Bộ đề ôn luyện kỳ thi Toán Quốc Tế HKIMO 2021 Khối 4 +5

Tài liệu ôn tập khối lớp 4 và khối lớp 5 ôn luyện cho kỳ thi Olympic Toán Học Quốc tế HKIMO. Nội dung thi là các dạng toán logic theo chuẩn chương trình học của Quốc Tế và các trường song ngữ, các trường Quốc tế tại Việt Nam

BỘ ĐỀ ÔN LUYỆN OLYMPIC TOÁN HỌC QUỐC TẾ HKIMO 2020

3 A stationery store had a batch of ball pens One halfand 7 morepens were sold

on the first day One half of the remaining part but 4fewerpens were sold on the

second day Finally 18ball pens are left How many ball pen(s) did thestationery store have originally?

4 7thMay is Monday Which day of the week is 5thAugust in the same year?

5 There are a total of 34 chickensand rabbitsin a farm.The animals have a total of

96 legs How many chicken(s) is / are there?

Arithmetic

6 Find the value of 1+2+3 + 59 + 60

7 Find the value of 333 x 333 + 111

8 Find the value of 14 x 14 + 2 x 14 x 86 + 86 x 86

9 Find the value of 11 x 10 – 10 x 9 + 9 x 8 – 8 x7 + 7 x 6 – 6 x 5 + 5 x 4 – 4 x 3 + 3 x 2 – 2 x 1

10 Find the value of 3 + 7 + 11 + + 55 + 59

Number Theory

11 If 9-digit number 20180901A is divisible by 18, find the value of A

12 Find the unit digit of A if

13 How many factors does 196 have?

14 The differenceof positive numbers A and B is 324 A is 3 times of B Find the

value of A

15 How many 2-digit numbers are neither divisible by 2 nor divisible by 3?

Geometry

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

17 The area of a rectangle is 48cm2 If sides of the rectangle are integers, how many different value(s) of perimeter of rectangle is/are there?

18 Find theperimeter of the figure shown below in cm

19 The perimeter of an equilateral triangleis 30cm Now Andy uses 4 triangles to

form a larger triangle What is the perimeter of the large trianglein cm?

20 How many right-angled triangle(s) is / are there in the figure below?

Combinatorics

21 Amy draws alltriangles which perimeter is 14cm and length of sides are

integers How many different types of triangle(s) does she draw?(triangles with sides (3,5,6) and (5,3,6) are regarded as the same type of triangle)

22 Two boys Bobby, Benny and three girls Grace, Gloria, Gaileare to be seated in

a row according to the rulethat a boy will not sit next to another boy and a girl will not sit next to another girl Find the number of the combination(s) satisfying the

above condition

23 At least how many of 119children that were born in the same month?

24 How many 2-digit number(s) is / are there such that the numberdoes

notcontaindigit “7” and no digits are repeated?

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

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

ĐỀ 2:

1 There are 27 transparent boxes, as shown in figure 1 Each layer contains 9 boxes as figure 2 shows Some black marbles are inserted into the boxes and view from different directions, the images are shown in the diagram below Find the

maximum number of black marbles inserted in these 27 boxes

2 Given 17th February 2015 is Tuesday, which day of the week will be 2015 days later?

3 John wrote a 2-digit number on a piece of paper and asked Peter to guess it Peter asked: “Is the number divisible by 9 and 12?”

John replied: “It is divisible by 9 but not 12”

Peter asked again: “Is the number less than 40?”

John replied: “Yes, it is less than 40.”

Peter asked again: “Is it an odd number?”

John said: “No, it is an even number.”

What is the number written by John?

4 What is the value of the number to represent “?” in the following table?

- 2 3 4 5 6

3 5 8 12 17 23

Figure 1 Figure 2 Top view Front view From the right

1st Group 2nd Group 3rd Group

6 Find the value of 720 ÷ 10 + 720 ÷ 6 + 720 ÷ 3 + 720 ÷ 1

7 Find the value of 1111 × 11111

8 Find the value of 7 + 15 + 23+ +87 + 95

9 Find the value of 11 × 61 + 22 × 12 + 33 × 5

10 Find the value of (2015 + 219 × 830) ÷ (831 × 220 + 965)

11 If a 5-digit number 20𝐴15 is divisible by 11, find the value of A

12 Find the unit digit of A if

13 How many three-digit even numbers that can be divisible by 7?

14 The sum of A and B is 2015 A is larger than B by 3 times Find the value of A

15 What is the largest three-digit number that can be divisible by 2 and 3?

16 If 3 identical squares are combined into a rectangle, the perimeter will be less

than the sum of that of the squares by 16 cm Find the perimeter of the square in cm

17 According to the pattern shown below, how many cubes are there in 10th layer?

18 The figure below is formed by 4 identical rectangles The areas of the big and

the small squares are 81cm2 and 25 cm2 respectively Find the perimeter of the

rectangle in cm.。

19 In the figure below, the area of parallelogram ABCD is 15 cm2 E is a point on

AD such that AE = 3cm CE is perpendicular to BC and is of 3 cm Find the area of

𝛥𝐸𝐶𝐷 in cm2

20 How many triangles are there in the figure below?

21 There are a balance and a set of weights, if this set is possible to measure all integral weights from 1kg to 30 kg, find the minimum number of the weights

22 Choose 3 numbers, without repetition, from 1, 4, 5, 6, 7, 9 to form a three-digit number How many even numbers are there?

23 Numbers are drawn from the 20 integers 1 to 20 At least how many numbers are drawn at random to ensure that there are two numbers whose product is 60?

24 How many 3-digit numbers with no repeated digits are there?

25 How many 3-digit numbers are there so that the product of digits is 12?

ĐỀ 3:

1 There are 20 problems in a mathematics competition The scores of each

problem are allocated in the following ways: 5 marks will be given for a correct

answer, 2 marks will be deducted for each incorrect or unanswered answer How many questions does Amy answer correctly if her score is 58?

2 If today is Tuesday, which day of the week will be 100 days later?

3 There are a total of 30 tigers and cranes in a zoo The animals have a total of 90 legs How many tigers are there?

4 John wrote a 3-digit number on a piece of paper and asked Peter to guess it Peter asked: “Is the number 173?”

John replied: “One of the digits is correct, the position of that digit is also correct.” Peter asked again: “Is the number 132?”

John replied: “Two digits are correct, but the positions of those digits are both

wrong.”

Peter asked again: “Is the number 324?”

John said: “All three digits are correct, but the digits are all in the wrong places.” What is the number written by John?

5 According to the pattern shown below, how many circles are there in the 25thgroup?

1st Group 2nd Group 3rd Group

6 Find the value of 1366 + 3987 − 1166 + 13 + 4895 − 126 + 876 − 445

7 Find the value of 22222 × 22222

8 Find the value of 5 + 10 + 15 + 20+ +125 + 130

9 Find the value of 13 × 27 + 26 × 11 + 39 × 17

10 Find the value of 360 ÷ 10 + 360 ÷ 6 + 360 ÷ 3 + 360 ÷ 1

11 If a 5-digit number 81𝐴76 is divisible by 9, find the value of A

12 Find the unit digit of A if 𝐴 = 3 × 3 × 3 × .× 3⏟

14 The sum of A and B is 168 A is larger than B by 6 times Find the value of A

15 What is the largest three-digit number that can be divisible by 6 and 8?

16 The figure below is formed by 3 squares with areas 16𝑐𝑚2, 36𝑐𝑚2 and 16𝑐𝑚2respectively Find the perimeter of the following figure

17 According to the pattern shown below, how many cubes are there in 10th layer?

18 The figure below is made of 16 equilateral triangles If the perimeter of figure

below is 36cm, what is the difference between before combination and after

combination?

19 The figure below is made of 2 squares Find the area of the shaded region below in cm2

20 How many rectangles are there in the figure below?

21 A flight of stairs has 10 steps David can go up for 1 step or 2 steps at a time How many ways are there for David to go up the stairs?

22 Choose 6 numbers, without repetition, from 1, 4, 5, 6, 7, 9 to form 2 three-digit numbers Find the maximum value of the difference between these 2 numbers

4 cm

6 cm

23 Numbers are drawn from the 80 integers 1 to 80 At least how many numbers are drawn at random to ensure that there are two numbers whose sum is 100?

24 Now there are 1-dollar, 2-dollar, 5-dollar and 10-dollar coins and 2 for each

type How many different amount can be formed?

25 How many 3-digits number are there so that the sum of digits is 8?

3 The age of Samuel 7 years later is equal to the age of Joseph 3 years later

Given that Joseph is 42 years old now, how old is Samuel now?

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

1 、8 、27 、64 、 、…

5 There are a total of 51 tigers and cranes in a zoo The animals have a total of

114 legs How many tiger(s) is / are there?

6 Today is 5th April, Friday Which day of the week is 31st August?

7 Find the value of

8 Find the value of 1+ 5 + 9 + + 81 + 85 + 89

9 Find the value of 1 + 2 + 3 + + 19 + 20 + 19 + + 3 + 2

10 Find the value of

12 Find the value of 3125 x 32 x 16 x 125

13 If a 9-digit number 20190405A is divisible by 9, find the value of A

14 If A is a 1-digit number, what is the value of A if the equation below is correct?

A + A = 10

15 6 students have 60 balloons in total and each of them has a different number of

balloons and at least 1 balloon At most how many balloon(s) does the student with the most balloons have?

16 The sum of seven consecutive numbers is 294 Find the smallest number

17 What is the smallest 4-digit number that can be divisible by 8 and 12?

18 Define the operation symbol a + b = a/b + a , find the value of

(12 + 4) + 3

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

20 The perimeter of a rectangle is 100 Find the maximum value of the area of this rectangle

21 The area of a rectangle is 60 If the sides of the rectangle are integers, how many different value(s) of the perimeter of this rectangle is / are there?

22 There are 7 points on a plane Given that any 3 points are not in collinear How many quadrilateral(s) can you draw?

23 It is known as the lengths of shorter sides for a right-angled triangle are 6 and 8 respectively Find the length of the longest length

24 Now there are 19 points How many line segment(s) can be formed?

25 A flight of stairs has 6 steps David can go up for 1 step, 2 steps or 3 steps each time How many way(s) is / are there for David to go up the stairs?

ĐỀ 5:

1 Alice has 51 marks and Peter has 65 marks in Mathematics exam If the

average mark of Alice, Peter and Mary is 58, what is Mary’s exam marks?

2 The sum of Alice’s and Peter’s ages is 47 If the age of Alice 2 years ago was the half of the age of Peter 6 years later, how old is Alice now?

3 A book store had a box of books One half but 6 more books were sold on the first day Two third of the remaining part and 2 less books were sold on the second day Finally 8 books are left How many book(s) did the book store have originally?

4 15th May is Friday Which day of the week is 1st April in the same year?

5 There are a total of 36 chickens and rabbits in a farm The animals have a total

of 80 legs How many chicken(s) is / are there?

6 Find the value of 31 + 32 + 33 + 34 + …+ 66 + 67

7 Find the value of 111 x 11 x 101

8 Find the value of 17 x 62 + 19 x 34

9 Find the value of 1000 : 2 + 1000 : 10 + 1000 : 100

10 Find the value of 2 + 4 + 6 + 8 +10 + + 28 + 30

11 If a 9-digit number 20190513A is divisible by 36, find the value of A

12 Find the unit digit of A if

13 How many factors does 513 have?

14 The sum of positive integers A and B is 783 A is 8 times of B Find the value

of A

15 How many 3-digit number(s) either divisible by 5 or divisible by 7 is / are

there?

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

17 The perimeter of a rectangle is 36cm If sides of the rectangle are integers, find

the minimum value of the area of the rectangle

18 The square below is cut into 5 squares and 1 rectangle If the perimeter of the

original square is 120cm, find the area of the rectangle

19 The perimeter of a square is 40cm Now Alice combine 36 square to a big

square What is the perimeter of the big square in cm?

20 How many right-angled triangle(s) is / are there in the figure below?

21 A flight of stairs has 8 steps Mary can go up for 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 Mary to go up the stairs??

22 99 students are either wearing XL, L, M, S or XS size uniforms At least how

many student(s) is / are wearing the same size of uniforms?

23 Alice has 40 $1 and $2 coins altogether She has $58 in total How many $1

coin(s) does she have?

24 How many 3-digit number(s) is / are there such that the number does not

contain digit “0” or “1”?

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

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

ĐỀ 6:

1 Given A and B are two non-zero digits and the 2-digit numbers formed by these two digits have the following properties:

a) AB is divisible by 17;

b) BA is a product of 2 odd prime numbers;

Find the 2-digit number BA

2 There are some chickens and rabbits, where the number of chickens is 28 more than the number of rabbits

If there are 338 legs in total, how many chickens are there?

3 A stationery store had a batch of ball pens One half and 5 more pens were sold

on the first day One half of the remaining part but 12 fewer pens were sold on the second day Finally 68 ball pens are left How many ball pen(s) did the stationery store have originally?

4 John wrote a 4-digit number on a piece of paper and asked Peter to guess it Peter asked: “Is the number 2435?”

John replied: “Two of the digits are correct The positions of those digits are also correct.”

Peter asked again: “Is the number 6397?”

John replied: “Two of the digits are correct The positions of those digits are all

wrong.”

Peter asked again: “Is the number 9823?”

John said: “All the digits are correct The positions of those digits are all wrong.” Given all the digits in that 4-digit number are all different, what is the number written

7 Find the value of

8 Find the value of 2018 + 2017 – 2016 + 2015 + 2014 – 2013 + 5 + 4 – 3 + 2 + 1

9 Find the value of 6672 – 3332

10 Find the value of

11 If 9-digit number 2018A902B is divisible by 144, find the value of A + B

12 Find the number of factors for 720

13 Define the operation symbol , find the value of

14 The sum of two positive numbers A and B is 96 Find the maximum value of the product of A and B

15 How many 3-digit number(s) that can be divisible by 4 but cannot be divisible

by 7 is / are there?

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

17 As shown in the figure below, area of ABC is 960 cm2 If D, E and F are midpoints of AC, BC and BE respectively, find the area of the shaded region in cm2

18 28 small cubes with side length 1cm are combined according to the pattern shown below, find the surface area in cm2

19 How many diagonals does a heptagon(7-sided polygon) have?

20 Find the area of the shaded region in the figure (Take 𝜋 = 22

7 )

21 Amy draws all triangles which perimeter is 16c m and length of sides are

integers How many different types of triangle(s) does she draw? (triangles with sides (5,5,6) and (5,5 ,6) are regarded as the same type of triangle)

22 How many digits are there from page numbers from page 23 to page 1028 of a book (inclusively)?

23 A t least how many of 253 children that was/were born in the same Chinese zodiac year (There are 12 different Chinese zodiac year)