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 ƠN LUYỆN OLYMPIC TỐN HỌC QUỐC TẾ HKIMO 2020 KHỐI 4, ĐỀ 1: Logical Thinking What is the value of the number to represent “?” in the following sequence? 2, 6, 12, 20, 30,？, 56, 72, … The sum of father’s and mother’s ages is 84 If the age of father years ago is the same as the age of mother years later, how old is father now? A stationery store had a batch of ball pens One halfand 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? 7thMay is Monday Which day of the week is 5thAugust in the same year? 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 Find the value of 1+2+3 + 59 + 60 Find the value of 333 x 333 + 111 Find the value of 14 x 14 + x 14 x 86 + 86 x 86 Find the value of 11 x 10 – 10 x + x – x7 + x – x + x – x +3x2–2x1 10 Find the value of + + 11 + + 55 + 59 Number Theory KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 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 times of B Find the value of A 15 How many 2-digit numbers are neither divisible by 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 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? KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 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? KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com ĐỀ 2: There are 27 transparent boxes, as shown in figure Each layer contains boxes as figure 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 Figure Top view Figure Front view From the right Given 17th February 2015 is Tuesday, which day of the week will be 2015 days later? John wrote a 2-digit number on a piece of paper and asked Peter to guess it Peter asked: “Is the number divisible by and 12?” John replied: “It is divisible by 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? What is the value of the number to represent “?” in the following table? - 12 17 23 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 17 29 46 69 14 31 60 106 175 20 51 111 217 ？ According to the pattern shown below, how many triangles are there in the 15th group? 1st Group 2nd Group 3rd Group Find the value of 720 ÷ 10 + 720 ÷ + 720 ÷ + 720 ÷ Find the value of 1111 × 11111 Find the value of + 15 + 23+ +87 + 95 Find the value of 11 × 61 + 22 × 12 + 33 × 10 Find the value of (2015 + 219 ì 830) ữ (831 ì 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 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 𝐴=⏟ 20 × 20 × 20 × .× 20 × ⏟ 15 × 15 × 15 × .× 15 100′𝑠 100′𝑠 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 times Find the value of A 15 What is the largest three-digit number that can be divisible by and 3? 16 If 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 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 cm Find the area of 𝛥𝐸𝐶𝐷 in cm2 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com A E D B 20 C 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 numbers, without repetition, from 1, 4, 5, 6, 7, to form a three-digit number How many even numbers are there? 23 Numbers are drawn from the 20 integers 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? KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com ĐỀ 3: There are 20 problems in a mathematics competition The scores of each problem are allocated in the following ways: marks will be given for a correct answer, marks will be deducted for each incorrect or unanswered answer How many questions does Amy answer correctly if her score is 58? If today is Tuesday, which day of the week will be 100 days later? 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? 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? According to the pattern shown below, how many circles are there in the 25th group? 1st Group 2nd Group 3rd Group Find the value of 1366 + 3987 − 1166 + 13 + 4895 − 126 + 876 − 445 Find the value of 22222 × 22222 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com Find the value of + 10 + 15 + 20+ +125 + 130 Find the value of 13 × 27 + 26 × 11 + 39 × 17 10 Find the value of 360 ÷ 10 + 360 ÷ + 360 ÷ + 360 ÷ 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 = ⏟ × × × .× × ⏟ × × × .× 100′𝑠 100′𝑠 13 The difference of A and B is 144 A is times of B Given A is greater than B, find the value of B 14 The sum of A and B is 168 A is larger than B by times Find the value of A 15 What is the largest three-digit number that can be divisible by and 8? 16 The figure below is formed by squares with areas 16𝑐𝑚2 , 36𝑐𝑚2 and 16𝑐𝑚2 respectively Find the perimeter of the following figure 17 According to the pattern shown below, how many cubes are there in 10th layer? KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 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 squares Find the area of the shaded region below in cm2 cm 20 cm How many rectangles are there in the figure below? 21 A flight of stairs has 10 steps David can go up for step or steps at a time How many ways are there for David to go up the stairs? 22 Choose numbers, without repetition, from 1, 4, 5, 6, 7, to form three-digit numbers Find the maximum value of the difference between these numbers 10 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com Find the value of 10 16 22 124 130 Find the value of 13 36 26 24 108 10 Find the value of 2400 10 2400 2400 2400 1 11 If a 5-digit number 81A76 is divisible by 36, find the value of A A 12 Find the unit digit of A if 100' s 100' s 13 The difference of A and B is 144 A is times of B Given A is greater than B, find the value of B 14 The sum of A and B is 168 A is larger than B by10 times Find the value of A 15 What is the smallest three-digit number that can be divisible by and 8? 2 16 The figure is formed by squares with areas 64 cm , 121 cm and 64 cm respectively Find the perimeter of the following figure 17 According to the pattern shown below, if there are 10 layers, how many cubes are there? 27 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 18 The figure below is made of 16 small equilateral triangles If the perimeter of figure below is 72cm, what is the difference between before combination and after combination? 19 The figure below is made of squares Find the area of the shaded region below in cm2 cm cm 28 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 20 How many rectangles are there in the figure below? 21 A flight of stairs has 10 steps David can go up for step, steps, or steps at a time How many ways are there for David to go up the stairs? 22 Choose numbers, without repetition, from 1, 4, 5, 6, 7, to form a three-digit number Find the minimum value of the difference between these numbers 23 Numbers are drawn from the 80 integers to 80 At least how many numbers are drawn at random to ensure that there are two numbers whose sum is 50? 24 If Andy goes from point A to point B, each step can only move up or move right How many methods are there? B 29 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com A 25 How many 4-digits number are there so that the sum of digits is 6? 30 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com ĐỀ 9: According to the pattern shown below, what is the number in the blank? 20 、15 、18 、18 、16 、21 、 、… There are some chickens and rabbits, where the number of chickens is 28 more than the number of rabbits If there are 320 legs in total, how many rabbits are there? A book store had a box of books One half but more books were sold on the first day Two third of the remaining part and less books were sold on the second day Finally books are left How many book(s) did the book store have originally? Alice needs 60 days to finish a project Peter needs 15 days to finish the same project If Alice and Peter the project together, how many day(s) they need to finish it? John said: “One digit is correct, but the position of that digit is wrong.” What is the number written by John? There are 10 blue balls, 18 red balls and green balls in a bag At least how many balls should be picked up to ensure there are balls for each colour? Find the value of 11 x 111 x 101 x 1001 Find the value of Find the value of 200 : 0.4 : 400 : 0.8 : 600 : 1.2 : 800 : 1.6 Find the value of 342 – x 32 31 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 10 Find the value of 11 If 10-digit number 20190513AB is divisible by 66 , find the value of A + B 12 Find the last 4-digits of (141 x 144 x 147 x x 186 x 189 x 192) 13 Define the operation symbol , find the value of 14 Teacher gives some candies to students If everyone gets candies, candies will be left If everyone gets candies, 18 candies will be left How many candies teacher have? 15 How many 3-digit number(s) that can be divisible by and of unit digit is / are there? 16 How many rectangle(s) is / are there in the figure below? 17 A square is formed by 128 rectangles that are 2cm x 1cm Find the decrease of total perimeter in cm 18 22 small cubes with side length 1cm are combined According to the pattern shown below, find the surface area in cm2 32 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 19 How many side(s) does a regular polygon have which has 45° exterior angle? 20 ABDF is square and ⊿FDE and ⊿BDC are right angle isosceles triangles Find the area of ACDE 21 A flight of stairs has steps Alice can go up for step, steps or steps each time The 4th step cannot be stepped on as it is destroyed How many way(s) is / are there for Alice to go up the stairs? 22 12 students join a competition Only top students can advance to next round How many different way(s) of advancing is / are there? 23 In a secondary school (a secondary school has groups), at least how many of 380 students that are in the same grade? 24 Numbers are drawn from 80 integers 20 to 99 At least how many number(s) is / are drawn at random to ensure that there are two numbers whose sum is 70? 25 If Alice goes from point A to point B, each step can only move up or move right How many way(s) is / are there? 33 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 34 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com ĐỀ 10: What is the value of the number to represent “?” in the following sequence? 1, 1, 1, 3, 5, 9, 17, 31, 57 , … According to the pattern shown below, how many circle(s) is / are there from 1st to 304th symbol counting from the left? ○ △ □ ○ ○ △ □ ○ ○ △ □ ○… Given there are 12 signs of the zodiac, at least how many student(s) is/ are there such that there exists at least 12 of them having the same sign of the zodiac? A stationery store had a batch of ball pens One half but few pens were sold on the first day One half of the remaining part and more pens were sold on the second day Finally 21 ball pens are left How many ball pen(s) did the stationery store have originally? There are a total of 73 tigers and cranes in a zoo The animals have a total of 200 legs How many crane (s) is / are there? Today is 5th April, Friday Which day of the week is 1st September? Find the value of 12x 41+ 24 x 11+ 36 x 12 Find the value of 20192 – 10192 Find the value of 1+ + + + 85 + 89 + 93 10 Find the value of 1+ + + + 99 + 100 + 99 + + 3+ 11 Find the value of 12345654321 : 12321 12 Find the value of 13 If a 8-digit number 2019405A is divisible by 18, find the value of A 35 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 14 Find the unit digit of A if 15 Define the operation symbol 16 The sum of A and B is 405 A is 26 times of B Find the value of A 17 What is the largest 4-digit number that can be divisible by and 12? , find the value of 18 The sum of seven consecutive odd numbers is 287 Find the smallest number 19 How many rectangle(s) is / are there in the figure below? 20 The perimeter of a rectangle is 120 Find the maximum value of the area of this rectangle 21 The area of a rectangle is 96 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 points on a plane Given that any points are not in collinear How many pentagon(s) can you draw? 23 It is known as the lengths of shorter sides for a right-angled triangle are and 12 respectively Find the length of the longest length 24 Now there are 20 points How many line segment(s) can be formed? 36 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com 25 How many 2-digit number(s) is / are there such that the number contains digit “8” and no digits are repeated? 37 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com ĐÁP ÁN THAM KHẢO Đề 1: Câu Câu Câu 42 11 21 circles 46 70 Sun 20 1830 1110 1000 00 16 17 18 96 80 12 22 10 13 23 14 486 24 15 30 25 13 18 13 64 23 17 392 225 1152 12344321 612 1100 14 15 16 17 18 19 1612 996 16 100 14 24 25 648 15 60 10 465 19 60 20 13 202013 Đề 2: Câu Monday Câu 11 12 Câu 21 22 40 10 20 15 Đề 3: Câu 14 Thursday 15 243 325 Câu 11 12 13 14 15 36 147 984 Câu 21 22 23 24 25 89 831 51 36 36 5000 16 40 493817284 2025 17 18 361 108 1300 19 14 10 576 20 210 38 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com Đề 4: Câu 57 101 38 Câu 11 12 13 13600 200000000 125 14 5 Sat 162 1035 15 16 17 18 45 39 1008 20 399 19 60 10 22 20 625 Câu 21 22 35 23 10 24 171 25 24 Đề 5: Câu Câu Câu 58 11 19 48 12 13 Wed 14 696 32 15 282 21 10 22 20 24 512 25 59 23 22 10 1813 123321 1700 610 240 16 17 18 19 20 70 17 720 240 20 Đề 6: Câu Câu Câu 15 11 21 75 234 2938 16 12 13 14 15 30 468 2304 193 22 23 24 25 2970 22 90 22 10 3/28 680403 334000 247/256 5720 16 17 96 600 18 94 19 14 20 504 Đề 7: 39 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com Câu Câu Câu 16 11 21 40 Monday 36 12 13 238 22 23 455 4 14 21 24 48 120 1260 493284 40400 1188 15 16 17 18 19 30 64 385 81 144 25 768 10 20 75 Đề 8: Câu 17 Wednesday 273 220 Câu 11 12 13 14 15 24 154 120 Câu 21 22 23 24 25 274 17 57 70 56 10 91 740725926 1,474 1,200 3,840 16 17 18 19 20 76 1,330 216 26 441 Đề 9: Câ u 10 55/8 900 17 1/5898 24 18 19 1023/102 20 6000 77 704 60 50 21 22 58 495 1 5 12344 4321 16 3 12 11 4 12 Câ u Câ u 14 24 66 40 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com Đề 10: Câu 105 152 Câu 11 12 1,002,001 405/406 Câu 21 22 21 133 13 23 15 108 14 24 190 10 46 Sun 1188 3038000 1128 9999 15 16 17 18 19 20 630 390 9984 35 90 900 25 17 41 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@daihocthudo.edu.vn hkimo.fe@gmail.com ... Email: hkimo@ daihocthudo.edu.vn hkimo. fe@gmail.com Câu Câu Câu 16 11 21 40 Monday 36 12 13 238 22 23 45 5 4 14 21 24 48 120 1260 49 32 84 4 040 0 1188 15 16 17 18 19 30 64 385 81 144 25 768 10 20 75 Đề. .. 58 49 5 1 5 12 344 43 21 16 3 12 11 4 12 Câ u Câ u 14 24 66 40 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: hkimo@ daihocthudo.edu.vn... 13 14 15 24 1 54 120 Câu 21 22 23 24 25 2 74 17 57 70 56 10 91 740 725926 1 ,47 4 1,200 3, 840 16 17 18 19 20 76 1,330 216 26 44 1 Đề 9: Câ u 10 55/8 900 17 1/5898 24 18 19 1023/102 20 6000 77 7 04 60