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 timo@daihocthu[.]
BỘ TÀI LIỆU ƠN THI VỊNG LOẠI TIMO 2020 Secondary – Secondary ĐỀ 1: KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com ĐỀ 2: Logical Thinking According to the pattern shown below, what is the number in the blank? 、 、 12 、 20 、 30 、 42 、 、 … There are chickens and rabbits on a farm with 210 legs in total The number of rabbits is twice that of chickens How many rabbit(s) is / are there? Right now, Alice is seven times as old as Peter years later, Alice is three times as old as 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 it take Mary to finish the project alone? There are pieces of red socks, pieces of white socks and 13 pieces of blue socks in a box At least how many piece(s) of sock(s) is / are needed to be drawn to be sure to get pairs of socks with the same colour? Arithmetic Find the value of 642 144 Find the value of 1 1 15 35 399 483 Find the value of 1 1 729 2187 Find the value of 102 112 122 … 192 202 10 Convert 56 into a decimal 99 Number Theory KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com 11 If the 11-digit number 20190513ABB is divisible by 33 and A B , find all possible values of A B 12 How many fraction(s) in lowest terms with a denominator of 120 is / are there? 13 We define the operation symbol ⨁ as follows: a b ab b Find the value a 1 of (2 6) 14 A 3-digit number N leaves a remainder when divided by 7, a remainder when divided by and a remainder when divided by 11 What is the largest possible value of N? 15 Given 20 consecutive natural numbers whose sum is 450 Find the second largest number Geometry 16 How many rectangle(s) is/are there in the figure below? 17 A rectangle is cut into equal squares If the area of the small square is 16cm 2, find the smallest possible difference between the total perimeters of all new squares and the perimeter of the original rectangle 18 Cubes with side length are combined to create layers as the pattern shown below Find the volume of the 4th layer 10 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com 12 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com ĐỀ 3: Logical Thinking According to the pattern shown below, what is the number in the blank? 、 12 、 30 、 56 、 90 、 132 、 、 … Peter goes 20km east, then goes 24km south, then goes 13km west How far is he now from the original position (the answer is given in km)? If abcd badc 7898 , calculate a b c d If x, y and z are all primes and x y y x z , find the minimum value of z There are 10 pieces of red socks, pieces of white socks and 22 pieces of blue socks in the box If you want to ensure that you get pairs of socks with different colour, at least how many piece(s) of sock(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) to 56 5x 14 89 is / are there? Let a and b be real numbers such that a2 b2 51 and a b Calculate a b 10 Factorise x2 2x 63 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 last digit of 52019 12019 32019 13 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com 14 Given x and x 1 , what is the value of x ? x x 15 The difference between two square numbers is 611 Find the sum of all possible values of the larger square? Geometry 16 A cube with side length 18 is divided into smaller cubes with side length Find the increase in the total surface area of the cubes after the division 17 An iron wire is bent in the shape of circles whose radii are both 21 If the wire is now bent in the shape of identical rectangles whose side lengths are integers, what is the maximum area of this rectangle? (Take 22/7) 18 Small cubes with side length are combined as the pattern shown below Find the volume of the 15th layer 19 The longest side of a triangle is 12 unit long Find the largest possible area of the triangle (the answer is in surd form) 20 In a regular polygon, the measure of an interior angle is times the measure of an exterior angle How many sides does the polygon have? Combinatorics 21 There are identical blue flags and identical white flags are put from left to right In how many way(s) can they be arranged? 22 There are five masses which are 1g, 2g, 5g, 7g and 9g respectively If we can combine the masses, 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 flowers in each vase? 14 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com 24 How many 5-digit number(s) can be formed from the digits 1, 2, 4, 7, 8, 9, given that the odd digits cannot be next to each other and none of the digits are repeated ? 25 Billy goes from point A to point B and he can only move up or move right In how many way(s) can Billy get to the destination if he must pass through point O? ĐỀ 4: Logical Thinking Let A and D be two non-zero digits which can form 2-digit numbers DA and AD with the following properties: a) DA can be expressed as a product of and another prime numbers; b) AD can be expressed as a product of and another prime numbers If A D , find the 2-digit number AD There are 340 people in the prom After being checked, the number of couple tickets is twice the number of single tickets How many people go to the prom alone? Right now, Amy is 10 years younger than her brother and their uncle’s age is three times the sum of the ages of Amy and her brother Six years later, their uncle’s age only doubles 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: “Is the number 2956?” John: “Three of the digits are correct The positions of those digits are all wrong.” Peter: “Is the number 7324?” John: “Two of the digits are correct The positions of those digits are all wrong.” Peter: “Is the number 4962?” John: “All the digits are correct The positions of those digits are all wrong.” 15 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com What was the number written by John? There are pieces of white chopsticks, pieces of yellow chopsticks and pieces of brown chopsticks mixed together You want to make sure that you get pairs of chopsticks, of which pairs have the same color At least how many piece(s) of chopstick(s) is/are needed to be taken? Arithmetic Calculate 2222 3333 + 3334 1111 Find the value of 1 1 2 46 2014 2016 2016 2018 Evaluate 66 65 – 65 64 + 64 63 – 63 62 + … + – + What is the value of 199 + 198 + … + 101 + 100 – 99 – 98 – 97 – … – 1? 10 Convert 235 into the decimals 999 Number Theory 11 If the 9-digit number A2018901B is divisible by 72, find the value of A B 12 Find the last digit of A if A = 82 + 182 + 282 + … + 20182 13 How many 3-digit number(s) divisible by and but not divisible by 7? 14 A 4-digit number leaves a remainder when divided by 3, a remainder when divided by and a remainder when divided by What is its largest possible value? 15 Given consecutive odd numbers whose sum is 144, find the largest number among them Geometry 16 How many rectangle(s) is/are there in the figure below? 16 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com 17 Given a square with the area of 225cm2, it is cut into equal rectangles Find the largest possible difference between the total perimeter of new rectangles and the perimeter of the original square 18 Ian uses identical cubes with side length 1cm to create shapes that has the pattern as shown below If the shape built by Ian has 12 layers, what is its volume? 19 A big rectangle is formed by 1001 squares with side length 2cm Find the minimum value of the perimeter in cm 20 The length of one side of a square is 12.56cm If the perimeter of the square is the same as that of a circle, what is the radius of the circle? (Take 3.14) Combinatorics 21 A flight of stairs has 10 steps David can take step or steps each time The 5th step cannot be stepped on as it is destroyed How many way(s) is/are there for David to go up the stairs? 17 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com 22 Amy needs 12 days to finish a task Paul needs 24 days to finish a task How many days they need to finish the task if they it together? 23 At what time between 5pm and 6pm the hour hand and the minute hand of the wall clock overlap? 24 Integral numbers are drawn from to 2020 At least how many number(s) is/are drawn at random to ensure that there are two numbers whose difference is 400? 25 Agnes goes from point A to point B In each step, she can only move up or move right In how many way(s) can Agnes that if she must pass through point O? ĐỀ 5: Logical Thinking 18 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com Given that A and C are non-zero digits, the 2-digit numbers formed by these two digits have the following properties: a) CA is a prime number; b) AC is a prime number; c) A C Find the 2-digit number AC Andy goes west for 5km, then goes north for 12km How far is he now from the original position? How many time(s) in a day does the minute hand of a wall clock coincide with the hour hand? There are 30 problems in a math test The scores of each problem are allocated in the following ways: mark will be given for a correct answer, marks will be given for a blank answer or a wrong answer Find the minimum number of candidate(s) to ensure that candidates will have the same score in the test There are 14 white chopsticks, yellow chopsticks and brown chopsticks mixed together If you want to draw chopsticks randomly and make sure that you get pairs with different colours, at least how many piece(s) of chopstick(s) is/are needed to be taken? (2 pieces of chopsticks with the same colour make a pair) Arithmetic Find the value of x if (4x + 5) + (3x - 19) = Given 1 24 48 252 , find the value of n3 1008n2 2n(2n 2) 1009 How many positive integral solution(s) 𝑦 is/are there if 13𝑦 + 47 < 2349? Let a and b be positive real numbers and satisfy the equations a2 b2 45 and ab 18 Find the value of a b 10 Factorise the polynomial x2 20 x 21 19 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com Number Theory 11 Find the number of positive factor(s) of 1002001 12 Find the value of 13 + 23 + 33 + …+ 1003 13 A is an integer such that A15 is a 17-digit number with as the unit digit Find A 14 Given x and x 1 14 , find the value of x x x 15 What is the remainder when 102018 is divided by 11? Geometry 16 A solid cuboid is formed by merging 2431 cubes with side length 1cm Find the smallest possible value of the total surface area of the cuboid 17 Given a right triangle, the length of a leg is 36cm and the length of the hypotenuse is 60cm Find the area of the triangle in cm2 18 Small cubes with side length 1cm are combined as the pattern shown below If the given solid has 11 layers with such pattern, find its surface area 19 The side length of an equilateral triangle is 6, find the area of the triangle 20 Find the area of the shaded region in the figure (Take 22 ) 20 KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020 Thailand Internationl Mathematical Olympiad - TIMO 2020 Đơn vị đồng tổ chức Việt Nam: Trường Đại học Thủ Đô Hà Nội FERMAT Education Email: timo@daihocthudo.edu.vn timo.fermat@gmail.com ... + 3334 1111 Find the value of 1 1 2 46 2014 2016 2016 20 18 Evaluate 66 65 – 65 64 + 64 63 – 63 62 + … + – + What is the value of 199 + 1 98 + … + 101 + 100 – 99 ... – 98 – 97 – … – 1? 10 Convert 235 into the decimals 99 9 Number Theory 11 If the 9- digit number A20 1 89 01B is divisible by 72, find the value of A B 12 Find the last digit of A if A = 82 + 182 ... 35 399 483 Find the value of 1 1 7 29 2 187 Find the value of 102 112 122 … 192 202 10 Convert 56 into a decimal 99 Number Theory KỲ THI OLYMPIC TOÁN HỌC QUỐC TẾ - TIMO 2020