Đáp án đề thi EIMC toán học QT IMC 2016 là lời giải của 2 đề thi dành cho các học sinh lứa tuổi lớp 6,7 (Elementary Mathematics International Contest) tham gia Kỳ thi toán học quốc tế IMC – International Mathematical Competition lần thứ 17 diễn ra từ ngày 14 tới ngày 20 tháng 8 tại Thái Lan.


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Individual Contest Time limit: 90 minutes English Version Instructions:  Do not turn to the first page until you are told to do so.  Write down your name, your contestant number and your team's name on the answer sheet.  Write down all answers on the answer sheet. Only Arabic NUMERICAL answers are needed.  Answer all 15 problems. Each problem is worth 10 points and the total is 150 points. For problems involving more than one answer, full credit will be given only if ALL answers are correct, no partial credit will be given. There is no penalty for a wrong answer.  Diagrams shown may not be drawn to scale.  No calculator, calculating device or protractor is allowed.  Answer the problems with pencil, blue or black ball pen.  All papers shall be collected at the end of this test. Elementary Mathematics International Contest 1. ABCDE is a regular pentagon of side length 1 m. There are 5, 15, 14, 9 and 17 students at the vertices A, B, C, D and E respectively. The teacher wants the same number of students at each vertex, so some of the students have to walk to other vertices. They may only walk along the sides. What is the minimum total length, in m, the students have to walk? 2. A, B and C run a 200-m race in constant speeds. When A finishes the race, B is 40 m behind A and C is 10 m behind B. When B finishes, C still has to run another 2 seconds. How many seconds does B still have to run when A finishes? 3. With each vertex of a 1 cm by 1 cm square as centre, circles of radius 1 cm are drawn, as shown in the diagram below. How much larger, in cm2 , is the area of the shaded region than the area of a circle of radius 1 cm? (Take π = 3.14) 4. How many multiples of 18 are there between 8142016 and 8202016? 5. In a basketball game, a foul shot is worth 1 point, a field shot is worth 2 points and a long-range shot is worth 3 points. Stephen makes 8 foul shots and 14 others. If he had made twice as many field shots and half as many long-range shots, he would have scored 7 extra points. How many points has Stephen actually scored? 6. John’s running speed is twice his walking speed. Both are constant. On his way to school one day, John walks for twice as long as he runs, and the trip takes 30 minutes. The next day, he runs for twice as long as he walks. How many minutes does the same trip take on the second day? 7. Jimmy has some peanuts. On the first day, he eats 13 peanuts in the morning and one tenth of the rest in the afternoon. On the second day, he eats 16 peanuts in the morning and one tenth of the rest in the afternoon. If he has eaten the same number of peanuts on both days, how many peanuts will he have left? 8. The sum of 49 different positive integers is 2016. What is the minimum number of these integers which are odd? 9. The sum of 25 positive integers is 2016. Find the maximum possible value of their greatest common divisor. 10. ABCD is the rectangle where AB = 12 cm and BC = 5 cm. E is a point on the opposite side of AB to C, as shown in the diagram below. If AE = BE and the area of triangle AEB is 36 cm2 , find the area, in cm2 , of triangle AEC. 11. Anna starts writing down all the prime numbers in order, 235711…. She stops after she has written down ten prime numbers. She now removes 7 of the digits, and treats what is left as a 9-digit number. What is the maximum value of this number? 12. Three two-digit numbers are such that the sum of any two is formed of the same digits as the third number but in reverse order. Find the sum of all three numbers. 13. The sum of two four-digit numbers is a five-digit number. If each of these three numbers reads the same in both directions, how many different four-digit numbers can appear in such an addition? 14. When 2016 is divided by 3, 5 and 11, the respective remainders are 0, 1 and 3. Find the smallest number with the same properties that can be made from the digits 2, 0, 1 and 6, using each at most once. 15. Each student writes down six positive integers, not necessarily distinct, such that their product is less than or equal to their sum, and their sum is less than or equal to 12. If no two students write down the same six numbers, at most how many students are there? A E D C B


Elementary Mathematics International Contest TEAM CONTEST Time：60 minutes English Version For Juries Use Only No. 1 2 3 4 5 6 7 8 9 10 Total Sign by Jury Score Score Instructions:  Do not turn to the first page until you are told to do so.  Remember to write down your team name in the space indicated on every page.  There are 10 problems in the Team Contest, arranged in increasing order of difficulty. Each question is printed on a separate sheet of paper. Each problem is worth 40 points. For Problems 1, 3, 5, 7 and 9, only answers are required. Partial credits will not be given. For Problems 2, 4, 6, 8 and 10, full solutions are required. Partial credits may be given.  The four team members are allowed 10 minutes to discuss and distribute the first 8 problems among themselves. Each student must attempt at least one problem. Each will then have 35 minutes to write the solutions of their allotted problem independently with no further discussion or exchange of problems. The four team members are allowed 25 minutes to solve the last 2 problems together.  No calculator or calculating device or electronic devices are allowed.  Answer must be in pencil or in blue or black ball point pen.  All papers shall be collected at the end of this test. TEAM CONTEST 17th August, 2016, Chiang Mai, Thailand Team： Score： Elementary Mathematics International Contest 1. Choose nine different ones of 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. Use each of them exactly once to form three equations, using each of addition, subtraction, multiplication and division at most once. What is the smallest number that we can leave out? Answer: TEAM CONTEST 17th August, 2016, Chiang Mai, Thailand Team： Score： Elementary Mathematics International Contest 2. The squares of a 3 by 3 table are labelled 1, 2, …, 9 as shown. In how many ways can we shade five of the squares so that no row or column is completely shaded? 1 2 3 4 5 6 7 8 9 Answer: ways TEAM CONTEST 17th August, 2016, Chiang Mai, Thailand Team： Score： Elementary Mathematics International Contest 3. The diagram shows an 8 by 8 board. An ant visits each of the 64 squares once and only once. It crawls from one to another of two squares sharing at least one corner. The order in which the squares are visited is marked with numbers, starting from 1 and ending at 64. Some of the marked numbers have been erased. Restore these erased numbers. 5 26 25 10 29 23 2 31 37 1 12 19 21 38 34 13 17 49 33 59 16 50 47 40 64 46 42 55 53 52 45 43 Answer: 5 26 25 10 29 23 2 31 37 1 12 19 21 38 34 13 17 49 33 59 16 50 47 40 64 46 42 55 53 52 45 43 TEAM CONTEST 17th August, 2016, Chiang Mai, Thailand Team： Score： Elementary Mathematics International Contest 4. In triangle ABC, AB = 7 cm and AC = 9 cm. D is a point on AB such that BD = 3 cm. E is a point on AC such that the area of the quadrilateral BCED is 5 7 of the area of triangle ABC. Find the length, in cm, of CE. Answer: cm A E D C B TEAM CONTEST 17th August, 2016, Chiang Mai, Thailand Team： Score： Elementary Mathematics International Contest 5. How many positive integers less than 100 are there such that the product of all positive divisors of such a number is equal to the square of the number? Answer: TEAM CONTEST 17th August, 2016, Chiang Mai, Thailand Team： Score： Elementary Mathematics International Contest 6. A hat is put on the head of each of 33 children. Each hat is red, white or blue. Each can see the hats of all other children except his or her own. Willem sees three times as many red hats as blue hats. Maxima sees twice as many white hats as blue hats. What is the colour of Maxima’s hat? Answer: TEAM CONTEST 17th August, 2016, Chiang Mai, Thailand Team： Score： Elementary Mathematics International Contest 7. Mary has a three-digit number. The first two digits are the same but different from the third digit. Myra has a one-digit number. It is the same as the last digit of Mary’s number. How many different four-digit numbers can be the product of Mary’s and Myra’s numbers? Answer: TEAM CONTEST 17th August, 2016, Chiang Mai, Thailand Team： Score： Elementary Mathematics International Contest 8. A 4 m by 4 m window on a wall is to be boarded up with eight identical 1 m by 2 m or 2 m by 1m wooden planks. In how many different ways can this be done? Two ways resulting in the same final diagram are not considered different. Answer: TEAM CONTEST 17th August, 2016, Chiang Mai, Thailand Team： Score： Elementary Mathematics International Contest 9. The 6 by 6 table in the diagram below is divided into 17 regions, each containing a number. Each of the 36 squares contains one of the numbers 1, 2, 3, 4, 5 and 6. All six numbers appear in every row and every column. The number in a white region is the number in the only square of the region. The number in a yellow region is the difference when the number in one of the squares is subtracted from the number in the other square. The number in a green region is the quotient when the number in one of the squares is divided by the number in the other square. The number in a red region is the sum of the numbers in all the squares of the region. The number in a blue region is the product of the numbers in all the squares of the region. Fill in the 36 numbers. 3 24 5 5 60 3 12 1 1 150 24 3 4 10 7 11 2 Answer: TEAM CONTEST 17th August, 2016, Chiang Mai, Thailand Team： Score： Elementary Mathematics International Contest 10. A polygon is said to be convex if each of its interior angles is less than 180° . What is the maximum number of sides of a convex polygon which can be dissected into squares and equilateral triangles of equal side lengths? Justify your answer. Answer:

2016 EMIC Answers Individual 1. 10 2. 6 3. 4 4. 3333 5. 42 6. 24 7. 99 8. 6 9. 72 10. 51 11. 737192329 12. 99 13. 72 14. 201 15. 14 Team 1. 4 2. 45 4. 1 4 2 5. 33 6. Red 7. 53 3. 8. 36 9. 10. 12 6 5 8 9 28 26 25 24 4 7 10 29 30 27 23 36 2 3 11 20 31 22 37 35 1 12 18 19 21 32 38 34 13 60 61 17 49 48 33 39 59 14 16 62 50 47 40 41 58 15 64 63 51 46 44 42 57 56 55 54 53 52 45 43 2 3 4 5 1 6 6 2 5 1 3 4 4 1 2 6 5 3 3 4 1 2 6 5 5 6 3 4 2 1 1 5 6 3 4 2