Tài liệu tham khảo môn Toán logic, Toán tiếng anh dành cho khối lớp 6 và khối lớp 7 và ôn luyện vòng loại và chung kết quốc gia của các kỳ thi Olympic Toán Quốc Tế như HKIMO, TIMO, ASMO, BBB ...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 6,7 ĐỀ 1: Logical Thinking According to the pattern shown below, what is the number in the blank? 、 、 12 、 20 、 30 、 42 、 、 … There are chickens and rabbits in a cage and the number of rabbits is two times of the number of chickens They have 210 legs in total How many rabbit(s) is / are there? The age of Alice is seven times that of Peter now years later, the age of Alice is three times that of Peter How old is Alice now? Alice needs 20 days to finish a project Peter needs 30 days to finish the same project If Alice, Peter and Mary the project together, it takes 10 days How long does Mary take to finish the project alone? There are pieces of red socks, pieces of white socks and 13 pieces of blue socks in a box If you want to draw pairs of socks with same colour, at least how many piece(s) of sock(sq ) is / are needed to be drawn? Arithmetic Find the value of 642 9144 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 102 112 122 192 202 10 Convert 56/99 into the decimals Number Theory 11 12 How many simplified fraction(s) with denominator 120 is / are there? 13 14 A 3-digit number has a remainder when divided by 7, has a remainder when divided by and has a remainder when divided by 11 What is such largest 3-digit number? 15 The sum of 20 consecutive numbers is 450 Find the second largest number Geometry 16 How many rectangle(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 17 A rectangle is cut into equal squares If the area of the small square is 16cm2 , how many cm at least does the perimeter increase in cm after cutting the rectangle? 18 Small cubes with side length are combined according to the pattern shown below Find the total volume 19 A square is formed by 150 rectangles that are 2cm 3cm Find the decrease of total perimeter 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 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 21 A flight of stairs has 12 steps Alice can go up for step, steps or 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 go up the stairs? 22 Alice, Mary and other classmates sit at a row Alice or Mary cannot sit at ends and they cannot sit next to each other How many sitting arrangements are there? 23 What is the nearest time the hour hand and the minute hand perpendicular to each other between 2pm and 3pm? 24 identical blue flags and identical red flags are put from left to right How many different permutation(s) is / are there? 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 if she must pass through the point O? 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: Logical Thinking According to the pattern shown below, what is the number in the blank? 、 12 、 30 、 56 、 90 、 132 、 、 … Peter goes east for 20km, then goes south for 24km, then goes west for 13km How far is he now from the original position? If x, y and z are all prime and 2xy yx 2z , find the minimum value ofz There are 10 pieces of red socks, pieces of white socks and 22 pieces of blue socks in the box If you want to get pairs of socks with different colour, at least how many piece(s) ofsock(s) is / are needed to be drawn? Arithmetic Find the value of x if (37 21x) (8x 67) If x and y are positive integers and 7x – 2y = 151, find the minimum value of |𝑥 − 𝑦| How many integral solution(s) is / are there for x if 56 5x14 89 ? Given a and b are positive real numbers and satisfy equations a2 b2 51 and a b , find the value of a b 10 Factorise x2 2x 63 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 Number Theory 11 Find the sum of all positive factor(s) of 513 12 From to 1000, how many integer(s) is / are divisible by and 5, but not divisible by 9? 13 Find the unit value of 52019 12019 32019 14 15 The difference between two square numbers is 611 Find the sum of the possible value of the larger square? Geometry 16 A cube with side length 18 is cut into cubes with side length Find the increase in total surface area of the cubes 17 An iron wire is bent into circles whose radius are 21 If the wire is now bent into identical rectangles where length is integer what is the maximum value of area of this rectangle? (Take 22/7 ) 18 Small cubes with side length are combined according to the pattern shown below Find the total volume of the 15th 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 19 The longest length of a triangle is 12, find the maximum area of the triangle (Answer in surd form) 20 An interior angle of a n-sided polygon is times of an exterior angle Find the value of n Combinatorics 21 There are identical blue flags and identical white flags are put from left to right How many way(s) of arrangement is / are there? 22 There are one 1g, one 2g, one 5g, one 7g and one 9g mass How many different weights can be measured? 23 In how many possible ways can 14 identical flowers be distributed to distinct vases with at least two in each vase? 24 Use 1, 2, 4, 7, 8, to form 5-digit numbers The odd digits cannot be next to each other How many number(s) can be formed in this way? 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 if she must pass through the point O? 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 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: Logical Thinking There are chickens and rabbits in a cage and the number of rabbits is twice the number of chickens They have 340 legs in total How many rabbits are there? Amy is 10 years younger than his brother Uncle’s age is three times of the sum of Amy’s and brother’s age now Six years later, Uncle’s age is two times of the sum of Amy’s and brother’s age Find the age of Amy now John wrote a 4-digit number on a piece of paper and asked Peter to guess it Peter asked: “Is the number 2956?” John replied: “Three of the digits are correct The positions of those digits are all wrong.” Peter asked again: “Is the number 7324?” John replied: “Two of the digits are correct The positions of those digits are all wrong.” Peter asked again: “Is the number 4962?” 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 by John? There are pieces of white chopsticks, pieces of yellow chopsticks and pieces of brown chopsticks mixed together If you want to get pairs of chopstick 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 in the dark, in which pairs of them are of the same color, at least how many piece(s) of chopstick(s) is / are needed to be taken? Arithmetic Find the value of 2222 x 3333 + 3334 x 1111 Find the value of 66 x 65 - 65 x 64 + 64 x 63 - 63 x 62+ +4 x - x + x Find the value of 199 + 198 + … + 101 + 100 – 99 – 98 – 97 - … - 10 Convert 235 999 into the decimals Number Theory 11 12 Find the last digit of A if A = 82 + 182 + 282 + … + 20182 13 How many digit number(s) that can be divisible by and but not divisible by 7? 14 A 4-digit number has remainder when divided by 3, has remainder when divided by and has remainder when divided by What is such largest digit number? 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 ĐỀ 10: Logical Thinking What is the value of the number to represent “?” in the following sequence? 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, ?, … According to the pattern shown below, how many circle(s) is / are there from 1th to 405th symbol counting from the right? It requires 30 minutes to cut a piece of wood into sections If the time required to cut into each section is the same, how many minute(s) is / are required to cut pieces of wood into 10 sections per piece? (Cannot cut sections at the same time) Find the average of the following 10 numbers 2017、 2028、 2008、 2005、 2016、 1989、 2044、 2037、 2011、 2035 There are pieces of white chopsticks, pieces of yellow chopsticks and pieces of black chopsticks mixed together If you want to get pairs of chopsticks with different colours in the dark, at least how many piece(s) of chopstick(s) is / are needed to be taken? Arithmetic 43 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 8 16 2048 Find the value of 11 2 33 9898 9999 100100 Find the value of 185 189 193 10 Find the value of 123456787654321 1234321 Number Theory 11 12 13 20 students have 2019 balloons in total and each of them has a different number of balloons and at least balloon At most how many balloon(s) does the student with the most balloons have? 14 The difference between A and B is 405 A is 28 times of B Find the value of A 15 Find the sum of all positive factor(s) of 2019 44 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 Geometry 16 How many rectangle(s) is / are there in the figure below? 17 The perimeter of a rectangle is 444 Find the maximum value of the area of this rectangle 18 The area of a rectangle is 600 If the sides of the rectangle are integers, how many different value(s) of the perimeter of this rectangle is / are there? 19 There are points on a plane Given that any points are not in collinear How many quadrilateral(s) can you draw? 20 Little cubes with side length 1cm are combined according to the pattern shown below, if there are 23 layers, find the volume 45 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 Question 20 Combinatorics 21 A flight of stairs has steps David can go up for step, steps or steps each time How many way(s) is / are there for David to go up the stairs? 22 According to the following sequence for first 3999th terms, how many odd number(s) is / are there? 1, 2, 3, 6, 11, 20, 37, 68, … 23 Given that A : B : , B : C : and C B 81 What is the value of A? 24 Find the smallest product by using all digits from to without repetition to form two 4-digit numbers 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? 46 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 11: Logical Thinking According to the pattern shown below, what is the 2019th symbol counting from the right? Find the average of the following 10 numbers 2019、 2028、 2008、 2005、 2012、 1989、 2048、 2037、 2011、 2033 Ice goes east for 20km, then goes north for 15km How far is she now from the original position? 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, 1mark will be given for a blank answer, marks will be given for a wrong answer Find the minimum number of candidate(s) to ensure that candidates will have the same scores in the competition There are 16 statements on a piece of paper There are incorrect statements at the front page only and there are correct statements at the back page Find the maximum number of correct statements Arithmetic Find the value of x if (3x 1) (5x 7) 47 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 How many positive integral solution(s) is / are there for x if 30x 17 2019 ? Given a and b are real numbers and satisfy equations a b and ab 13 , find the value of a2 b2 10 Find the value of 8 16 1024 Number Theory 11 Find the number of positive factor(s) of the product of 2017 and 2019 12 13 14 15 29 students have 2019 balloons in total and each of them has a different number of balloons and at least balloon At most how many balloon(s) does the student with the most balloons have? 48 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 Geometry 16 A solid cuboid is formed by merging 2019 cubes with side length 1cm Find the minimum value of the total surface area of the cuboid 17 A triangle has sides with lengths 40, 42 and 58 Find the area of the triangle 18 Little cubes with side length 1cm are combined according to the pattern shown below, if there are 23 layers, find the volume Question 18 19 There are points on a plane Given that any points are not in collinear How many pentagon(s) can you draw? 20 The area of a rectangle is 610 If the sides of the rectangle are integers, how many different value(s) of the perimeter of this rectangle is / are there? Combinatorics 49 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 21 Find the number of the combination(s) arranging boys in a row 22 A flight of stairs has steps David can go up for step, steps or steps each time How many way(s) is / are there for David to go up the stairs? 23 Given that A : B : , B : C : and C A 280 What is the value of A? 24 Choose three digits, without repetition, from 2, 5, 6, 7, to form 3- digit numbers How many number(s) can be divisible by 4? 25 If Amy goes from point A to point B, each step can only move up or move right How many way(s) is / are there? 50 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 10 Đáp án 56 42 21 60 20 2800 11 23 4374 2585 0.56 Câu 11 12 13 14 15 16 17 18 19 20 Đáp án or 10 32 18 830 31 64 40 15 1380 Câu 21 22 23 24 25 Đáp án 136 1440 20 28 2: 27 11 Đề 2: Câu Đáp án 10 182 25 25 13 25 -8 29 17 Câu 11 12 13 14 15 16 17 18 19 20 Đáp án 800 44 527 94536 3888 484 271 36√3 18 Câu 21 22 23 24 25 x 7x 9 51 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 84 22 45 252 10 Đề 3: Câu Đáp 62 68 án Câu 11 12 13 Đáp án 15 129 Câu 21 22 23 Đáp án 25 : 27 10 11110000 252 1009 2178 10000 0.235 15 16 17 18 19 20 9922 25 106 256 2300 360 6249 10 14 24 25 1118 18 11 Đề 4: Câu 10 Đáp án 71 13 22 31 10 177 (2x + 3)(2x + 7) Câu 11 12 13 14 15 16 17 18 19 20 Đáp án 27 25502500 13 4 1102 864 566 9√3 896 52 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 21 22 Đáp 142857 án : 32 23 24 25 45 42 28 11 Đề 5: Câu Đáp n or án n=-29 10 4 80 28 (3x + 5)(2x - 3) 18 19 20 Câu 11 12 13 14 15 16 17 Đáp án 50 101 21h 53 20 Câu 21 22 23 24 Đáp án 500 840 72 3900 11600 28 25 48 1330 Đề 6: Câu 10 Đáp án 43 50 441 3906 3030000 986 18 19 27 Câu 11 12 13 14 15 16 17 18 19 20 53 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 2688 222 2017 102 Câu 21 22 23 24 25 Đáp án 96 625 990 144 24 28 36 24 954 and 1980 Đề 7: Câu Đáp án 11 125 Team C Saturday -5 4290 Câu 11 12 13 14 15 16 Đáp án 1440 25 4031 Câu 21 22 23 24 25 1832 30030 126 52 164 Đáp án 10 0.432 17 18 19 20 64 130 Đề 8: Câu 10 54 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 C 217 543 15 177146 1570000 1072 10 779 990 Câu 11 12 13 14 15 16 17 18 19 20 Đáp án 3030 27 2015 59 198 128 242 12 8002 Câu 21 22 23 24 25 Đáp án 86 406 153 72 780 Câu Đáp án C 216 8521 8 3080 1296 36 Câu 11 12 13 14 15 16 17 18 19 20 Đáp án 12 35 4374 4031 13 18 150 462 12 726 Câu 21 22 23 24 25 Đáp án 144 872 1785 108 205 Đề 9: 10 1023 511 512 55 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 10 Đáp 193 án 203 108 2019 4094 2018 2019 Câu 11 12 13 14 15 16 17 18 19 20 Đáp án 1829 420 2696 45 12321 12 70 4324 Câu 21 22 23 24 25 Đáp án 81 2000 126 2465422 15 338350 4753 100020001 Đề 11: Câu 10 Đáp án 1009 2019 25 42 10 -1 -4 66 38 2047 Câu 11 12 13 14 15 16 17 18 19 20 Đáp án 23 1613 5390 840 4324 56 Câu 21 22 23 24 25 Đáp án 24 149 560 18 20 56 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 57 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 ... the value of 2222 x 3333 + 3334 x 1111 Find the value of 66 x 65 - 65 x 64 + 64 x 63 - 63 x 62 + +4 x - x + x Find the value of 199 + 198 + … + 101 + 100 – 99 – 98 – 97 - … - 10 Convert 235 999... 182 25 25 13 25 -8 29 17 Câu 11 12 13 14 15 16 17 18 19 20 Đáp án 800 44 5 27 945 36 3888 484 271 36? ??3 18 Câu 21 22 23 24 25 x 7? ??x 9 51 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng... 26 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 7: Logical Thinking