Khối 8 + khối 9 tài liệu TIMO 2020

1 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ TIMO 2020 Thailand Internationl Mathematical Olympiad TIMO 2020 Đơn vị đồng tổ chức tại Việt Nam Trường Đại học Thủ Đô Hà Nội và FERMAT Education Email timo@daihocthu[.]

BỘ TÀI LIỆU ÔN THI VÒNG LOẠI TIMO 2020

Secondary 2 – Secondary 3

ĐỀ 1:

2

4

6

8

ĐỀ 2:

Logical Thinking

1 According to the pattern shown below, what is the number in the blank?

2 、 6 、 12 、 20 、 30 、 42 、 、 …

2 There are chickens and rabbits on a farm with 210 legs in total The number of

rabbits is twice that of chickens How many rabbit(s) is / are there?

3 Right now, Alice is seven times as old as Peter 6 years later, Alice is three times

as old as Peter How old is Alice now?

4 Alice needs 20 days to finish a project Peter needs 30 days to finish the same

project If Alice, Peter and Mary do the project together, it takes 10 days How long does it take Mary to finish the project alone?

5 There are 5 pieces of red socks, 9 pieces of white socks and 13 pieces of blue

socks in a box At least how many piece(s) of sock(s) is / are needed to be drawn to

be sure to get 4 pairs of socks with the same colour?

Arithmetic

6 Find the value of 642  9  144

7 Find the value of 1 1 1 1 1

10

11 If the 11-digit number 20190513ABB is divisible by 33 and A B, find all possible values of AB

12 How many fraction(s) in lowest terms with a denominator of 120 is / are there?

13 We define the operation symbol ⨁ as follows:

14 A 3-digit number N leaves a remainder 4 when divided by 7, a remainder 2 when

divided by 9 and a remainder 5 when divided by 11 What is the largest possible

value of N?

15 Given 20 consecutive natural numbers whose sum is 450 Find the second largest

number

Geometry

16 How many rectangle(s) is/are there in the figure below?

17 A rectangle is cut into 6 equal squares If the area of the small square is 16cm2, find the smallest possible difference between the total perimeters of all new squares and the perimeter of the original rectangle

18 Cubes with side length 1 are combined to create layers as the pattern shown

below Find the volume of the 4th layer

19 Given 150 rectangles of dimension 2cm  3cm, they all combine to form a new square Find the decrease in the total perimeter (the answer is given in cm)

20 The area of a circle is 12.56cm If the radius of the circle is the same as the

diagonal of a square, what is the area of the square? (Take  3.14 )

Combinatorics

21 A flight of stairs has 12 steps Alice can climb up 1 step, 2 steps or 3 steps each

time The 4th and 8th step cannot be stepped on as it is destroyed How many way(s)

is / are there for Alice to climb up the stairs?

22 Alice, Mary and 5 other classmates sit in a row Alice and Mary can neither sit

at 2 ends nor sit next to each other How many seating arrangements are there?

23 What is the first time after 2pm that the hour hand and the minute hand of the

wall clock are perpendicular to each other?

24 3 identical blue flags and 3 identical red flags are placed from left to right How

many different permutation(s) is/are there?

25 Alice goes from point A to point B and in each step, she can only move up or

move right How many way(s) is / are there if she must pass through point O?

12

ĐỀ 3:

Logical Thinking

1 According to the pattern shown below, what is the number in the blank?

2 、 12 、 30 、 56 、 90 、 132 、 、 …

2 Peter goes 20km east, then goes 24km south, then goes 13km west How far is he

now from the original position (the answer is given in km)?

3 If abcd badc  7898, calculate a  b c d

4 If x, y and z are all primes and 2x y  y x  2z , find the minimum value of z

5 There are 10 pieces of red socks, 6 pieces of white socks and 22 pieces of blue

socks in the box If you want to ensure that you get 2 pairs of socks with different colour, at least how many piece(s) of sock(s) is/are needed to be drawn?

Arithmetic

6 Find the value of x if (37  21x)  (8x  67)  0

7 If x and y are positive integers and 7x – 2y = 151, find the minimum value of

x  y

8 How many integral solution(s) to 56  5x  14  89 is / are there?

9 Let a and b be real numbers such that a2 b2  51 and a  3  b Calculate a  b

10 Factorise x2 2x  63

Number Theory

11 Find the sum of all positive factor(s) of 513

12 From 1 to 1000, how many integer(s) is/are divisible by 3 and 5, but not divisible

by 9?

13 Find the last digit of 52019 12019 32019

14

14 Given x 0 and 1

5

x x

  , what is the value of 4 14

x x

 ?

15 The difference between two square numbers is 611 Find the sum of all possible

values of the larger square?

Geometry

16 A cube with side length 18 is divided into smaller cubes with side length 6 Find

the increase in the total surface area of the cubes after the division

17 An iron wire is bent in the shape of 2 circles whose radii are both 21 If the wire

is now bent in the shape of 3 identical rectangles whose side lengths are integers, what is the maximum area of this rectangle? (Take  22/7)

18 Small cubes with side length 1 are combined as the pattern shown below Find

the volume of the 15th layer

19 The longest side of a triangle is 12 unit long Find the largest possible area of the

triangle (the answer is in surd form)

20 In a regular polygon, the measure of an interior angle is 8 times the measure of

an exterior angle How many sides does the polygon have?

Combinatorics

21 There are 6 identical blue flags and 3 identical white flags are put from left to

right In how many way(s) can they be arranged?

22 There are five masses which are 1g, 2g, 5g, 7g and 9g respectively If we can

combine the masses, how many different weights can be measured?

23 In how many possible ways can 14 identical flowers be distributed to 3 distinct

vases with at least two flowers in each vase?

24 How many 5-digit number(s) can be formed from the digits 1, 2, 4, 7, 8, 9, given

that the odd digits cannot be next to each other and none of the digits are repeated ?

25 Billy goes from point A to point B and he can only move up or move right In

how many way(s) can Billy get to the destination if he must pass through point O?

ĐỀ 4:

Logical Thinking

1 Let A and Dbe two non-zero digits which can form 2-digit numbers DA and

AD with the following properties:

a) DA can be expressed as a product of 2 and another prime numbers;

b) AD can be expressed as a product of 2 and another prime numbers

If A D , find the 2-digit number AD

2 There are 340 people in the prom After being checked, the number of couple

tickets is twice the number of single tickets How many people go to the prom alone?

3 Right now, Amy is 10 years younger than her brother and their uncle’s age is three

times the sum of the ages of Amy and her brother Six years later, their uncle’s age only doubles the sum of Amy’s and brother’s age Find the age of Amy now

4 John wrote a 4-digit number on a piece of paper and asked Peter to guess it

Peter: “Is the number 2956?”

John: “Three of the digits are correct The positions of those digits are all wrong.” Peter: “Is the number 7324?”

John: “Two of the digits are correct The positions of those digits are all wrong.” Peter: “Is the number 4962?”

John: “All the digits are correct The positions of those digits are all wrong.”

16

What was the number written by John?

5 There are 7 pieces of white chopsticks, 6 pieces of yellow chopsticks and 5 pieces

of brown chopsticks mixed together You want to make sure that you get 3 pairs of chopsticks, of which 2 pairs have the same color At least how many piece(s) of chopstick(s) is/are needed to be taken?

11 If the 9-digit number A2018901B is divisible by 72, find the value of A B

12 Find the last digit of A if A = 82 + 182 + 282 + … + 20182

13 How many 3-digit number(s) divisible by 2 and 3 but not divisible by 7?

14 A 4-digit number leaves a remainder 1 when divided by 3, a remainder 2 when

divided by 5 and a remainder 3 when divided by 7 What is its largest possible value?

15 Given 8 consecutive odd numbers whose sum is 144, find the largest number

among them

Geometry

16 How many rectangle(s) is/are there in the figure below?

17 Given a square with the area of 225cm2, it is cut into 4 equal rectangles Find the largest possible difference between the total perimeter of 4 new rectangles and the perimeter of the original square

18 Ian uses identical cubes with side length 1cm to create shapes that has the pattern

as shown below If the shape built by Ian has 12 layers, what is its volume?

19 A big rectangle is formed by 1001 squares with side length 2cm Find the

minimum value of the perimeter in cm

20 The length of one side of a square is 12.56cm If the perimeter of the square is

the same as that of a circle, what is the radius of the circle? (Take  3.14)

Combinatorics

21 A flight of stairs has 10 steps David can take 1 step or 2 steps each time The 5th

step cannot be stepped on as it is destroyed How many way(s) is/are there for David

to go up the stairs?

18

22 Amy needs 12 days to finish a task Paul needs 24 days to finish a task How

many days do they need to finish the task if they do it together?

23 At what time between 5pm and 6pm do the hour hand and the minute hand of

the wall clock overlap?

24 Integral numbers are drawn from 1 to 2020 At least how many number(s) is/are

drawn at random to ensure that there are two numbers whose difference is 400?

25 Agnes goes from point A to point B In each step, she can only move up or move

right In how many way(s) can Agnes do that if she must pass through point O?

ĐỀ 5:

Logical Thinking

1 Given thatA and C are non-zero digits, the 2-digit numbers formed by these two

digits have the following properties:

a) CA is a prime number;

b) AC is a prime number;

c) A C 4

Find the 2-digit number AC

2 Andy goes west for 5km, then goes north for 12km How far is he now from the

original position?

3 How many time(s) in a day does the minute hand of a wall clock coincide with

the hour hand?

4 There are 30 problems in a math test The scores of each problem are allocated in

the following ways: 1 mark will be given for a correct answer, 0 marks will be given for a blank answer or a wrong answer Find the minimum number of candidate(s) to ensure that 2 candidates will have the same score in the test

5 There are 14 white chopsticks, 6 yellow chopsticks and 4 brown chopsticks mixed

together If you want to draw chopsticks randomly and make sure that you get 2 pairs with different colours, at least how many piece(s) of chopstick(s) is/are needed to be taken? (2 pieces of chopsticks with the same colour make a pair)

Arithmetic

6 Find the value of x if (4x + 5) + (3x - 19) = 0

7 Given 1 1 1 1 252

824 48 2 (2n n 2) 1009

 , find the value of n3 1008n2 1

8 How many positive integral solution(s) 𝑦 is/are there if 13𝑦 + 47 < 2349?

9 Let a and b be positive real numbers and satisfy the equations a2 b2 45 and 18

ab Find the value of ab

10 Factorise the polynomial 4x2 20x21

20

Number Theory

11 Find the number of positive factor(s) of 1002001

12 Find the value of 13 + 23 + 33 + …+ 1003

13 A is an integer such that A15 is a 17-digit number with 7 as the unit digit Find A

14 Given x 0 and 2 12

14

x x

  , find the value of 1

x x



15 What is the remainder when 102018 is divided by 11?

Geometry

16 A solid cuboid is formed by merging 2431 cubes with side length 1cm Find the

smallest possible value of the total surface area of the cuboid

17 Given a right triangle, the length of a leg is 36cm and the length of the hypotenuse

is 60cm Find the area of the triangle in cm2

18 Small cubes with side length 1cm are combined as the pattern shown below If

the given solid has 11 layers with such pattern, find its surface area

19 The side length of an equilateral triangle is 6, find the area of the triangle

20 Find the area of the shaded region in the figure (Take 22

7 )

Combinatorics

21 Given a 6-digit number, if the rightmost digit is now put at the leftmost, the new

number formed is 5 times the original Find the original number

22 In between 3 o’clock and 4 o’clock, when does the hour hand and minute hand

form a right angle?

23 In how many possible ways can 8 identical balls be distributed to 3 distinct

boxes?

24 Choose three digits (without repetition) from 1, 2, 3, 5, 8 and 9 to form a 3-digit

numbers How many number(s) can be divisible by 6?

25 If Amy goes from point A to point B In each step, she can only move up or move

right How many way(s) is/are there?

22

KEY

Đề 1:

24

Đề 2:

Câu 1 2 3 4 5 6 7 8 9 10 Đáp

1123

1

4374 2585 0.(56)

Câu 11 12 13 14 15 16 17 18 19 20 Đáp

x  7

x  9)

Câu 11 12 13 14 15 16 17 18 19 20 Đáp

án 62 68 4 6249 10 11110000

252

1009 2178 10000 0.(235)

Câu 11 12 13 14 15 16 17 18 19 20

án 71 13 22 32 17 2 1 177 9

(2x + 3) (2x + 7)

Câu 11 12 13 14 15 16 17 18 19 20 Đáp