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20202021 AUSTRALIAN MATHEMATICS COMPETITION Instructions and Information General 1. Do not open the booklet until told to do so by your teacher. 2. NO calculators, maths stencils, mobile phones or other calculating aids are  permitted. Scribbling paper, graph paper, ruler and compasses are permitted,  but are not essential. 3. Diagrams are NOT drawn to scale. They are intended only as aids. 4. There are 25 multiple-choice questions, each requiring a single answer, and  5 questions that require a whole number answer between 0 and 999. The  questions generally get harder as you work through the paper. There is no  penalty for an incorrect response. 5. This is a competition not a test; do not expect to answer all questions. You are  only competing against your own year in your own country/Australian state so  dif  erent years doing the same paper are not compared. 6. Read the instructions on the answer sheet carefully. Ensure your name, school  name and school year are entered. It is your responsibility to correctly code  your answer sheet. 7. When your teacher gives the signal, begin working on the problems.  The answer sheet 1. Use only lead pencil. 2. Record your answers on the reverse of the answer sheet (not on the question  paper) by FULLY colouring the circle matching your answer. 3. Your answer sheet will be scanned. The optical scanner will attempt to read  all markings even if they are in the wrong places, so please be careful not to  doodle or write anything extra on the answer sheet. If you want to change an  answer or remove any marks, use a plastic eraser and be sure to remove all  marks and smudges. Integrity of the competition The AMT reserves the right to re-examine students before deciding whether  to grant of  cial status to their score. Reminder You may sit this competition once, in one division only, or risk no score. Copyright © 2021 Australian Mathematics Trust | ACN 083 950 341 DAT E TIME ALLOWED 75 minutes Junior Ye a r s 7– 8 (AUSTRALIAN  SCHOOL YEARS) 4–6 August

Junior Division Questions 1 to 10, 3 marks each 1. 2021 −1202 = (A) 719 (B) 723 (C) 819 (D) 823 (E) 3223 2. What is the pe\bimete\b of this figu\be? (A) 28 units (B) 26 units (C) 24 units (D) 20 units (E) 21 units 1 unit 3. The a\bea of this t\biangle is (A) 10 cm 2 (B) 12 cm 2 (C) 12 .5 cm 2 (D) 15 cm 2 (E) 16 cm 2 6 cm 4 cm \b. On the numbe\b line below, the f\baction 3 8 lies between P Q R S T U 0 1 2 1 (A) Pand Q (B)Qand R (C)Rand S (D)Sand T (E)Tand U 5. Which of the following is closest to 2021? (A) 202 ×100 (B) 22 ×1000 (C) 20. 2× 100 (D) 10 ×20. 2 (E) 100 ×2.2 J2 6. In the diagram, ABis parallel to EF and DE is parallel to BC. What is the value of x\b (A) 43 (B) 47 (C) 133 (D) 135 (E) 137 D A E C B E F 43 ◦ x◦ 7. Mister Meow attempted the calculation 5 ×2 + 4, but accidentally swapped the multiplication and addition symbols. His answer was (A) too low by 2 (B) too low by 1(C) still correct (D) too high by 1 (E) too high by 2 8. Dad puts a cake in the oven at 11:49 am. The recipe says to bake it for 75 minutes. When should the cake come out of the oven\b (A) 1:04 pm (B) 12:34 pm (C) 1:54 pm (D) 1:19 pm (E) 12:04 pm 9. Damon made up a joke and sent it as a text message to three people in his class. These three each sent it to three other people in the class. No-one receiving the joke had seen it before. Including Damon, how many people now know the joke\b (A) 9 (B) 11 (C) 13 (D) 15 (E) 16 10. I am shuffling a deck of cards but I accidentally drop a card on the ground every now and then. After a while, I notice that I have dropped five cards. From above, the five cards look like one of the following pictures. Which picture could it be\b (A) ♠ ♠ ♠ ♠ ♠ ♠ ♠ 7♠ 7♠ ♥ ♥ 2♥ 2♥ ♠ ♠ ♠ ♠ 4♠ 4♠ 1 1 1 1 1 1 1 1 1 9♣ 9♣ ♦ ♦ ♦ ♦ ♦ ♦ 6♦ 6♦ ♥ ♥ 2♥ 2♥ (B) ♠ ♠ ♠ ♠ ♠ ♠ ♠ 7♠ 7♠ ♥ ♥ 2♥ 2♥ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 9♣ 9♣ ♦ ♦ ♦ ♦ 4♠ 4♠ ♦ ♦ ♦ ♦ ♦ ♦ 6♦ 6♦ ♠ ♠ ♠ ♠ ♠ ♠ ♠ 7♠ 7♠ ♥ ♥ 2♥ 2♥ (C) ♠ ♠ ♠ ♠ ♠ ♠ ♠ 7♠ 7♠ 9♣ 9♣ ♦ ♦ ♦ ♦ 4♠ 4♠ ♦ ♦ ♦ ♦ ♦ ♦ 6♦ 6♦ ♥ ♥ 2♥ 2♥ ♠ ♠ ♠ ♠ ♠ ♠ ♠ 7♠ 7♠ ♥ ♥ 2♥ 2♥ (D) ♠ ♠ ♠ ♠ 4♠ 4♠ ♦ ♦ ♦ ♦ ♦ ♦ ♦ 7♠ 7♠ 9♣ 9♣ ♥ ♥ 2♥ 2♥ ♦ ♦ ♦ ♦ ♦ ♦ 6♦ 6♦ ♠ ♠ ♠ ♠ ♠ ♠ ♠ 7♠ 7♠ (E) ♠ ♠ ♠ ♠ 4♠ 4♠ ♦ ♦ ♦ ♦ ♦ ♦ ♦ 7♠ 7♠ 9♣ 9♣ ♥ ♥ 2♥ 2♥ ♦ ♦ ♦ ♦ ♦ ♦ 6♦ 6♦ 2021 AUSTRALIAN MATHEMATICS COMPETITION JUNIOR

J2 6.In the diagram, ABis parallel to EF and DE is parallel to BC. What is the value of x\b (A) 43 (B) 47 (C) 133 (D) 135 (E) 137 D A E C B E F 43 ◦ x◦ 7. Mister Meow attempted the calculation 5 ×2 + 4, but accidentally swapped the multiplication and addition symbols. His answer was (A) too low by 2 (B) too low by 1(C) still correct (D) too high by 1 (E) too high by 2 8.Dad puts a cake in the oven at 11:49 am. The recipe says to bake it for 75 minutes. When should the cake come out of the oven\b (A) 1:04 pm (B) 12:34 pm (C) 1:54 pm (D) 1:19 pm (E) 12:04 pm 9. Damon made up a joke and sent it as a text message to three people in his class. These three each sent it to three other people in the class. No-one receiving the joke had seen it before. Including Damon, how many people now know the joke\b (A) 9 (B) 11 (C) 13 (D) 15 (E) 16 10.I am shuffling a deck of cards but I accidentally drop a card on the ground every now and then. After a while, I notice that I have dropped five cards. From above, the five cards look like one of the following pictures. Which picture could it be\b (A) ♠ ♠♠♠ ♠♠ ♠ 7♠ 7♠ ♥ ♥ 2♥ 2♥ ♠ ♠ ♠ ♠ 4♠ 4♠ ♣ ♣ ♣ ♣♣♣ ♣ ♣ ♣ 9♣ 9♣ ♦ ♦ ♦ ♦♦ ♦ 6♦ 6♦ ♥ ♥ 2♥ 2♥ (B) ♠ ♠♠♠ ♠ ♠ ♠ 7♠ 7♠ ♥ ♥ 2♥ 2♥ ♣ ♣ ♣ ♣♣♣ ♣ ♣ ♣ 9♣ 9♣ ♠ ♠ ♠ ♠ 4♠ 4♠ ♦ ♦ ♦ ♦ ♦ ♦ 6♦ 6♦ ♠ ♠ ♠ ♠ ♠ ♠ ♠ 7♠ 7♠ ♥ ♥ 2♥ 2♥ (C) ♠ ♠♠♠ ♠♠ ♠ 7♠ 7♠ ♣ ♣ ♣ ♣♣♣ ♣ ♣ ♣ 9♣ 9♣ ♠ ♠ ♠ ♠ 4♠ 4♠ ♦ ♦ ♦ ♦ ♦ ♦ 6♦ 6♦ ♥ ♥ 2♥ 2♥ ♠ ♠♠♠ ♠♠ ♠ 7♠ 7♠ ♥ ♥ 2♥ 2♥ (D) ♠ ♠ ♠ ♠ 4♠ 4♠ ♠ ♠♠♠ ♠♠ ♠ 7♠ 7♠ ♣ ♣ ♣ ♣♣♣ ♣ ♣ ♣ 9♣ 9♣ ♥ ♥ 2♥ 2♥ ♦ ♦ ♦ ♦♦ ♦ 6♦ 6♦ ♠ ♠ ♠ ♠ ♠♠ ♠ 7♠ 7♠ (E) ♠ ♠ ♠ ♠ 4♠ 4♠ ♠ ♠♠♠ ♠♠ ♠ 7♠ 7♠ ♣ ♣ ♣ ♣♣♣ ♣ ♣ ♣ 9♣ 9♣ ♥ ♥ 2♥ 2♥ ♦ ♦ ♦ ♦♦ ♦ 6♦ 6♦ 2021 AUSTRALIAN MATHEMATICS COMPETITIONJUNIOR

J3 Questions 11 to 20, 4 marks each 11. To feed a horse, Kim mixes three bags of oats with one bag containing 2\b% lucerne and 8\b% oats. If all the bags have the same volume, what percentage of the combined feed mixture is lucerne? (A) 3 (B) 5(C) 6 (D) 2\b (E) 6\b 12.Three squares with perimeters 12 cm, 2\b cm and 16 cm are joined as shown. What is the perimeter of the shape formed? (A) 34 cm (B) 4\b cm (C) 41 cm (D) 42 cm (E) 48 cm 13. The odometer in my car measures the total distance travelled. At the moment, it reads 199 786 kilometres. I’m interested in when the odometer reading is a palin- drome, so that it reads the same backwards as forwards. How many more kilometres of travel will this take? (A) 25 (B) 125 (C) 15(D) 2\b5 (E) 2\b\b5 14.A square has an internal point Psuch that the perpen- dicular distances from Pto the four sides are 1 cm, 2 cm, 3 cm, and 4 cm. How many otherinternal points of the square have this property? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 P 15. How many different positive whole numbers can replace the to make this a true statement? 1\b+ 1 3< 1 (A) 3 (B) 4(C) 5 (D) 6 (E) 7 16.Three blocks with rectangular faces are placed together to form a larger rectangular prism. All blocks have side lengths which are whole numbers of centimetres. The areas of some of the faces are shown, as is the length of one edge. In cubic centimetres, what is the volume of the combined prism? (A) 36\b (B) 54\b (C) 6\b\b (D) 72\b (E) 9\b\b 4230 28 27 3 J4 17. I have four consecutive odd numbers. The largest is one less than t\bice the smallest. Which of the follo\bing is the largest of the four numbers? (A) 9 (B) 11 (C) 13 (D) 15 (E) 21 18. This is a square \bith sides of 10 metres. From the constructions sho\bn, \bhich of the areas is the largest? (A) A (B)B (C)C (D)D (E)E 3m 4m 4m 6m AB C D E 19. Sandy, Rachel and Thandie collect toy cars. Altogether they have 300 cars. Rachel has gro\bn up and decides to give her cars a\bay. If she gives them all to Sandy, then Sandy \bill have 180. If she gives them all to Thandie, then Thandie \bill have 200. Ho\b many cars does Rachel have? (A) 80 (B) 90(C) 100 (D) 110 (E) 120 20. A standard dice numbered 1 to 6 \bith opposite sides adding to 7 is placed on a 2 by 2 square as sho\bn. The dice is rolled over one edge onto each of the four base squares in turn and then back on to the original square, as indicated by the arro\bs. Which side of the dice is no\b facing up\bards? (A) (B) (C) (D) (E) Questions 21 to 25, 5 marks each 21. Leonhard is designing a puzzle for Katharina. It has nine squares in a 3 ×3 grid and a number of clues. Each clue is a number 1, 2 or 3 placed in one of the squares. Katharina then has to find a solution by placing 1, 2 or 3 in each of the remaining squares so that no ro\b or column has a repeated number. What is the smallest number of clues that Leonhard could include so that his puzzle has exactly one solution? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 2021 AUSTRALIAN MATHEMATICS COMPETITION JUNIOR

J3 Questions 11 to 20, 4 marks each 11. To feed a horse, Kim mixes three bags of oats with one bag containing 2\b% lucerne and 8\b% oats. If all the bags have the same volume, what percentage of the combined feed mixture is lucerne? (A) 3 (B) 5(C) 6 (D) 2\b (E) 6\b 12. Three squares with perimeters 12 cm, 2\b cm and 16 cm are joined as shown. What is the perimeter of the shape formed? (A) 34 cm (B) 4\b cm (C) 41 cm (D) 42 cm (E) 48 cm 13. The odometer in my car measures the total distance travelled. At the moment, it reads 199 786 kilometres. I’m interested in when the odometer reading is a palin- drome, so that it reads the same backwards as forwards. How many more kilometres of travel will this take? (A) 25 (B) 125 (C) 15(D) 2\b5 (E) 2\b\b5 14. A square has an internal point Psuch that the perpen- dicular distances from Pto the four sides are 1 cm, 2 cm, 3 cm, and 4 cm. How many otherinternal points of the square have this property? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 P 15. How many different positive whole numbers can replace the to make this a true statement? 1\b+ 1 3< 1 (A) 3 (B) 4(C) 5 (D) 6 (E) 7 16.Three blocks with rectangular faces are placed together to form a larger rectangular prism. All blocks have side lengths which are whole numbers of centimetres. The areas of some of the faces are shown, as is the length of one edge. In cubic centimetres, what is the volume of the combined prism? (A) 36\b (B) 54\b (C) 6\b\b (D) 72\b (E) 9\b\b 4230 28 27 3 J4 17. I have four consecutive odd numbers. The largest is one less than t\bice the smallest. Which of the follo\bing is the largest of the four numbers? (A) 9 (B) 11 (C) 13 (D) 15 (E) 21 18.This is a square \bith sides of 10 metres. From the constructions sho\bn, \bhich of the areas is the largest? (A) A (B)B (C)C (D)D (E)E 3m 4m 4m 6m AB C D E 19. Sandy, Rachel and Thandie collect toy cars. Altogether they have 300 cars. Rachel has gro\bn up and decides to give her cars a\bay. If she gives them all to Sandy, then Sandy \bill have 180. If she gives them all to Thandie, then Thandie \bill have 200. Ho\b many cars does Rachel have? (A) 80 (B) 90(C) 100 (D) 110 (E) 120 20.A standard dice numbered 1 to 6 \bith opposite sides adding to 7 is placed on a 2 by 2 square as sho\bn. The dice is rolled over one edge onto each of the four base squares in turn and then back on to the original square, as indicated by the arro\bs. Which side of the dice is no\b facing up\bards? (A) (B) (C) (D) (E) Questions 21 to 25, 5 marks each 21. Leonhard is designing a puzzle for Katharina. It has nine squares in a 3 ×3 grid and a number of clues. Each clue is a number 1, 2 or 3 placed in one of the squares. Katharina then has to find a solution by placing 1, 2 or 3 in each of the remaining squares so that no ro\b or column has a repeated number. What is the smallest number of clues that Leonhard could include so that his puzzle has exactly one solution? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 2021 AUSTRALIAN MATHEMATICS COMPETITIONJUNIOR

J5 22.Grandma and Grandpa took their three grandchildren to the cinema. They purchased 5 seats in a ro\b. Each grandparent \banted to sit next to t\bo of the grandchildren. Ho\b many such seating arrangements are possible? (A) 8 (B) 12 (C) 30 (D) 3(E) 60 23.I have a 4 by 4 by 4 cube made up from 64 unit cubes. I paint 3 faces of the larger cube. Then I pull the cube apart. Which of the follo\bing could be the number of unit cubes \bith no paint on them? (A) 16 (B) 21(C) 24 (D) 28 (E) 36 24.Ben and Jerry each roll a standard dice. If Ben rolls higher than Jerry, he \bins; other\bise Jerry \bins. What is the probability that Ben \bins? (A) 1 6 (B) 1 3 (C) 5 12 (D) 17 36 (E) 1 2 25. In the diagram, P QR is isosceles, \bith PQ =QR .S is a point on PRand Tis a point on PQsuch that QT=QS , and ∠S QR = 20 ◦. The size of ∠T SP, in degrees, is (A) 10 (B) 12 (C) 15 (D) 20 (E) 24 P Q R S T 20 ◦ x◦ For questions 26 to 30, shade the answer as an integer \brom 0 to 999 in the space provided on the answer sheet. Questions 26–30 are worth 6, 7, 8, 9 and 10 marks, respectively. 26. Starting \bith a 43 ×47 rectangle of paper, Sadako cuts the paper to remove the largest square possible. With the remaining rectangle, she again cuts it to remove the largest square possible. She continues doing this until the remaining piece is a square. What is the total perimeter of all the squares Sadako has at the end? J6 27. There are 14 chairs equally spaced around a circular table, and nu\bbered fro\b 1 up to 14. How \bany ways are there to choose two chairs that are not opposite each other? 28. A swi\b\bing \bedley consists of 100 \betres of each of butterfly, backstroke, breast- stroke and freestyle, in that order. I swi\b freestyle 3 ti\bes faster than breaststroke, and butterfly twice as fast as breaststroke, and \by backstroke is half as fast as \by freestyle. It takes \be 6 \binutes to swi\b the full \bedley. To the nearest \betre, how far will I have swu\b after 4 \binutes? 29. An ant’s walk starts at the apex of a regular octahedron as shown. It walks along five edges, never retracing its path. It visits each of the other five vertices exactly once. In how \bany different ways can the ant do this? 30. Consider a 15 ×15 grid of unit squares. In the square in row aand colu\bn b, we write the nu\bb er a× b. We then colour the squares black and white in a checkerboard fashion, so that the square labelled 225 is coloured white. The diagra\b shows the parts of the grid near each corner. What are the last three digits of the su\b of the nu\bbers in the white squares? 1 3 4 3 9 28 28 196 15 45 15 45 225 2 2 6 6 14 42 14 42 30 30 210 210 2021 AUSTRALIAN MATHEMATICS COMPETITION JUNIOR

J5 22. Grandma and Grandpa took their three grandchildren to the cinema. They purchased 5 seats in a ro\b. Each grandparent \banted to sit next to t\bo of the grandchildren. Ho\b many such seating arrangements are possible? (A) 8 (B) 12 (C) 30 (D) 3(E) 60 23. I have a 4 by 4 by 4 cube made up from 64 unit cubes. I paint 3 faces of the larger cube. Then I pull the cube apart. Which of the follo\bing could be the number of unit cubes \bith no paint on them? (A) 16 (B) 21(C) 24 (D) 28 (E) 36 24. Ben and Jerry each roll a standard dice. If Ben rolls higher than Jerry, he \bins; other\bise Jerry \bins. What is the probability that Ben \bins? (A) 1 6 (B) 1 3 (C) 5 12 (D) 17 36 (E) 1 2 25. In the diagram, P QR is isosceles, \bith PQ =QR .S is a point on PRand Tis a point on PQsuch that QT=QS , and ∠S QR = 20 ◦. The size of ∠T SP, in degrees, is (A) 10 (B) 12 (C) 15 (D) 20 (E) 24 P Q R S T 20 ◦ x◦ For questions 26 to 30, shade the answer as an integer \brom 0 to 999 in the space provided on the answer sheet. Questions 26–30 are worth 6, 7, 8, 9 and 10 marks, respectively. 26. Starting \bith a 43 ×47 rectangle of paper, Sadako cuts the paper to remove the largest square possible. With the remaining rectangle, she again cuts it to remove the largest square possible. She continues doing this until the remaining piece is a square. What is the total perimeter of all the squares Sadako has at the end? J6 27. There are 14 chairs equally spaced around a circular table, and nu\bbered fro\b 1 up to 14. How \bany ways are there to choose two chairs that are not opposite each other? 28. A swi\b\bing \bedley consists of 100 \betres of each of butterfly, backstroke, breast- stroke and freestyle, in that order. I swi\b freestyle 3 ti\bes faster than breaststroke, and butterfly twice as fast as breaststroke, and \by backstroke is half as fast as \by freestyle. It takes \be 6 \binutes to swi\b the full \bedley. To the nearest \betre, how far will I have swu\b after 4 \binutes? 29. An ant’s walk starts at the apex of a regular octahedron as shown. It walks along five edges, never retracing its path. It visits each of the other five vertices exactly once. In how \bany different ways can the ant do this? 30. Consider a 15 ×15 grid of unit squares. In the square in row aand colu\bn b, we write the nu\bb er a× b. We then colour the squares black and white in a checkerboard fashion, so that the square labelled 225 is coloured white. The diagra\b shows the parts of the grid near each corner. What are the last three digits of the su\b of the nu\bbers in the white squares? 1 3 4 3 9 28 28 196 15 45 15 45 225 2 2 6 6 14 42 14 42 30 30 210 210 2021 AUSTRALIAN MATHEMATICS COMPETITIONJUNIOR

Check out Problemo Student, a free problem-solving platform that allows you to explore mathematics and algorithmics problems at your own pace. ✔ Packed with problems covering a range of topics. ✔ Designed for school Years 3–12. Need a challenge? Test yourself a level or two up! ✔ Try one question or work your way through a series based on your chosen topic. ✔ Work at your own pace and on your own device – anytime, anywhere. ✔ Get instant feedback and detailed solutions. ✔ Explore computer-programmer thinking with the Computation and Algorithms Discovery Series. Jump into more maths problem solving without the ticking clock Find out more at app.problemo.edu.au/student Junior Ye a r s 7– 8 (AUSTRALIAN  SCHOOL YEARS)