Bộ đề ôn luyện kỳ thi Olympic Toán Quốc Tế HKIMO 2021 Khối 8, 9 dành cho các học sinh làm quen và ôn luyện về các dạng toán logic, tham gia thi các kỳ thi toán Tiếng anh Olympic Quốc Tế, các trường dạy theo chương trình học châu âu. Yêu cầu đã phải có vốn tiếng anh cơ bản
BỘ ĐỀ ƠN LUYỆN KỲ THI OLYMPIC TỐN HỌC HKIMO 2020 KHỐI 8, Đề 1: Logical Thinking Given B and C are two non-zero digits and the 3-digit numbers formed by these two digits have the following properties: BCB is divisible by 7; CBB is divisible by 9; BBC has an odd number of factors Find the 3-digit number CCB Andy goes west for 31km, then goes north for 18km and goes east for 7km How far is he now from the original position? If abcd dcba 16225 , calculate a b c d There are 28 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 given for a blank answer or wrong answer Find the minimum number of candidate(s) to ensure that candidates will have the same scores in the competition There are pieces of white chopsticks, 10 pieces of yellow chopsticks and pieces of brown chopsticks mixed together Close your eyes If you want to get a pair of chopsticks that is not white, at least how many piece(s) of chopstick(s) is / are needed to be taken? Algebra Find the value of y if | x y | | x y | 2 Factorize x y 4y 2x How many integral solution(s) is / are there for x if 33 5x 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 1 Find the value of 1 16 4072324 a 2 10 Given that a is a real number and a # If 1 a 1 , find the value of a Number Theory 8012 11 Find the remainder when 2018 is divided by 12 x is a positive integer such that the remainder of x divided by is If x>23, what is the minimal value of x? 13 Find the last digit of A if A 14 Find the sum of all positive factors of 2040 15 If (x-2) is a factor of polynomial 4x 77x c 2017 42018 Find the value of c Geometry 16 In the figure below, D is a point on AB E is on the extension of AC DE DE A D AC intersects BC at F If EF , DB Find the value of A E 17 For three points on a coordinate plane A (7, 8) , B(2, 5) and C(3, 6), find the area of the triangle formed by using those three points as vertices 18 An iron wire is bent into a circle which radius is 21 If the wire is now bent into a rectangle, what is the maximum value of area of it? ( T ake 22 ) 12n 6 Find n? An interior angle of a n-sided polygon is o 19 20 How many diagonal(s) does a convex decagon(10-sided polygon) have? 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 Find the number of the combination(s) arranging boys and girls in a row 22 identical Mathematics books and identical English books are put on the bookshelf How many way(s) of arrangement is/are there? 23 A fair 6-face die is thrown times Find the probability that the product of numbers obtained is 12 24 Amy draws all triangles which perimeter is 26cm and length of sides are integers How many different types of triangle(s) does she draw? (triangles with sides (4,5,6) and (4,6,5) are regarded as the same type of triangle) 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 Given A and B are two non-zero digits and the 3-digit numbers formed by these two digits have the following properties: BBA is divisible by 17; A A B is divisible by 7; A BA has an odd number of factors Find the 3-digit number BA B Given that the mean, median, range and the only mode of 120 integers are also 70 If A is the largest integer among those 120 integers, find the maximum value of A Andy goes west for 13km, then goes north for 26km, goes east for 61km and goes south for 46km How far is he now from the original position? There are 28 problems in a mathematics competition The scores of each problem are allocated in the following ways: marks will be given for a correct answer, mark will be deducted from a wrong answer and marks will be given for a blank answer Find the minimum number of candidate(s) to ensure that candidates will have the same scores in the competition here are 10 pieces of white chopsticks, pieces of yellow chopsticks and pieces of brown chopsticks mixed together Close your eyes If you want to get pairs of chopsticks that are not yellow, at least how many piece(s) of chopstick(s) is / are needed to be taken? Algebra , Let and be the roots of the equation x 2018x If and re the roots of the equation x Sx find the value of S 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 Factorize xy yz zx x 2y y 2z z 2x Simplify One integral roots of equation x 6x a 5x is Given that a>0 find the 30 12 other integral root 10 Find the greatest value of 901 x 40x 401 Number Theory 11 Given that 81A 02B is a 6-digit number which is divisible by 36 and B>A, find the value of A+B 12 Find the remainder for 122122 divided by 13 How many positive integer(s) x that is less than 8012 is / are there so that X 2018 is an integer? 14 Now is June Which month is it after 1515 months? 15 It is given that x and y are real numbers such that x y and x y 10 , find the value of x y Geometry 16 Find the area enclosed by the x-axis, y-axis and the straight line 2y 3x 12 17 Find the maximum value of s inx cos x 18 Given that an interior angle of an n-sided regular polygon is 17 times an exterior angle Find the value of n 19 A triangle in the figure below has sides with lengths 8, 10 and 12 Find the value of the radius of inscribed circle of that triangle 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 Find the shortest distance from the straight line y x to 2x 3y Combinatorics 21 Find the number of the combination(s) arranging girls and boys in a circle 22 There are identical Mathematics books, identical Chinese books and identical English books, how many different arrangement(s) is/are there? 23 A fair 6-face die is thrown times Find the probability that the sum of numbers obtained is a multiple of 24 When does the hour hand and minute hand form a straight line in between o’clock and o’clock? 25 Given a, b, c is a set of integers and all of them are greater than Find the number of solution set(s) of a b c 19 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 If a polygon has its sum of interior angles smaller then 2020 0, what is the maximum number of sides of the polygon? Let x,y and z be non-negative numbers Suppose x+y=10 and y+z=8 Let S=x+z What is the sum of the maximum and the minimum value of S? Gautam is brother of Bala Divya is sister of Gautam Matthew is brother of Tina Tina is daughter of Bala Who is the uncle of Matthew? A particular month has Saturdays The first and the last day of the month are not Saturdays What day is the first fay of the month? 3 of the workers are women, and of the people are married What is the maximum possible number of unmarried women in the office Algebra 448 people work at an office Suppose a 0, b and b c bc 2020 Find the value of a b ab If | x | x y and | y | y x , find the value of x y Suppose x y Find the value of x4 xy3 x3 y 3x2 y 3xy y Factorize x4 y y x y 10 Given that x and y are positive integers such that 56 x y 59 and 0.9 0.91 , find the value of y x Number Theory 11 Find the last two digits of 2020 20202 20203 20202020 12 The sum of m 1 1 is in its lowest 3 3 5 13 14 15 14 15 16 n terms Find the value of m n 13 N pieces of candy are made and packed into boxes, with each box containing 45 pieces If N is a non-zero perfect cube and 45 is one of its factors, what is the least possible number of boxes that can be packed? 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 How many zeroes does the number 50 49 48 3 1 end with? 15 When 15 is added to a number x, it becomes a square number When 74 is subtracted from x, the result is again a square number Find the number x Geometry 16 The figure ABCDEF is a regular hexagon Evalute the quotient Area of hexagon ABCDEF Area of triangle ACD 17 Five identical rectangles of area cm2 are arranged into a large rectangle as shown below Find the perimeter of the large rectangle Question 17 18 A television set has a 30-inch screen size, i.e the length of the diagonal of the screen is 30 inches The aspect ratio of the screen is 4:3 Find the height of the screen (in inches) Leave your answer to the nearest whole number if necessary (Note: The aspect ratio of the screen refers to the ratio of the width of the screen to its height) 19 Let AA’ and BB’ be two line segments which are perpendicular to A’B’ The length of AA’, BB’ and A’B’ are 680, 2000 and 2010 respectively Find the minimal length of AX+XB Where X is a point between A’ and B’ Question 19 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 In triangle ABC, DC=2BD, ABC 450 and ADC 60o Find ACB in degrees Question 20 Combinatorics 21 A painter is to paint the five circles of the Olympic flag He cannot remember the colours to use for any of the circles, but he knows they should all be different He has eight colours of paint available In how many ways can he paint the circles on the flag? 22 Using the digits 1, 2, 3, only once to form a 4-digit number, how many of them are divisible by 11? 23 Suppose a number is chosen at random from the set {0,1,2,3,…,202} What is the probability that the number is a perfect cube? 24 2020 students are taking a test which comprises ten true or false questions Find the minimum number of answer scripts required to guarantee two scripts with identical answers? 25 Given that n is an odd integer less than 1000 and the product of all its digits is 252 How many such integers 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 4: Logical Thinking For the Maths Olympiad, each school can send a maximum of students Suppose 316 students took part in the Maths Olympiad What is the smallest number of schools that could have sent students to the Olympiad? We use n! to mean the product of all integers from to n In the range 22 x 62 , how many number are there such that ( x 1)! is not divisible by x? A car travels from A to B at a speed of 40 km/h then returns, using the same road, from B to A at a speed of 60 km/h What is the average speed for the round trip? The numbers x, y, z and w have an average equal to 25 The average of x, y and z is equal to 27 Find w A rectangular garden in Mrs Dorothy’s house has a length of 100 meters and a width of 50 meters A square swimming pool is to be constructed inside the garden Find the length of one side of the swimming pool if the remaining area (not occupied by the pool) is equal to the area of the rectangular garden Algebra Let and , 0 be the roots of the equation x 20 x 20 If and are roots of the equation 20 x Sx , find the value of S Factorize ( xy 4)2 (2 x y)2 Simplify 23 15 Find the value of a such that the two equations x ax and x x a have one common real root 10 Find the greatest value of 2020 x 10 x 30 Number Theory 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 A Monday B C D E F Tuesday G Wednesday Friday Saturday Thursday Sunday Among them only Thursday satisfies the condition that “three statements are correct” 6 a 2 a a a a 2 a 3 a 3a 2 a 3 a 2 a a a a a a (rejected) a2 4 a 3 a a a a a a a a 8 (rejected) 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 a 4 a a a a a a 3a 16 16 a 3 The sum of all possible value is ( 16 ) 6 5775 1 10 12 14 10 (i )(i 2)(i 4) i 1 10 (i 6i 8i ) i 1 10 10 10 i 1 i 1 i 6i 8i i 1 55 10 11 21 55 5775 0.087912 88 87912 91 1001 999999 0.087912 91 13 x y 2 x 9, y x y 13 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 10 47 x2 x x 7 x x 7 x x 49 x x 47 x 11 Since and the number should be an odd number, the unit digit should be 12 The remainder of 2015 is The remainder of 20152 is So we conclude that the pattern of the remainder of 20152015 will be 8,1,8,1, Since 2015 is an odd number, the remainder will be 13 x 11 y 13 11 y 13 x yx x x yx 2x y yx 2x y 1 yx 2x y y x 3x y x 2, y 58 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 4,062,240 1 x y 2015 xy 2015 x 2015 y 4060225 4060225 ( x 2015)( y 2015) 4060225 The greatest value of x 4060225 2015 4062240 15 61 xy x y 41 x y 1 y 41 x 1 y 1 42 x y xy 330 xy x y 330 Only x and y satisfy the two equations, so x2 y 52 62 25 36 61 Geometry 16 192 Consider the area of the rectangle should be 2015, the perimeter of that rectangle will be minimum if the sum of the length and the width is minimum Consider 2015 1 2015 403 13153 31 65 The minimum perimeter is 31 65 192 17 (9-3)x(7-2) – ( 0,5 x (6-3) x (7-3) + 0,5 x (3-2) x (6-3) + 0,5 x (9-6) x (7-2) ) = 15 18 130 Add straight lines one the graph, as shown in the figure below Consider the area of the square we have 4a (3a a ) 1600 10a 1600 a 160 59 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 AG (3a) (2a) By Pythagoras Theorem we have 13a AG 130 19 CE DG AB 1 ED GA BC CE 1 1 ED CE 3 ED CD DE 88 32 32 8 1 8 S ACD SGCD SGCD 20 Let h be the length of the height of ABC corresponding to BC We have h2 h2 h h AB AB Combinatorics 60 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 1,511 Following the question the answer can be found by counting numbers in the form of 3n k (n is an even number and k is not multiple of 3) [2015] [ 2015 2015 2015 2015 2015 2015 ] [ ] [ ] [ ] [ ] [ ] 2015 671 223 74 24 1511 3 3 3 22 270,270 There are C1015 3003 methods of choosing teammates and for each combination there are 10 choices for captain and choices for vice-captain So the answer is 300310 270270 23 252 Number of solutions for equation x1 x2 Inequality x1 x2 x5 10 is C49 126 x5 10 is equivalent to the equation x1 x2 x5 x6 10 where x6 is also a positive integer So the number of solutions is C59 126 As a result there are 126 126 252 solutions 24 The number of possible routes of passing is shown as the table below Star After 1st Pass After 2nd Pass After 3rd Pass After 4th Pass After 5th Pass t A A B C D E A D A D 25 32 5-digit number: 1 3 1 12 61 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 4-digit number: 1 3 12 3-digit number: 1 2-digit number: 1 There are 32 numbers in total Đề 8: 89 To make sure 25 balls with same colours are drawn, we need to consider the worst possible outcome that all the green and yellow balls and also 24 blue balls and 24 red balls are drawn So 20 20 24 24 1 89 pens should be drawn to make sure she draw 25 balls with same colour 513 Each number is the sum of a cubic number and 13 23 33 28 43 65 53 126 63 217 73 344 83 513 1 2 50 51 52 1 2 2 2 2 69 50 51 52 69 69 69 69 1 40 2 2 40 50 50 51 52 69 69 1.25 1.725 2 2 50 51 52 69 The integral part of is 2 2 50 51 52 69 69 Team F The arrangement of competition is shown in the following table 62 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 Day Contest Contest Contest A vs B C vs E D vs F A vs F B vs E C vs D A vs E B vs D C vs F A vs D B vs C E vs F A vs C B vs F D vs E Monday Converting their statements, we will have the following table: A B C D E F G Monday Tuesday Wednesday Friday Saturday Sunday Thursday Among them only Monday satisfies the condition that “only one statement is correct” Algebra 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 a 2 a a a a 4a 14 4(2) 14 6 3 a 2 a a a a a a a a 2a 10 2(3) 10 4 4 a 3 a a a a a a a a 4 5 a 4 a a a a a a a a 2a 2(4) 4 a 5 a a a a a a a a 4a 14 4(5) 14 6 The minimum value of a a a a is Denote x 13 12 13 12 , x 13 12 x x 12 x 13 ( x 13)( x 1) x 1 64 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 x (2 x ) 32 32 8 x x x x 2x x 23 x3 13 x y 54 y y 40 x y 14 y y 20 y5 y 4 (rejected) y xy 30 x4 x y 13 10 322 x2 x x 7 x x 7 x 3 x 3x 343 x x x3 343 3(7) x 322 Number Theory 11 15 x2 x 16 11 x2 x2 4x x2 16 15 x2 ( x 2)2 ( x ) 15 x The expression attains its minimum when x 65 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 12 12 112 a b 12 28 a 2a b b a b 28 2 13 x x 58 x x 205 ( x 3) ( x 3) 142 Consider the value of ( x 3)2 ( x 3)2 142 is equal to the sum of distance between (3,7), (x, 0) and ( 3 , 14 ), (x , 0), the minimum value attains when (3 , 7), (x , 0) and ( 3 , 14 ) lie on a straight line Hence 14 111 14 , x 33 3 x x (mod17) x (mod17) 2x (mod 7) x (mod 7) x (mod17) x 17 p 17 p (mod 7) p (mod 7) p (mod 7) p 7k x 17(7 k 6) 119k 111 The least positive integral solution of x is 111 15 24 As a b and a c are odd numbers, and the difference between two odd numbers should be even, we have a Without generosity, let b c (2 b)(2 c) 75 (b 2)(c 2) 52 (b 2, c 2) (1, 75) / (3, 25) / (5, 15) (b, c) (3, 77) / (5, 27) / (7,17) Only a number pair (7,17) satisfies the requirement, hence b c 24 66 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 81 FBO FCO 180 90 BFO ~ OFC BF OF OF FC BF FC OF AD ( ) =81 17 (5 2)a Rotate ABP to CBP ' around B for 90 clockwise as in the figure below Then BP ' BP 2a CP ' AP a Connect PP ' PBP ' PBC CBP ' PBC ABP 90 Therefore PP ' PB P ' B (2a)2 (2a) 2a Since PP '2 P ' C 8a a 9a PC , PP ' C 90 APB CP ' B BP ' P PBP ' 90 45 135 Area of the square BC a 4a 2(a)(2a) cos135 (5 2)a 67 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 D A P B C 18 4030 FE / / BC and FD / / AC , ABC AFE BFD BAC ABC , AFE and FBD are all isosceles triangles and hence CE DF BD Perimeter of DCEF 2(CE DC ) 2( BD DC ) BC 4030 19 37 CAM ABC CAM 74 37 20 3 Let h be the length of the height of ABC corresponding to BC 2 We have h h h 3 tan 60 tan 45 BC BC 3(1 3) 3 The area of ABC Combinatorics 68 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 21 231 Every term in the expansion of (a b c)20 is in the form of a n b n c n where n1 , n2 , n3 are all non-negative and n1 n2 n3 20 Number of unlike terms is C222 231 22 If one of them does not sit on his own chair, there should be another student not sitting on his chair So there does not exist a case that only one boy does not sit on his own chair 23 252 a1 x1 a2 x2 Denote a3 x3 , a4 x4 a5 x5 we have x1 x2 x1 x2 x5 20 a1 a2 x5 20 a1 a2 a5 10 and a5 10 Number of solutions for equation a1 a2 Inequality x1 x2 a5 10 is C49 126 x5 20 is equivalent to the equation a1 a2 a5 a6 10 where a6 is also a positive integer So the number of solutions is C59 126 As a result there are 126 126 252 solutions 24 Separate the numbers into groups: {1 , , , 8} {3 , 6} {5 , 10} {7} {9} If any numbers are in the same group, then one of them is a multiple of another By the pigeonhole principle, if numbers are chosen from them, then there are of them in the same group 69 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 25 729 For every ball it has choices, so the number of ways for balls to be put into different boxes is 36 729 70 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 ... 54 99 66 6534 99 9 666 665334 99 99 6666 66653334 99 999 66666 6666533334 15 17 Let the square number be n p 64 n p n 64 p (n 8) (n 8) n ? ?8 1 n? ?9 p 17 16 700 Let... 30 26 57 11 -7 (x-y-1)(x+y+3) 12 20 19 4036 10 -2 11 12 31 13 14 6 480 15 1 38 16 17 18 19 20 21 22 23 24 25 37 1 0 89 12 35 40320 56 14 32 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - HKIMO 2020 Đơn vị đồng... Email: hkimo@ daihocthudo.edu.vn hkimo. fe@gmail.com Đề 2 84 8 1 38 52 1 69 13 20 18 x y y z z x 10 11 12 13 14 15 16 33 -2 90 1 7 78 September - 18 12 17 18 36 19 20 20 13 13 21 720 22 693 0