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20202022 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. 5. This is a competition and not a test so don’t worry if you can’t answer all the questions. Attempt as many as you can — there is no penalty for an incorrect a n s we r. 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 Your answer sheet will be scanned. To make sure the scanner reads your paper correctly, there are some DOs and DON’Ts: DO: • use only a lead pencil • record your answers on the answer sheet (not on the question paper) • for questions 1–25, fully colour the circle matching your answer — keep within the lines as much as you can • for questions 26–30, write your 3-digit answer in the box — make sure your writing does not touch the box • use an eraser if you want to change an answer or remove any marks or smudges. D O N O T: • doodle or write anything extra on the answer sheet • colour in the QR codes on the corners of the answer sheet. 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 © 2022 Australian Mathematics Trust | ACN 083 950 341 DAT E TIME ALLOWED 75 minutes Junior Ye a r s 7– 8 (AUSTRALIAN SCHOOL YEARS) 3–5 August
Junior Division Questions 1 to 10, 3 marks each 1. What is the perimeter of this rhombus? (A) 20 cm (B) 24 cm (\b) 28 cm (D) 32 cm (E) 36 cm 9 cm 9 cm 2. The temperature in the mountains was 4 ◦\b but dropped overnight by 7 ◦\b. What was the temperature in the morning? (A) 3 ◦\b (B) 11 ◦\b (\b) −3 ◦\b (D) −4 ◦\b (E) −11 ◦\b 3. What is the value of 20 ×22? (A) 42 (B) 440 (\b) 2022 (D) 2220 (E) 4400 \b. Which spinner is twice as likely to land on red as white? (A) (B) (\b) (D) (E) 5. Russell’s tuba lesson started at 4:28 pm and finished at 5:05 pm. How long was the lesson? (A) 23 minutes (B) 27 minutes (\b) 33 minutes (D) 37 minutes (E) 43 minutes 6. What fraction of the square is shaded? (A) 1 3 (B) 1 4 (\b) 2 5 (D) 1 2 (E) 3 8 1 2 2 2 7. What is the value of 10 3−11 2? (A) 8 (B) 22 (\b) 279 (D) 779 (E) 879 8.Which one of these fractions lies between 4 and 5 on the number line? (A\b 7 2 (B\b 15 4 (C\b 165 (D\b 174 (E\b 185 9. In the triangle P QRshown, PQ=PR and ∠QP R = 48 ◦. What is ∠P QR? (A\b 60 ◦ (B\b 66 ◦ (C\b 72 ◦ (D\b 78 ◦ (E\b 84 ◦ P Q R 48 ◦ 10. What is the time and day one-quarter of a week after midday on Sunday? (A\b 6 am Tuesday (B\b 9 pm Tuesday (C\b midday Monday (D\b 3 am Wednesday (E\b 6 pm Monday Questions 11 to 20, 4 marks each 11. These three coins have a number on each side. The two numbers on each coin multiply to 60. What is the sum of the three hidden numbers? 30 30 30 555 444 (A\b 17 (B\b 21(C\b 29 (D\b 31 (E\b 39 12.Three different squares are arranged as shown. The perimeter of the largest square is 36 cm. The area of the smallest square is 9 cm 2. What is the perimeter of the medium-sized square? (A\b 12 cm (B\b 18 cm (C\b 24 cm (D\b 30 cm (E\b 32 cm 13.Australia uses 160 million litres of petrol each day. There is enough petrol stored to last 60 days. How much more petrol does Australia need to buy to have enough stored for 90 days? (A\b 4 million litres (B\b 4.8 million litres (C\b 480 million litres (D\b 160 million litres (E\b 4800 million litres 2022 AUSTRALIAN MATHEMATICS COMPETITION JUNIOR
8.Which one of these fractions lies between 4 and 5 on the number line? (A\b 7 . (B\b 15 4 (C\b 16 5 (D\b 17 4 (E\b 18 5 9. In the triangle P QRshown, PQ=PR and ∠QP R = 48 ◦. What is ∠P QR? (A\b 60 ◦ (B\b 66 ◦ (C\b 72 ◦ (D\b 78 ◦ (E\b 84 ◦ P Q R 48 ◦ 10. What is the time and day one-quarter of a week after midday on Sunday? (A\b 6 am Tuesday (B\b 9 pm Tuesday (C\b midday Monday (D\b 3 am Wednesday (E\b 6 pm Monday Questions 11 to 20, 4 marks each 11. These three coins have a number on each side. The two numbers on each coin multiply to 60. What is the sum of the three hidden numbers? 303030 555 444 (A\b 17 (B\b 21(C\b 29 (D\b 31 (E\b 39 12.Three different squares are arranged as shown. The perimeter of the largest square is 36 cm. The area of the smallest square is 9 cm 2. What is the perimeter of the medium-sized square? (A\b 12 cm (B\b 18 cm (C\b 24 cm (D\b 30 cm (E\b 32 cm 13.Australia uses 160 million litres of petrol each day. There is enough petrol stored to last 60 days. How much more petrol does Australia need to buy to have enough stored for 90 days? (A\b 4 million litres (B\b 4.8 million litres (C\b 480 million litres (D\b 160 million litres (E\b 4800 million litres 2022 AUSTRALIAN MATHEMATICS COMPETITION JUNIOR
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22.On a large grid, rows and columns are numbered as shown. All squares in row 1 are shaded. E\bery second square in row 2 is shaded. E\bery third square in row 3 is shaded, and so on. As a result each column has certain squares shaded. For instance, the shaded squares in column 6 are the three squares shown and one more in row 6. In column 105, how many squares are shaded? • • • • • • • • • • • ∙∙∙ . . . (A) 3 (B) 4(C) 8 (D) 12 (E) 35 23.Usually Andrew walks home from school in 24 minutes. Last Monday, he walked for the first 15 minutes but then it started to rain, so he ran the rest of the way home. His running speed is 1.5 times his usual walking speed. How many minutes did it take him to get from school to home? (A) 18 (B) 20(C) 21 (D) 22 (E) 23 24.The single-digit unit fractions are 1 n,1 3,1 4,1 5,1 6,1 7,1 8 and 1 9. How many pairs of these fractions are there where the first fraction minus the second fraction is bigger than 1 10 . (A) 5 (B) 10 (C) 15 (D) 20 (E) 25 25.In the grid shown, the numbers 1 to 6 are placed so that when joined in ascending order they make a trail. The trail mo\bes from one square to an adjacent square but does not mo\be diagonally. In how many ways can the numbers 1 to 6 be placed in the grid to gi\be such a trail? 1 45 2 36 (A) 16 (B) 20(C) 24 (D) 28 (E) 36 For questions 26 to 30, shade the answer as an integer from 0 to 999 in the s\bace \brovided on the answer sheet. Questions 26–30 are worth 6, 7, 8, 9 and 10 marks, res\bectively. 26. In the sum shown, the symbols ♦,♥ and represent three different digits. What is the three\bdigit number represented by ♦♥ ? ♦♥ + ♦♥ ♥♥♦ 27. The digits 1, 2, 3, 4, 5, 6, 7, 8 are separated into two groups of 4 each. Each group is formed into a four\bdigit number and the two numbers are added. Finally, the digits in this sum are added together. An example of this is 3541 + 7628 = 11169, with digit sum 1 + 1 + 1 + 6 + 9 = 18. What is the difference between the largest possible and smallest possible digit sums? 28. A surf club consists of three types of members: trainees, paddlers and legends. There are 20 trainees, which is less than half the membership. There are twice as many legends as paddlers. After a surf rescue they received a $1000 donation to be divided among the members. All the donation was shared and every member received a whole number of dollars, at least $2. Each paddler received 6 times as much as each trainee. Each legend received $5 more than each paddler. How many members are in the surf club? 29. Horton has a regular hexagon of area 60. For each choice of three vertices of the hexagon, he writes down the area of the triangle with these three vertices. What is the sum of the 20 areas that Horton writes down? 30. In how many ways can 100 be written as the sum of three different positive integers? Note that we do not consider sums formed by reordering the terms to be different, so that 34 + 5 + 61 and 61 + 34 + 5 are treated as the same sum. 2022 AUSTRALIAN MATHEMATICS COMPETITION JUNIOR
22.On a large grid, rows and columns are numbered as shown. All squares in row 1 are shaded. E\bery second square in row 2 is shaded. E\bery third square in row 3 is shaded, and so on. As a result each column has certain squares shaded. For instance, the shaded squares in column 6 are the three squares shown and one more in row 6. In column 105, how many squares are shaded? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 ∙∙∙ . . . (A) 3 (B) 4(C) 8 (D) 12 (E) 35 23. Usually Andrew walks home from school in 24 minutes. Last Monday, he walked for the first 15 minutes but then it started to rain, so he ran the rest of the way home. His running speed is 1.5 times his usual walking speed. How many minutes did it take him to get from school to home? (A) 18 (B) 20(C) 21 (D) 22 (E) 23 24. The single-digit unit fractions are 1 2,1 3,1 4,1 5,1 6,1 7,1 8 and 1 9. How many pairs of these fractions are there where the first fraction minus the second fraction is bigger than 1 10. (A) 5 (B) 10 (C) 15 (D) 20 (E) 25 25. In the grid shown, the numbers 1 to 6 are placed so that when joined in ascending order they make a trail. The trail mo\bes from one square to an adjacent square but does not mo\be diagonally. In how many ways can the numbers 1 to 6 be placed in the grid to gi\be such a trail? •• - € -‚€ •-ƒ€ • -„€ • - € For questions 26 to 30, shade the answer as an integer from 0 to 999 in the s\bace \brovided on the answer sheet. Questions 26–30 are worth 6, 7, 8, 9 and 10 marks, res\bectively. In the sum shown, the symbols ♦,♥ and represent three different digits. What is the three\bdigit number represented by ♦♥ ? ♦♥ + ♦♥ ♥♥♦ 27. The digits 1, 2, 3, 4, 5, 6, 7, 8 are separated into two groups of 4 each. Each group is formed into a four\bdigit number and the two numbers are added. Finally, the digits in this sum are added together. An example of this is 3541 + 7628 = 11169, with digit sum 1 + 1 + 1 + 6 + 9 = 18. What is the difference between the largest possible and smallest possible digit sums? 28. A surf club consists of three types of members: trainees, paddlers and legends. There are 20 trainees, which is less than half the membership. There are twice as many legends as paddlers. After a surf rescue they received a $1000 donation to be divided among the members. All the donation was shared and every member received a whole number of dollars, at least $2. Each paddler received 6 times as much as each trainee. Each legend received $5 more than each paddler. How many members are in the surf club? 29. Horton has a regular hexagon of area 60. For each choice of three vertices of the hexagon, he writes down the area of the triangle with these three vertices. What is the sum of the 20 areas that Horton writes down? 30. In how many ways can 100 be written as the sum of three different positive integers? Note that we do not consider sums formed by reordering the terms to be different, so that 34 + 5 + 61 and 61 + 34 + 5 are treated as the same sum. 2022 AUSTRALIAN MATHEMATICS COMPETITION JUNIOR
CORRECTLY RECORDING YOUR ANSWER (QUESTIONS 1–25) Only use a lead pencil to record your answer. When recording your answer on the sheet, fi ll in the bubble completely. The example below shows the answer to Question 1 was recorded as ‘B’\ . DO NOT record your answers as shown below. They cannot be read accurately by the scanner and you may not receive a mark for the question. Use an eraser if you want to change an answer or remove any pencil marks or smudges. DO NOT cross out one answer and fi ll in another answer, as the scanner cannot determine which one is your answer. Correct CORRECTLY WRITING YOUR ANSWER (QUESTIONS 26–30) For questions 26–30, write your answer in the boxes as shown below. 2 + 3 = 20 + 21 = 200 + 38 = WRITING SAMPLES 0 12 3 45 6 78 9 Your numbers MUST NOT touch the edges of the box or go outside it. The number one must only be written as above, otherwise the scanner migh\ t interpret it as a seven. DO NOT doodle or write anything extra on the answer sheet or colour in the QR \ codes on the corners of the answer sheet, as this will interfere with the scanner. Incorrect Incorrect Incorrect Incorrect Incorrect Incorrect this one! 1 digit 2 digits 3 digits 54 l 2 3 8 0 Correct l Correct 3 Correct 4 6 Correct 7 9 Correct 1 Incorrect 3 0 6 9 4 7 Correct Correct 2 Correct 5 Correct 8 Correct 5 2 8 2 36 5 4 0 5 8 1 Junior Ye a r s 7– 8 (AUSTRALIAN SCHOOL YEARS)