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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 4–6 August Senior Ye a r s 1 1 –1 2 (AUSTRALIAN  SCHOOL YEARS)

Senior Division Questions 1 to 10, 3 marks each 1. Each small triangle is the same size. What fraction of the largest triangle is shaded? (A) 1 e (\b) 1 3 (C) 1 4 (D) 2 5 (E) 3 8 2. When 2021 is divided by 7 the remainder is (A) 2 (\b) 3(C) 4 (D) 5 (E) 6 3.What is 12 1−12 −1 −12 0? (A) 0 (\b) 1(C) 10 11 Ea (D) 11 (E) 11 1 12 \b.In this diagram, what is the size of the angles marked θ? (A) 70 ◦ (\b) 75 ◦ (C) 80 ◦ (D) 85 ◦ (E) 90 ◦ 150 ◦ θ θ θ 5. The value of (20×21) + 21 .θ is (A) 20 (\b) 21(C) 22 (D) 41 (E) 42 6.Henry’s electric scooter took him 1.5 km in 3 minutes and 45 seconds. What was the average speed of Henry’s trip in kilometres per hour? (A) 20 (\b) 21(C) 24 (D) 25 (E) 30 7.8 3× 3 6 65 = (A) 6 (\b) 48 (C) 72 (D) 128 (E) 256 8.The product of recurring decimals 0. ˙ 3 and 0. ˙ 6 is the recurring decimal 0. ˙ x. What is the value of x? (A) 1 (\b) 2(C) 5 (D) 7 (E) 9 S2 9. The parallelogram shown has an area of 48 square units. The \balue of ais (A) 7 (B) 8(C) 9 (D) 10 (E) 11 P(1, 2) Q(a,2) R S (4, 8) 10. Mer\bin is allowed to paint the four walls and the ceiling of his rectangular bedroom as he wishes, sub ject to the following constraints. He paints each surface in one of three colours. He cannot paint two adjacent surfaces the same colour. He decides to use red, white and green. How many different ways can he paint his room? (A) 2 (B) 3(C) 6 (D) 12 (E) 24 Questions 11 to 20, 4 marks each 11. Here is a list of fractions which, when written in simplest form, ha\be a denominator less than 6: 1 2, T1 T,2 5, 3 32 3,3 4,4 5 The list is in ascending order, but three fractions are omitted. The sum of these three fractions is (A) 1 (B) 2(C) 21 20 (D) 27 20 (E) 29 20 12.In autumn, Tilly’s meadow changes rapidly with the weather. The number of flowering plants starts at 150 000 but they are dying off so each week the number hal\bes. At the same time, the number of fungi starts at 20 and triples each week. To the nearest week, how long will it be until the fungi outnumber the flowering plants? (A) 3 weeks (B) 5 weeks (C) 8 weeks (D) 11 weeks (E) 13 weeks 13. A formula in physics is gi\ben as: r= mV og If qwas trebled, mwas hal\bed and rand Bremained the same, then Vwould (A) increase by a factor of 6 (B) decrease by a factor of 5 (C) stay the same (D) double (E) increase by a factor of 3 2021 AUSTRALIAN MATHEMATICS COMPETITION SENIOR

S2 9.The parallelogram shown has an area of 48 square units. The \balue of ais (A) 7 (B) 8(C) 9 (D) 10 (E) 11 P(1, 2) Q(a,2) R S (4, 8) 10. Mer\bin is allowed to paint the four walls and the ceiling of his rectangular bedroom as he wishes, sub ject to the following constraints. He paints each surface in one of three colours. He cannot paint two adjacent surfaces the same colour. He decides to use red, white and green. How many different ways can he paint his room? (A) 2 (B) 3(C) 6 (D) 12 (E) 24 Questions 11 to 20, 4 marks each 11. Here is a list of fractions which, when written in simplest form, ha\be a denominator less than 6: 1 5, ,1 3,2 5, , ,2 3,3 4,4 5 The list is in ascending order, but three fractions are omitted. The sum of these three fractions is (A) 1 (B) 2(C) 21 20 (D) 27 20 (E) 29 20 12.In autumn, Tilly’s meadow changes rapidly with the weather. The number of flowering plants starts at 150 000 but they are dying off so each week the number hal\bes. At the same time, the number of fungi starts at 20 and triples each week. To the nearest week, how long will it be until the fungi outnumber the flowering plants? (A) 3 weeks (B) 5 weeks (C) 8 weeks (D) 11 weeks (E) 13 weeks 13. A formula in physics is gi\ben as: r= mV qB If qwas trebled, mwas hal\bed and rand Bremained the same, then Vwould (A) increase by a factor of 6 (B) decrease by a factor of 5 (C) stay the same (D) double (E) increase by a factor of 3 2021 AUSTRALIAN MATHEMATICS COMPETITIONSENIOR

S3 14.In this diagram, AD= 12 and AB Cand CDE are right-angled isosceles triangles. \bhe area of triangle BDEis 9. What is the area of triangle AB D? (A) 36 (B) 50 (C) 54 (D) 60 (E) 72 A D C 15.\bhe numbers 1 40,2 30,3 20 and 4 10, in increasing order, are (A) 1 40, 4 10,2 30,3 20 (B) 1 40,3 20,4 10,2 30 (C) 4 10,3 20,2 30,1 40 (D) 1 40,2 30,3 20,4 10 (E) 1 40,2 30,4 10,3 20 16. In the rectangle AB C D, the lengths marked x, y and zare positive integers. \briangle AE Dhas an area of 12 square units and triangle BCEhas an area of 21 square units. How many possible values are there for z? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 A B C D E z xy 17. A square is divided into three congruent isosceles triangles and a shaded pentagon, as shown. What fraction of the square’s area is shaded? (A) 1 D (B) 1 4 (C) √2 6 (D) √3 8 (E) 2 −√ 3 18. A non-standard dice has the numbers 2, 3, 5, 8, 13 and 21 on it. \bhe dice is rolled twice and the numbers are added together. What is the probability that the resulting sum is also a value on the dice? (A) 1 (B) 5 12 (C) 1 3 (D) 1 6 (E) 2 9 21 B E S4 19. Ifaand bare positive numbers, then a 2+ 1 b2 × b 2+ 1 a2 is equal to \bA) a b+ b a \bB) a 2b2+ 1 a2b2 \bC) ab+2+ 1 ab \bD) a+ b+ 1 a+ 1 b \bE) ab+ 1 ab 20. The quadrilateral shown is cut into two equal areas by the dashed line. What is the ratio a:b ? \bA) 2:1 \bB) 7:3 \bC) 5:2 \bD) 4:3 \bE) 3:2 6 1 1 4 a b Questions 21 to 25, 5 marks each 21. Positive integers xand ysatisfy the equation x 2+2 xy+2 y 2+2y= 1988 What is the largest possible value of x+ y? \bA) 33 \bB) 38\bC) 42 \bD) 46 \bE) 47 22. What fraction of the area of the diagram is shaded? \bA) 1 4 \bB) 4 9 \bC) 5 12 \bD) 1 2 \bE) 5 18 23. Sebastien is playing with a square paper serviette with side length 24 centimetres. He folds it in half along a diagonal to obtain a triangle AB Cwith a right angle at A. He then folds the triangle so that Cends up on line ABat some point D. Suppose that the fold created meets BCat the point X. Sebastien then folds Bto meet Xand notices that the fold created passes through the point D. The distance in centimetres between the points Aand Dis \bA) 6 √ 2 \bB) 12\b √ 3 − 1) \bC) 10 \bD) 12 \bE) 24\b √ 2 − 1) 2021 AUSTRALIAN MATHEMATICS COMPETITION SENIOR

S3 14. In this diagram, AD= 12 and AB Cand CDE are right-angled isosceles triangles. \bhe area of triangle BDEis 9. What is the area of triangle AB D? (A) 36 (B) 50 (C) 54 (D) 60 (E) 72 A D C 15. \bhe numbers 1 40,2 30,3 20 and 4 10, in increasing order, are (A) 1 40, 4 10,2 30,3 20 (B) 1 40,3 20,4 10,2 30 (C) 4 10,3 20,2 30,1 40 (D) 1 40,2 30,3 20,4 10 (E) 1 40,2 30,4 10,3 20 16. In the rectangle AB C D, the lengths marked x, y and zare positive integers. \briangle AE Dhas an area of 12 square units and triangle BCEhas an area of 21 square units. How many possible values are there for z? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 12 A B C D E z xy 17. A square is divided into three congruent isosceles triangles and a shaded pentagon, as shown. What fraction of the square’s area is shaded? (A) 1 3 (B) 1 4 (C) √ 2 6 (D) √ 3 8 (E) 2 −√ 3 = = = = 18. A non-standard dice has the numbers 2, 3, 5, 8, 13 and 21 on it. \bhe dice is rolled twice and the numbers are added together. What is the probability that the resulting sum is also a value on the dice? (A) 1 9 (B) 5 12 (C) 1 3 (D) 1 6 (E) 2 9 21 B E S4 19. Ifaand bare positive numbers, then a2+ 1 b2 × b2+ 1 a2 is equal to \bA) a b+ b a \bB) a 2b2+ 1 a2b2 \bC) ab+2+ 1 ab \bD) a+ b+ 1 a+ 1 b \bE) ab+ 1 ab 20. The quadrilateral shown is cut into two equal areas by the dashed line. What is the ratio a:b ? \bA) 2:1 \bB) 7:3 \bC) 5:2 \bD) 4:3 \bE) 3:2 6 1 1 4 a b Questions 21 to 25, 5 marks each 21. Positive integers xand ysatisfy the equation x 2+2 xy+2 y 2+2y= 1988 What is the largest possible value of x+ y? \bA) 33 \bB) 38\bC) 42 \bD) 46 \bE) 47 22.What fraction of the area of the diagram is shaded? \bA) 1 4 \bB) 4 9 \bC) 5 12 \bD) 1 2 \bE) 5 18 23. Sebastien is playing with a square paper serviette with side length 24 centimetres. He folds it in half along a diagonal to obtain a triangle AB Cwith a right angle at A. He then folds the triangle so that Cends up on line ABat some point D. Suppose that the fold created meets BCat the point X. Sebastien then folds Bto meet Xand notices that the fold created passes through the point D. The distance in centimetres between the points Aand Dis \bA) 6 √ 2 \bB) 12\b √ 3 − 1) \bC) 10 \bD) 12 \bE) 24\b √ 2 − 1) 2021 AUSTRALIAN MATHEMATICS COMPETITIONSENIOR

S5 24.The fraction a 5 is positive and in lowest terms, so that aand bare positive with no common factors. When \b add the integer nto both the numerator and denominator of the fraction a b, the result is double the original fraction. When \b subtract nfrom both the numerator and denominator, the result is triple the original value. The value of nis (A) 13 (B) 18(C) 21 (D) 24 (E) 28 25.A cube has an internal point Psuch that the perpen- dicular distances from Pto the six faces of the cube are 1 cm, 2 cm, 3 cm, 4 cm, 5 cm and 6 cm. How many otherinternal points of the cube have this property? (A) 5 (B) 11 (C) 23 (D) 47 (E) infinitely many P 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. A 70 cm long loop of string is to be arranged into a shape consisting of two adjacent squares, as shown on the left. The side of the smaller square must lie entirely within the side of the larger one, so the example on the right is not allowed. allowed not allowed What is the minimum area of the resulting shape, in square centimetres? 27. How many pairs (m, n ) exist, wheremand nare different divisors of 2310 and n divides m? Both 1 and 2310 are considered divisors of 2310. S6 28. A grid that measures 20 squares tall and 21 squares wide has each o\b its squares painted either green or gold. The diagram shows part o\b the grid, including the top-le\bt corner. The pattern \bollows these rules: • All squares in the le\btmost column are gold. • Only the second-le\btmost square in the top row is green. • For every triplet o\b squares in this orientation , the number o\b gold squares is odd. How many o\b the 20 ×21 = 420 squares are painted green? 29. Starting with a paper rectangle measuring 1 ×√ 2 metres, Sadako makes a single cut to remove the largest square possible, leaving a rectangle. She repeats this process with the remaining rectangle, producing another square and a smaller rectangle. Since √ 2≈ 1.41421356 is irrational, she can in theory keep doing this \borever, pro- ducing an infinite sequence o\b paper squares. To the nearest centimetre, what would be the total perimeter o\b this infinite pile o\b squares? 30. An elastic band is wound around a deck o\b playing cards three times so that three horizontal stripes are \bormed on the top o\b the deck, as shown on the le\bt. Ignoring the different ways the rubber band could overlap itsel\b, there are essentially two different patterns it could make on the under side o\b the deck, as shown on the right. 4 ♣ 4 ♣ ♣ ♣ ♣ ♣ or 4 ♣ 4 ♣ ♣ ♣ ♣ ♣ Treating two patterns as the same i\b one is a 180 ◦rotation o\b the other, how many different patterns are possible on the under side o\b the deck i\b the rubber band is wound around to \borm seven horizontal stripes on top? 2021 AUSTRALIAN MATHEMATICS COMPETITION SENIOR

S5 24. The fraction a b is positive and in lowest terms, so that aand bare positive with no common factors. When \b add the integer nto both the numerator and denominator of the fraction a b, the result is double the original fraction. When \b subtract nfrom both the numerator and denominator, the result is triple the original value. The value of nis (A) 13 (B) 18(C) 21 (D) 24 (E) 28 25. A cube has an internal point Psuch that the perpen- dicular distances from Pto the six faces of the cube are 1 cm, 2 cm, 3 cm, 4 cm, 5 cm and 6 cm. How many otherinternal points of the cube have this property? (A) 5 (B) 11 (C) 23 (D) 47 (E) infinitely many P 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. A 70 cm long loop of string is to be arranged into a shape consisting of two adjacent squares, as shown on the left. The side of the smaller square must lie entirely within the side of the larger one, so the example on the right is not allowed. e00u,o3 suF e00u,o3 g5eF qi F5o aqsqapa ehoe ut F5o hoip0Fqsm i5enod qs i‹peho rosFqaoFhoi† 27. How many pairs (m, n ) exist, wheremand nare different divisors of 2310 and n divides m? Both 1 and 2310 are considered divisors of 2310. S6 28.A grid that measures 20 squares tall and 21 squares wide has each o\b its squares painted either green or gold. The diagram shows part o\b the grid, including the top-le\bt corner. The pattern \bollows these rules: • All squares in the le\btmost column are gold. • Only the second-le\btmost square in the top row is green. • For every triplet o\b squares in this orientation ( aco wsiCoe W\b hWlr ndsteon fn Wrrb 2W, itw3 W\b aco pv ×21 = 420 squares are painted green? 29. Starting with a paper rectangle measuring 1 ×√ 2 metres, Sadako makes a single cut to remove the largest square possible, leaving a rectangle. She repeats this process with the remaining rectangle, producing another square and a smaller rectangle. Since √ 2 ≈ 1.41421356 is irrational, she can in theory keep doing this \borever, pro- ducing an infinite sequence o\b paper squares. To the nearest centimetre, what would be the total perimeter o\b this infinite pile o\b squares? 30. An elastic band is wound around a deck o\b playing cards three times so that three horizontal stripes are \bormed on the top o\b the deck, as shown on the le\bt. Ignoring the different ways the rubber band could overlap itsel\b, there are essentially two different patterns it could make on the under side o\b the deck, as shown on the right. 4 ♣ 4 ♣ ♣ ♣ ♣ ♣ or 4 ♣ 4 ♣ ♣ ♣ ♣ ♣ Treating two patterns as the same i\b one is a 180 ◦rotation o\b the other, how many different patterns are possible on the under side o\b the deck i\b the rubber band is wound around to \borm seven horizontal stripes on top? 2021 AUSTRALIAN MATHEMATICS COMPETITIONSENIOR

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 Senior Ye a r s 1 1 –1 2 (AUSTRALIAN  SCHOOL YEARS)