trị nhỏ nhất; Total surface area: Diện tích tồn phần.

8

KỲ THI OLYMPIC TỐN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


20. The figure below is a rectangle overlapped by a quarter of a circle and a triangle.
Find the area of the shaded region. (Take  


22
)
7

Square: Hình vng; Overlapped: bị phủ bởi; A quarter of a circle: Một phần tư
cung trịn; Triangle: Hình tam giác; Area: Diện tích.

Combinatorics
21. Choose 5 digits without repetition from 1, 4, 6, 7, 8, 9 to form 5-digit numbers. The
odd digits cannot be next to each other. How many number(s) can be formed in this
way?
Digits: Chữ số; Without repetition: Không lặp lại; Odd: Lẻ.
22. What is the nearest time while the hour hand and the minute hand are perpendicular
to each other between 10am and 11am?
Perpendicular: Vng góc.
23. 40 cards are marked from 1 to 40 and 2 are drawn at random. Find the probability
that two cards drawn are both multiple of 3 or 7.
At random: Ngẫu nhiên; Probability: Xác suất; Multiple: Bội.

9

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


24. In how many possible ways can 9 identical flowers be distributed to 3 distinct vases
with at least one in each vase?
Identical: y hệt nhau; Distributed: Được chia; Distinct: Phân biệt (Khác nhau).

25. 7 identical black books and 10 identical white books are put from left to right. How
many different permutation(s) is / are there?
Identical: y hệt nhau; Permutation: Cách sắp xếp.
ĐỀ 3: (Đề TIMO Khối 6 Vòng Chung Kết Quốc Tế 2015 – 2016)
Logical Thinking
1. Peter cuts a circle into parts by straight lines. At least how many straight lines are
required to ensure he can get 14 parts?
Circle: Hình tròn; Straight lines: Đường thẳng; Parts: phần.
2. Split 3, 12, 14, 26, and 91 into 2 groups of numbers such that they have equal product.
Find the sum of the numbers of the group that contains 14.
Split: Chia; Equal product: Tích bằng nhau; Sum: Tổng.
3. There are some $1 and $2 coins. The number of $1 coins is twice that of $2 coins. The
total value of coins is $100. Find the number of $1 coins.
Twice: Gấp đôi; Total value: Tổng giá trị.

10

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


4. John wrote a 3-digit number on a piece of paper and asked Peter to guess it.
Peter asked: “Is the number divisible by 21?”
John replied: “It is divisible by 21.”
Peter asked: “Is the number divisible by 9?”
John replied: “It is divisible by 9.”
Peter asked again: “Is it a square number?”
John said: “Yes, it is a square number.”

What is the number written by John?
Divisible: Chia hết; Square number: Số chính phương.
5. A class has 40 students. 26 students join the badminton club, 25 students join the judo
club and 20 students join the dance club. Each student joins at least one club. Given that
23 students join two clubs, how many students join all three clubs?
Arithmetic
6. Find the value of 1  5  25  125  625  3125 .
7. Find the value of 20152 10152 .
8. Find the value of 1  1  2  3 

 144  233  377 .

(Remark: Starting from the third number, each number is the sum of the previous two
numbers.)
Sum: Tổng; Previous two numbers: Hai số đằng trước.
9. Find the value of


2 2 2
2
2
2
2
2
2
.
    




3 15 35 63 99 143 195 255 323



10. Convert 0.148 into the simplest fraction.
Convert: Đổi; Simplest fraction: Phân số tối giản.
11

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


Number Theory
11. Find the sum of all positive factors of 2015.
Sum: Tổng; Positive factos: Ước số dương.
12. If a 5-digit number 2015A is divisible by 8, find the value of A.
Divisible: Chia hết.
13. Counting from 1 to 999, how many numbers are divisible by 6 or 9?
Divisible: Chia hết.
14. When a 4-digit integer is divided by 5, 13, or 31, the remainder in each division is 2.
Find the least possible value of that integer.
4-digit integer: Số nguyên có 4 chữ số; Remainder: Số dư; Division: Phép chia; Least
possible value: Giá trị nhỏ nhất có thể.
15. Find the smallest 3-digit number that can be divisible by 3 different prime numbers?
Prime numbers: Số nguyên tố.
Geometry
16. In the figure below, the side length of the square is 7cm. Find the area of the shaded
region. (Take  


22
).
7

Side length: Độ dài cạnh; Square: Hình vng; Area: Diện tích; Shaded region: Miền
được tơ đậm.

12

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


17. The side length of square ABCF is 18cm. Extend BC to D and connect AD. It is
known that the area of triangle AEF is 90cm2 less than that of triangle DCE. Find the
length of CD (in terms of cm).
Side length: Độ dài cạnh; Square: Diện tích; Extend: Kéo dài; Connect: Nối; Area: Diện
tích; Triangle: Tam giác; Less than: Nhỏ hơn.

18. Rectangle ABCD is divided into 8 parts whose areas are shown below. Find the area
of shaded region.
Rectangle: Hình chữ nhật; Divided: bị chia ra; Area: Diện tích; Shaded region: Phần
được tơ đậm.
A
20

9


D

8
30

6
3

B

27

C
13

KỲ THI OLYMPIC TỐN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


19. Paint a cube with edge length of 4cm with red colour. If it is divided into smaller
cubes with edge length of 1cm afterwards, find the numbers of small cubes with only 1
face being painted.
Cube: Lập phương; Edge length: Độ dài cạnh; Face: Mặt của hình lập phương.
20. Given a cuboid. If its length is increased by 4cm, its volume is increased by 720cm3.
If its width is decreased by 2cm, its volume is decreased by 264cm3; if its height is
increased by 4cm, its volume is increased by 660cm3. Find the surface area and the
volume of the original cuboid.

Cuboid: Hình hộp chữ nhật; Length: Chiều dài; Width: Chiều rộng; Height: Chiều cao;
Increased: Được tăng thêm; Volume: Thể tích;; Surface area: Diện tích tồn phần.
Combinatorics
21. Counting from 1 to 10000, how many numbers are there that have exactly two digits
“5” and only one digit “4”?
Digit: Chữ số.
22. How many 4-digit numbers whose product of all the digits is an odd number are
there?
Product: Tích; Digit: Chữ số; Odd: Lẻ.
23. There are 11 boys and 4 girls in a class. Choose 3 boys and 2 girls to form a study
group. How many ways to form such a group are there?

14

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


24. The attached figure is made up of 5 grids. A colour is filled into each grid, and two
adjacent grids do not share the same colour. If there are in total 4 colours to choose from
to fill in the grids, how many ways are there to colour the figure?
Grid: Ô; Adjacent: Kề nhau.

Question 24
25. Arrange 5 letters A, B, C, D and E in a row. If the pair A and B and the pair C and E
must be adjacent regardless of order, find the number of permutations.
Row: Hàng; Adjacent: Kề nhau; Permutation: Cách sắp xếp.
ĐỀ 4: (Đề TIMO Khối 7 Vòng Chung Kết Quốc Tế 2015 – 2016)

Logical Thinking
1. Amy bought flowers from a market. It is known that a rose, a lily and a tulip cost $16,
$8 and $1 respectively. She used $300 to buy 30 flowers and bought at least 1 for each
type. How many roses did she buy?
Respectively: Lần lượt; At least: Ít nhất.
2. According to the pattern shown below, what is the next number?
1、2、3、6、11、20、37、68、…
Pattern: Quy luật.

15

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


3. Find the integer part of

1
1 1 1
  
10 11 12



1
19

.


Integer part: Phần nguyên.
4. There are six teams named A, B, C, D, E and F, participating in a tournament. In 5
days, each team will play one game on each day. They play against another team once in
the tournament and there are 3 games every day. Given that:
1)

Team A wins Team B on the first day.

2)

Team C is defeated by Team D on the second day.

3)

Team E wins Team A on the third day.

4)

Team B wins Team C on the fourth day.

Which team does Team A play with on the fifth day？
5. There are 7 people arguing which day of the week is today. Their statements are
quoted as below:
A:

Yesterday was Wednesday.

B:


Tomorrow will be Tuesday.

C:

Tomorrow will be Wednesday.

D:

The day after tomorrow will be Tuesday.

E:

Today is Tuesday.

F:

Today is neither Monday, Tuesday nor Sunday.

G:

Today is not Saturday.

Given that only one statement is correct, which day of the week is today?

16

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc



Arithmetic
6. Given a  2  a  3  7 , find the sum of all possible values of a.
Sum: Tổng; All possible values: Tất cả các giá trị có thể.
7. Find the value of 1 2  3  2  3 4  3 4  5 
8. Convert

 10 1112 .

16
into recurring decimals.
37

Convert: Đổi; Recurring decimals: Số thập phân vơ hạn tuần hồn.
9. Let a be a positive integer satisfying the constraint 3a  7  a  19 . How many possible
values for a are there?
Positive integer: Số nguyên dương; Constraint: Điều kiện.
1 1
3 9

10. Find the value of 1   

1

27

.
Number Theory


11. If n is a positive integer and n  7, n 2  2, n  2 are all prime numbers, find the value of n.
Positive integer: Số nguyên dương; Prime numbers: Số nguyên tố.
12. Find the remainder when 20152015 is divided by 8.
Remainder: Số dư; Divided: Bị chia.
13. Given that x and y are natural numbers and

7 x 11
  . When y attains its minimum,
10 y 14

find the value of x.
Natural numbers: Số tự nhiên; Miminum: Giá trị nhỏ nhất.

17

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


14. How many simplest fractions with denominator 2015 are there, given that it is
positive smaller than 1?
Simplest fraction: Phân số tối giản; Denominator: Mẫu số; Positive: Dương.
15. Given positive integers x and y satisfying equations xy  x  y  19 and x 2 y  xy 2  84 .
Find the value of x 2  y 2 .
Positive integers: Số nguyên dương; Equations: Phương trình.
Geometry
16. Combine 2015 equilateral triangles with side length of 1 unit to form a quadrilateral.
Find the maximum perimeter of that quadrilateral.

Equilateral triangles: Tam giác đều; Side length: Độ dài cạnh; Quadrilateral: Tứ giác;
Maximum perimeter: Chu vi lớn nhất.
17. ABCDEFGH shown in the graph is a cube. If the distance between of A and F is 4 3 ,
find the volume of the cube.
Graph: Hình vẽ; Cube: Hình lập phương; Distance: Khoảng cách; Volume: Thể tích.

Question 17

18

KỲ THI OLYMPIC TỐN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


18. In the diagram given below, ABCD is a square whose side length is 20 units. If
CD : DE  EF : FA  1: 2 , find the length of AG.

Diagram: Hình vẽ; Square: Hình vng; Side length: Độ dài cạnh.

Question 18
19. A circle is inscribed in a square and another smaller square is inscribed in that circle.
If the side length of the larger square is 4cm, find the area of the smaller square in cm2.
Circle: Hình trịn; Inscribe: Nội tiếp; Square: Hình vng; Side length: Độ dài cạnh;
Area: Diện tích.
20. It is known that the lengths of two legs of a right-angled triangle are in the ratio of
1: 7 . If the length of the hypotenuse is 10cm, find the area of the triangle in cm2.

Legs: Cạnh góc vuông; Right-angled triangle: Tam giác vuông; Ratio: Tỉ số;

Hypotenuse: Cạnh huyền; Area: Diện tích.

19

KỲ THI OLYMPIC TỐN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


Combinatorics
21. A set of integers chosen from 1 to 2015 has the property that none of its members are
10 times another. What is the maximum number of members such a collection can have?
Set of integers: Tập các số nguyên; Property: Tính chất; 10 times: Gấp 10 lần; Maximum
number of members: Số phần tử lớn nhất.
22. A basketball team has 15 members. How many different ways are there to send 10
teammates to participate in a competition in which one of them is the captain?
23. Given that x1 , x2 , , x5 are all positive integers, how many solutions are there to the
inequality x1  x2 

 x5  10 ?

Positive integers: Số nguyên dương; Solutions: Nghiệm; Inequality: Bất đẳng thức.
24. Five students, A, B, C, D and E are passing a ball given that initially A holds the ball.
If A passes the ball to D at the 5th pass, how many possible ways of passing the ball are
there?
25. Using 0,2,4,6 and 8 as digits, how many positive integers less than 10000 can be
formed? (Each digit cannot be used more than once to form a number).
Digits: Chữ số; Positive integers: Số nguyên dương; Less than: Nhỏ hơn; More than
once: Nhiều hơn một lần.


20

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


ĐỀ 5: (Đề TIMO Khối 6 Vòng Chung Kết Quốc Tế 2016 – 2017)
Logical Thinking
1. There are 4 statements on a piece of paper as shown below, and only one of them is
correct. Write down the letter before the correct statement on the answer sheet.

2. Split 14, 24, 60, 65, 77, and 143 into 2 groups of 3 numbers such that the product of
numbers in each group is the same. Find the sum of the numbers of the group that
contains 14.
Split: Chia; Product: Tích; The same: Bằng nhau; Sum: Tổng.
3. They caught 12 animals including spiders and crickets with 90 legs in total. Find the
number of spiders knowing that a cricket has 6 legs and a spider has 8 legs.
In total: Tổng số.

21

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc



4. John wrote a 3-digit number on a piece of paper and asked Peter to guess it.
Peter asked: “Is the number 123?”
John replied: “One of the digits is correct, the position of that digit is also correct.”
Peter asked again: “Is the number 456?”
John replied: “Two digits are correct, but the positions of those digits are both wrong.”
Peter asked again: “Is the number 354?”
John said: “All three digits are correct, but the digits are all in the wrong places.”
What is the number written by John?
Digit: Chữ số; Position of the digits: Vị trí của chữ số.
5. There are 51 students in Grade 6, among them 22, 19 and 18 students like playing
basketball, volleyball and football respectively. There are 5 students like playing
basketball and volleyball; 7 students like playing volleyball and football; 10 students like
playing basketball and football. 3 students like all three sports. Find the number of
students who likes none of the sports.
Arithmetic
6. Find the value of 2  6  18  54  162  486  1458  4374  13122  39366 118098 .
7. Find the value of 12852  2852 .
8. Find the value of 1  4  5  9 

 157  254  411.

(Remark: Starting from the third number, each number is the sum of the previous two
numbers.)
Sum: Tổng; Previous two numbers: Hai số liền trước.
9. Find the value of

1 1 1 1
1
1
1

1
1
  
 
   .
2 6 12 20 30 42 56 72 90

22

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


 

10. Convert 0.786 into the simplest fraction.
Convert: Đổi; Simplest fraction: Phân số tối giản.
Number Theory
11. Find the sum of all positive factors of 2018.
Sum: Tổng; All positive factors: Tất cả các ước số dương.
12. When a natural number is divided by 20, the remainder is 4. When the number is
divided by 39, the remainder is also 4. When that number is divided by 65, what is the
remainder?
Natural number: Số tự nhiên; Divided: Bị chia; Remainder: Số dư.
13. Write 9 as the sum of some positive integers (e.g. (8,1) / (2,7) / (4, 2,3)... ) so that the
product among those numbers is the largest. Find the value of that product.
Sum: Tổng; Positive integers: Số nguyên dương; Product: Tích.
14. When a 4-digit positive integer is divided by 3, 11, or 61, the remainder of each

division is 2. Find the least possible value of such an integer.
4-digit positive integer: Số nguyên dương có 4 chữ số; Divided: Bị chia; Remainder: Số
dư; Division: Phép chia; Least possible value: Giá trị nhỏ nhất có thể.
15. A book has pages numbered 1, 2, 3, and so on. A page was torn from the book and
the sum of all remaining page numbers is 761. What is the page number of the torn page?
Numbered: Được đánh số; Sum: Tổng; Remaining page numbers: Số trang còn lại.
Geometry

23

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


16. In the attached figure, the perimeter of the shaded region is formed by arcs of 3 semicircles. The diameters of those semi-circles are 13m, 50m and 63m. Given that 3 centers
of the circles lie on the dotted line as shown below, find the perimeter of the shaded
region in meter. (Take  

22
)
7

Perimeter: Chu vi; Shaded region: Phần được tô đậm; Arcs: Cung; Semi-circles: Nửa
đường trịn; Diameters: Đường kính; Centers: Tâm; Lie: Nằm trên; Dotted line: Đường
nét đứt.

17. The figure below is formed by two squares ABFG and BCDE. If the length of CD and
GA are 16cm and 18cm respectively, find the area of CEG in cm2.

Squares: Hình vng; Respectively: Lần lượt; Area: Diện tích.
G

F
E

A

B

D

C

24

KỲ THI OLYMPIC TỐN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc


18. The attached figure shows triangle ABC . P and Q are points lying on AB and BC
respectively such that AP : PB  1:1 and BQ : QC  6 : 5 . If the area of APQ is 66cm2,
find the area of ABC in cm2.
Triangle: Tam giác; Points: Điểm; Lying on: Nằm trên; Respectively: Lần lượt; Area:
Diện tích.
A

P


B

Q

C

Question 18
19. The figure below shows a cubic container whose side length is 30cm, and it is filled to
a depth of 16cm with water . A cuboid iron bar with a square base is inserted into the
container so that the base of the cuboid touches the base of the container. If the water level
rose 2cm, find the side length of the square base of the iron bar in cm.

25

KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020
Thailand International Mathematical Olympiad – Timo 2020
Đơn vị đồng tổ chức tại Việt Nam: Trường Đại học Thủ Đô Hà Nội và FERMAT Education.
Email: hoặc