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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 Intermediate Ye a r s 9 –1 0 (AUSTRALIAN  SCHOOL YEARS) 4–6 August

Intermediate Division Questions 1 to 10, 3 marks each 1. Each edge of this star is 2 cm long. What is its perimeter? (A\b 5 cm (B\b 10 cm (C\b 15 cm (D\b 20 cm (E\b 25 cm \b.The value of 2000 −200 + 20 −2 is (A\b 1778 (B\b 1782 (C\b 1818 (D\b 1822 (E\b 1888 3. What is the value of ain the diagram? (A\b 35 (B\b 45(C\b 55 (D\b 65 (E\b 75 145 ◦ a◦ 4.What is 50% more than 1 a? (A\b 1 4 (B\b 5 8 (C\b 3 2 (D\b 3 4 (E\b 50.5 5. 1+3+5+7+9 • •-•••ƒ \b 1 2 (B\b 5 6 (C\b 11 12 (D\b 9 10 (E\b 63 256 6. Square AB C D has centre O. The shaded area is 16 square units. What is the length of the side of the square? (A\b 4 (B\b 8(C\b 16 (D\b 32 (E\b 64 DA B C O I2 7. On the number line, which number is halfway between 10 2and 10 4\b (A) 500 (B) 550 (C) 1010 (D) 2021 (E) 5050 8. To feed a horse, Kim mixes three bags of oats with one bag containing 20% lucerne and 80% oats. If all the bags have the same volume, what percentage of the combined feed mixture is lucerne\b (A) 3 (B) 5(C) 6 (D) 20 (E) 60 9. I have a solid block of wood in the shape of a cylinder. The top and bottom faces meet the curved side at right angles. Suppose that I slice the cylinder along a plane to create two smaller blocks of wood. Which of the following could notbe the shape of the re- sulting faces created by the slice\b (A) (B) (C) (D) (E) 10. Diya timed herself cycling laps around her suburb. After five laps, her stopwatch indicated a time of 18 minutes and 15 seconds. What was Diya’s average time per lap\b (A) 3 minutes and 3 seconds (C) 3 minutes and 27 seconds (B) 3 minutes and 15 seconds (D) 3 minutes and 39 seconds (E) 3 minutes and 51 seconds Questions 11 to 20, 4 marks each 11. I have four consecutive odd numbers. The largest is one less than twice the smallest. Which of the following is the largest of the four numbers\b (A) 9 (B) 11 (C) 13 (D) 15 (E) 21 12. On a compact disc, uncompressed music data is stored as 44 100 samples for each second of music, where each sample requires 4 bytes of data. Which of the following is closest to the number of bytes required to store 5 minutes of music on the disc\b (A) 1 million (B) 5 million (C) 10 million (D) 50 million (E) 100 million 2021 AUSTRALIAN MATHEMATICS COMPETITION INTERMEDIATE

I2 7.On the number line, which number is halfway between 10 2and 10 4\b (A) 500 (B) 550 (C) 1010 (D) 2021 (E) 5050 8. To feed a horse, Kim mixes three bags of oats with one bag containing 20% lucerne and 80% oats. If all the bags have the same volume, what percentage of the combined feed mixture is lucerne\b (A) 3 (B) 5(C) 6 (D) 20 (E) 60 9.I have a solid block of wood in the shape of a cylinder. The top and bottom faces meet the curved side at right angles. Suppose that I slice the cylinder along a plane to create two smaller blocks of wood. Which of the following could notbe the shape of the re- sulting faces created by the slice\b (A) (B) (C) (D) (E) 10. Diya timed herself cycling laps around her suburb. After five laps, her stopwatch indicated a time of 18 minutes and 15 seconds. What was Diya’s average time per lap\b (A) 3 minutes and 3 seconds (C) 3 minutes and 27 seconds (B) 3 minutes and 15 seconds (D) 3 minutes and 39 seconds (E) 3 minutes and 51 seconds Questions 11 to 20, 4 marks each 11. I have four consecutive odd numbers. The largest is one less than twice the smallest. Which of the following is the largest of the four numbers\b (A) 9 (B) 11 (C) 13 (D) 15 (E) 21 12.On a compact disc, uncompressed music data is stored as 44 100 samples for each second of music, where each sample requires 4 bytes of data. Which of the following is closest to the number of bytes required to store 5 minutes of music on the disc\b (A) 1 million (B) 5 million (C) 10 million (D) 50 million (E) 100 million 2021 AUSTRALIAN MATHEMATICS COMPETITIONINTERMEDIATE

I3 13.In the figure, the value of xis (A) 30 (B) \b0(C) 50 (D) 60 (E) 70 x◦ 3x ◦ (x + 30) ◦ 14. What is the equation of the line passing through (0, 0) that bisects the square in the diagram? (A) y= x 3 (B) y= x 2 (C) y= x 4 (D) y=2 x (E)y=3 x 1 12 15.A standard dice numbered 1 to 6 with opposite sides adding to 7 is placed on a 2 by 2 square as shown. 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 arrows. Which side of the dice is now facing upwards? (A) (B) (C) (D) (E) 16.The two spinners shown are spun and the numbers that the arrows point to when they stop are recorded. For example, the numbers here are 3 and 6. What is the probability that the sum of the two numbers is even? 1 3 n t h e (A) 1 3 (B) 3 h (C) 3 n (D) 2 I (E) 5 t3 17. The area of the shaded region is given by (A) ab+ac (C) bc+a 2− ab −ac (B) a√ b2+ c 2 (D) ab+ac −bc (E) ab+ac −a 2 ac b a a c b a I4 18. Ifkand nare positive integers, and √ 10n +k= k, then the smallest possi\ble value for kis (A) 3 (B) 4(C) 5 (D) 6 (E) 10 19. Two squares are drawn as shown. The smaller square covers 1 8 of the larger square and the larger square covers 2 9 of the smaller square. What is the ratio of the side length of the larger square to the side length of the smaller square? (A) 3:2 (B) 7:3(C) 7:4 (D) 5:3 (E) 4:3 20. Six identical darts fit inside a regular hexagon as shown. Each dart has three interior angles of 30 ◦, and one of 270 ◦. What fraction of the large hexagon is shaded? (A) 1 2 (B) 1 3 (C) 2 5 (D) 4 9 (E) 3 8 Questions 21 to 25, 5 marks each 21. We want to place num\bers into each of the \blank squares in this diagram so that each of the num\bers we place is the average of those in the squares directly connected to it. What num\ber should we put in the middle square of the top row? (A) 5 3 (B) 3 2 (C) 10 9 (D) 119 (E) 116 1 2 4 3 22. To set the timer on his microwave oven, Rick enters the digits of the hours, minutes and seconds in order from left to right. For example, entering ‘12345’ sets the timer to 1 hour 23 minutes 45 seconds, while entering ‘408’ sets it to 4 minutes 8 seconds. One day, Rick accidentally missed the last digit and the timer finished 4 minutes and 42 seconds earlier than he was expecting. What was the missing digit? (A) 3 (B) 4(C) 5 (D) 6 (E) 7 2021 AUSTRALIAN MATHEMATICS COMPETITION INTERMEDIATE

I3 13. In the figure, the value of xis (A) 30 (B) \b0(C) 50 (D) 60 (E) 70 x ◦ 3 x ◦ (x + 30) ◦ 14. What is the equation of the line passing through (0, 0) that bisects the square in the diagram? (A) y= x 3 (B) y= x 2 (C) y= x 4 (D) y=2 x (E)y=3 x 1 12 15.A standard dice numbered 1 to 6 with opposite sides adding to 7 is placed on a 2 by 2 square as shown. 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 arrows. Which side of the dice is now facing upwards? (A) (B) (C) (D) (E) 16. The two spinners shown are spun and the numbers that the arrows point to when they stop are recorded. For example, the numbers here are 3 and 6. What is the probability that the sum of the two numbers is even? 1 3 2 t h e (A) 1 2 (B) 3 8 (C) 3 4 (D) 2 3 (E) 5 12 17. The area of the shaded region is given by (A) ab+ac (C) bc+a 2− ab −ac (B) a√ b 2+ c 2 (D) ab+ac −bc (E) ab+ac −a 2 ac b a a c b a I4 18. Ifkand nare positive integers, and √ 10n +k= k, then the smallest possi\ble value for kis (A) 3 (B) 4(C) 5 (D) 6 (E) 10 19.Two squares are drawn as shown. The smaller square covers 1 3 of the larger square and the larger square covers 2 9 of the smaller square. What is the ratio of the side length of the larger square to the side length of the smaller square? (A) 3:2 (B) 7:3(C) 7:4 (D) 5:3 (E) 4:3 20.Six identical darts fit inside a regular hexagon as shown. Each dart has three interior angles of 30 ◦, and one of 270 ◦. What fraction of the large hexagon is shaded? (A) 1 2 (B) 1 3 (C) 2 5 (D) 4 9 (E) 3 8 Questions 21 to 25, 5 marks each 21. We want to place num\bers into each of the \blank squares in this diagram so that each of the num\bers we place is the average of those in the squares directly connected to it. What num\ber should we put in the middle square of the top row? (A) 5 3 (B) 3 2 (C) 10 9 (D) 11 9 (E) 11 6 1 8 2 9 22. To set the timer on his microwave oven, Rick enters the digits of the hours, minutes and seconds in order from left to right. For example, entering ‘12345’ sets the timer to 1 hour 23 minutes 45 seconds, while entering ‘408’ sets it to 4 minutes 8 seconds. One day, Rick accidentally missed the last digit and the timer finished 4 minutes and 42 seconds earlier than he was expecting. What was the missing digit? (A) 3 (B) 4(C) 5 (D) 6 (E) 7 2021 AUSTRALIAN MATHEMATICS COMPETITIONINTERMEDIATE

I5 23.I build a large cube from unit cubes. Then I completely paint a number of faces of the large cube. \bhen I dismantle the large cube, I find that I have 288 unit cubes without any paint on them. How many faces of the large cube were painted? (A) 1 (B) 2(C) 3 (D) 4 (E) 5 24.The product 1− 1 22 1− 1 32 1− 1 42 ∙∙∙ 1− 1 15 2 is equal to (A) 7 13 (B) 8 15 (C) 9 16 (D) 10 21 (E) 13 24 25. Three artificial islands Razz, Sazz and Tazz were constructed in a shallow sea, each with a coastline of 12 km. 12 π Razz 3 3 3 3 Sazz 4 4 4 Tazz Around each island is a fishing zone, consisting of all points in the sea within 1 km of the island. \bhich islands have a fishing zone of the largest area? (A) Razz only (B) Sazz only(C) Tazz only (D) Sazz and Tazz (E) All three have the same area 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. In Australian Rules football, a team scores six points for a ‘goal’ and one point for a ‘behind’. During a game, Vladislav likes to record his team’s score with a sequence of sixes and ones. There are exactly three distinct sequences which give a final score of 7 points, namely 6,1 and 1,6 and 1,1,1,1,1,1,1. How many different sequences give a final score of 20 points? I6 27. What is the smallest natural number nsuch that the number N = 100000 ×100002 ×100006 ×100008 + n is a per\bect square? 28. I have a large supply o\b matchsticks in \bour colours: red, yellow, blue and green. I use them to make squares where each side is one matchstick long. I count two squares as the same i\b one can be rotated and/or reflected to match the shape and colour o\b the other. How many different squares can be created? 29. Bluey divides the number 499 by each o\b the numbers 1, 2,3 ,..., 499 and records the remainders in order. So her sequence begins: 0,1 ,1 ,3 ,4 ,1 ,... Let Mbe the sum o\b these 499 remainders. Jean-Luc divides the number 500 by each o\b the numbers 1, 2,3 ,..., 500 and records the remainders in order. So his sequence begins: 0,0 ,2 ,0 ,0 ,2 ,... Let Nbe the sum o\b these 500 remainders. What is the difference between the numbers Mand N? 30. Tyler has a large number o\b square tiles, all the same size. He has \bour times as many blue tiles as red tiles. He builds a large rectangle using all the tiles, with the red tiles \borming a boundary 1 tile wide around the blue tiles. He then breaks up this rectangle and uses the tiles to make two smaller rectangles. Like the large rectangle, each o\b the smaller rectangles has \bour times as many blue tiles as red tiles, and the red tiles \borm a boundary 1 tile wide around the blue tiles. How many blue tiles does Tyler have? 2021 AUSTRALIAN MATHEMATICS COMPETITION INTERMEDIATE

I5 23. I build a large cube from unit cubes. Then I completely paint a number of faces of the large cube. \bhen I dismantle the large cube, I find that I have 288 unit cubes without any paint on them. How many faces of the large cube were painted? (A) 1 (B) 2(C) 3 (D) 4 (E) 5 24. The product 1− 1 2 2 1− 1 3 2 1− 1 4 2 ∙∙∙ 1− 1 15 2 is equal to (A) 7 13 (B) 8 15 (C) 9 16 (D) 10 21 (E) 13 24 25. Three artificial islands Razz, Sazz and Tazz were constructed in a shallow sea, each with a coastline of 12 km. 12 π Razz 3 3 3 3 Sazz 4 4 4 Tazz Around each island is a fishing zone, consisting of all points in the sea within 1 km of the island. \bhich islands have a fishing zone of the largest area? (A) Razz only (B) Sazz only(C) Tazz only (D) Sazz and Tazz (E) All three have the same area 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. In Australian Rules football, a team scores six points for a ‘goal’ and one point for a ‘behind’. During a game, Vladislav likes to record his team’s score with a sequence of sixes and ones. There are exactly three distinct sequences which give a final score of 7 points, namely 6,1 and 1,6 and 1,1,1,1,1,1,1. How many different sequences give a final score of 20 points? I6 27. What is the smallest natural number nsuch that the number N = 100000 ×100002 ×100006 ×100008 + n is a per\bect square? 28. I have a large supply o\b matchsticks in \bour colours: red, yellow, blue and green. I use them to make squares where each side is one matchstick long. I count two squares as the same i\b one can be rotated and/or reflected to match the shape and colour o\b the other. How many different squares can be created? 29. Bluey divides the number 499 by each o\b the numbers 1, 2,3 ,..., 499 and records the remainders in order. So her sequence begins: 0,1 ,1 ,3 ,4 ,1 ,... Let Mbe the sum o\b these 499 remainders. Jean-Luc divides the number 500 by each o\b the numbers 1, 2,3 ,..., 500 and records the remainders in order. So his sequence begins: 0,0 ,2 ,0 ,0 ,2 ,... Let Nbe the sum o\b these 500 remainders. What is the difference between the numbers Mand N? 30. Tyler has a large number o\b square tiles, all the same size. He has \bour times as many blue tiles as red tiles. He builds a large rectangle using all the tiles, with the red tiles \borming a boundary 1 tile wide around the blue tiles. He then breaks up this rectangle and uses the tiles to make two smaller rectangles. Like the large rectangle, each o\b the smaller rectangles has \bour times as many blue tiles as red tiles, and the red tiles \borm a boundary 1 tile wide around the blue tiles. How many blue tiles does Tyler have? 2021 AUSTRALIAN MATHEMATICS COMPETITIONINTERMEDIATE

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 Intermediate Ye a r s 9 –1 0 (AUSTRALIAN  SCHOOL YEARS)