Khối 6 + Khối 7 - Tài liệu TIMO

BỘ TÀI LIỆU ÔN THI VÒNG LOẠI TIMO 2020 Primary 6 - Secondary 1 ĐỀ 1: 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: [email protected] hoặc [email protected].

2 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: [email protected] hoặc [email protected].

3 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: [email protected] hoặc [email protected].

4 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: [email protected] hoặc [email protected].

5 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: [email protected] hoặc [email protected].

6 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: [email protected] hoặc [email protected].

ĐỀ 2: 7 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: [email protected] hoặc [email protected].

Logical Thinking 1. What is the value of the number to represent “?” in the following sequence? 2, 6, 12, 20, 30, ? , 56, 72, … 2. The sum of father’s and mother’s ages is 84. If the age of father 5 years ago is the same as the age of mother 3 years later, how old is father now? 3. A stationery store had a batch of ball pens. One half and 7 more pens were sold on the first day. One half of the remaining part but 4 fewer pens were sold on the second day. Finally 18 ball pens are left. How many ball pen(s) did the stationery store have originally? 4. 7th May is Monday. Which day of the week is 5th August in the same year? 5. There are a total of 34 chickens and rabbits in a farm. These 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 333 111. 8. Find the value of 14 14  2 14  86  86  86 . 9. Find the value of 1110 10  9  9  8  8 7  7  6  6  5  5 4  4  3  3 2  2 1. 10. Find the value of 3  7 11  55  59 . Number Theory 11. If 9-digit number ̅2̅0̅̅1̅̅8̅0̅̅9̅̅0̅1̅̅���̅��� is divisible by 18, find the value of A. 12. Find the unit digit of ������ if 13. How many factors does the number 196 have? 8 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: [email protected] hoặc [email protected].

14. The difference of 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 the perimeter of the figure shown below in cm. 19. The perimeter of an equilateral triangle is 30cm. Now Andy uses 4 triangles to form a larger triangle. What is the perimeter of the large triangle in cm? 20. How many right-angled triangle(s) is / are there in the figure below? 9 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: [email protected] hoặc [email protected].

Combinatorics 21. Amy draws all triangles which perimeter is 14cm and lengths 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, Gaile are to be seated in a row according to the rule that 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 conditions. 23. At least how many of 119 children that were born in the same month? 24. How many 2-digit number(s) is / are there such that the number does not contain digit “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 3: 10 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: [email protected] hoặc [email protected].

Logical Thinking 1. According to the pattern shown below, how many triangles are there in the 15th group? 1st Group 2nd Group 3rd Group 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? -23 4 5 6 3 5 8 12 17 23 4 9 17 29 46 69 11 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: [email protected] hoặc [email protected].

5 14 31 60 106 175 6 20 51 111 217 ？ 5. 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. Arithmetic 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). Number Theory 11. If a 5-digit number 20������15 is divisible by 11, find the value of A. 12. Find the unit digit of ������ if ������ = 2⏟0 × 20 × 20 ×. . .× 20 × ⏟15 × 15 × 15 ×. . .× 15. 100′������ 100′������ 13. How many three-digit even numbers that can be divisible by 7? 12 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: [email protected] hoặc [email protected].

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? Geometry 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 ������������������������ in cm2. 20. How many triangles are there in the figure below? 13 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: [email protected] hoặc [email protected].

Combinatorics 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 and 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 4: Logical Thinking 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: “Is the number 173?” 14 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: [email protected] hoặc [email protected].

John: “One of the digits is correct, the position of that digit is also correct.” Peter: “Is the number 132?” John: “Two digits are correct, but the positions of those digits are both wrong.” Peter: “Is the number 324?” John: “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 25th group? 1st Group 2nd Group 3rd Group Arithmetic 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. Number Theory 11. If a 5-digit number 81������76 is divisible by 9, find the value of A. 12. Find the unit digit of ������ if ������ = 3⏟× 3 × 3 ×. . .× 3 × ⏟6 × 6 × 6 ×. . .× 6. 100′������ 100′������ 13. The difference of A and B is 144. A is 5 times of B. Find the value of B. 15 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: [email protected] hoặc [email protected].

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? Geometry 16. The figure below is formed by 3 squares with areas 16������������2, 36������������2 and 16������������2 respectively. Find the perimeter of the following figure in cm. 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 36, what is the difference between the perimeter before combination and after combination? 16 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: [email protected] hoặc [email protected].

19. The figure below is made of 2 squares. Find the area of the shaded region below in cm2. 6 cm 4 cm 20. How many rectangles are there in the figure below? Combinatorics 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 and 9 to form 2 three- digit numbers. Find the maximum value of the difference between these 2 numbers. 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 of money can be formed? 25. How many 3-digits numbers are there so that the sum of digits is 8? ĐỀ 5: 17 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: [email protected] hoặc [email protected].

Logical Thinking 1. What is the value of the number to represent “?” in the following sequence? 1, 1, 1, 3, 5, 9, 17, 31, ? , … 2. According to the pattern shown below, how many circle(s) is / are there from 1st to 203rd symbol counting from the left? ○ △ □ ○ ○ △ □ ○ ○ △ □ ○… 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? Arithmetic 6. Today is Friday 5th April. Which day of the week is 31st August? 7. Find the value of 90 :1 90 :10  90 : 5  90 : 2. 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 1 4  7 10 13 16 19  22  25  28  31 34  37  40  41. 11. Find the value of 272 12 15 272  272  23 . 12. Find the value of 3125 32 16 125. Number Theory 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? 18 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: [email protected] hoặc [email protected].

������ + ������ = 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. Geometry 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? Combinatorics 22. There are 7 points on a plane. Given that there are no 3 collinear points, how many quadrilateral(s) can you draw? 23. It is known that the lengths of shorter sides in a right-angled triangle are 6 and 8 respectively. Find the length of the longest side. 24. Given 19 distinct points on a plane, 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? 19 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: [email protected] hoặc [email protected].

20 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: [email protected] hoặc [email protected].

KEY Đề 1: 21 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: [email protected] hoặc [email protected].

22 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: [email protected] hoặc [email protected].

Đề 2: 7 8 9 10 Câu 1 2 3 4 5 6 42 46 70 Sun 20 1830 111000 10000 60 465 Câu 11 12 13 14 15 16 17 18 19 20 6 4 9 486 30 96 5 80 60 13 Câu 21 22 23 24 25 4 12 10 64 16 Đề 3: 1 2 3456 7 8 9 10 Câu 9 Monday 18 392 225 1152 12344321 612 1100 1 Câu 11 12 13 14 15 16 17 18 19 20 5 0 64 1612 996 16 100 18 3 15 Câu 21 22 23 24 25 4 40 17 648 15 Đề 4: Câu 1 2 3 4 5 6 7 8 9 10 14 Thursday 15 243 325 9400 493817284 1755 1300 576 Câu 11 12 13 14 15 16 17 18 19 20 5 6 36 147 984 40 361 108 14 210 Câu 21 22 23 24 25 89 831 51 36 36 6 78 9 10 Đề 5: 2 3 45 Câu 1 57 101 38 125 6 Sat 162 1035 399 20 Câu 11 12 13 14 15 16 17 18 19 20 23 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: [email protected] hoặc [email protected].

13600 200000000 6 5 45 39 1008 20 60 625 24 25 Câu 21 22 23 171 24 6 35 10 24 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: [email protected] hoặc [email protected].