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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 3–5 August Senior Ye a r s 1 1 –1 2 (AUSTRALIAN SCHOOL YEARS)
Senior Division Questions 1 to 10, 3 marks each 1. The temperature in the mountains was 4 ◦C but dropped overnight by 7 ◦C. \bhat was the temperature in the morning? (A) 3 ◦C (B) 11 ◦C (C) −3 ◦C (D) −4 ◦C (E) −11 ◦C 2. The rectangle shown has area 135. \bhat is the value of hin the diagram? (A) 7 (B) 9(C) 11 (D) 13 (E) 15 15h Area = 135 3. The value of 5 1+4 2+3 3+2 4+1 5is (A) 20 (B) 30(C) 35 (D) 50 (E) 65 \b. 1 he − 1 22 = (A) 1 2 (B) 1 42 (C) − 1 440 (D) − 1 44 (E) 1 220 5. Three vertices of a rectangle are the points (1, 4), (7, 4) and (1 ,8). At which point do the diagonals of the rectangle cross? (A) (4, 6) (B) (3, 2) (C) (3, 1) (D) (5, 6) (E) (7,8) 6.\bhat fraction of this trapezium is shaded? (A) 1 2 (B) 1 3 (C) 1 4 (D) 1 5 (E) 4 15 2 12 1 11 7. The sum of the numbers on these six cards is 4 √ 5. Lily remo\bes one of them. What is the largest possible sum of the remaining fi\be cards? (A) 4 √ 5+3 √ 7 (C) 4 √ 5− 2√ 7 (B) 4 √ 5+7 √ 7 (D) 4 √ 5− 5√ 7 (E) 4 √ 5+4 √ 7 −4√ 7 −3√ 7 v√ 7 f√ 7 √5 c√ 5 8. In this diagram, what is the \balue of x? (A) 65 (B) 70 (C) 75 (D) 80 (E) 85 x ◦ x◦140 ◦ 40 ◦ 9.The smaller of these dice has three zeroes and three ones on its faces and the larger has the numbers 1, 3, 5, 7, 9, and 11 on its faces. Both dice are rolled once and the numbers showing on top are added. What is the probability of obtaining a sum of 12? (A) 0 (B) 1 6 (C) 1 9 (D) 1 12 (E) 1 36 7 hh e 0 h h 10. The \balue of 24 −1 2 + 17 −1 4 is closest to (A) 0.2 (B) 0.5(C) 0.7 (D) 1(E) 1.2 Questions 11 to 20, 4 marks each 11. The operation is defined by p q=2 p− q. Each of pand qis an integer from −6 to +6. How many pairs of \balues (p, q ) will ha\bep q= q p? (A) 6 (B) 13 (C) 36 (D) 49 (E) 169 12. In the diagram, the lengths of the sides of the triangle are 8, 9 and 13 centimetres. The centres of the circles are at the \bertices of the triangle and the circles just touch. The radius, in centimetres, of the largest circle is (A) 6 (B) 6.5 (C) 7 (D) 7.5 (E) 8 2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR
7.The sum of the numbers on these six cards is 4 √ 5. Lily remo\bes one of them. What is the largest possible sum of the remaining fi\be cards? (A) 4 √ 5+3√ 7 (C) 4 √ 5 − 2√ 7 (B) 4 √ 5+7√ 7 (D) 4 √ 5 − 5√ 7 (E) 4 √ 5+4√ 7 − 4√ 7 − 3√ 7 5√ 7 2√ 7 √5 3√ 5 8. In this diagram, what is the \balue of x? (A) 65 (B) 70 (C) 75 (D) 80 (E) 85 x◦ x◦140 ◦ 40 ◦ 9.The smaller of these dice has three zeroes and three ones on its faces and the larger has the numbers 1, 3, 5, 7, 9, and 11 on its faces. Both dice are rolled once and the numbers showing on top are added. What is the probability of obtaining a sum of 12? (A) 0 (B) 1 − (C) 1 9 (D) 1 12 (E) 1 36 7 11 3 0 1 1 10. The \balue of 24 −11+ 17 −14 is closest to (A) 0.2 (B) 0.5(C) 0.7 (D) 1(E) 1.2 Questions 11 to 20, 4 marks each 11. The operation is defined by p q=2 p− q. Each of pand qis an integer from −6 to +6. How many pairs of \balues (p, q ) will ha\bep q= q p? (A) 6 (B) 13 (C) 36 (D) 49 (E) 169 12.In the diagram, the lengths of the sides of the triangle are 8, 9 and 13 centimetres. The centres of the circles are at the \bertices of the triangle and the circles just touch. The radius, in centimetres, of the largest circle is (A) 6 (B) 6.5 (C) 7 (D) 7.5 (E) 8 2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR
13.The four integers 3, 4, 8, 11 have their mean and range calculated\b A fifth integer is then included that is different from the other four\b This doesn’t change the range, but the mean is now an integer\b What is this new mean? (A) 4 (B) 5(C) 6 (D) 7 (E) 8 14.Two triangles AB C and BCD are both right angled as shown\b Lines ABand CD are parallel\b Also AC= 1 and AB= 2\b What is the length of CD? (A) √ 2 (B) 3 2 (C)√ 3 (D) 16 9 (E) 5 3 A C B D > 2 > 1 15. The value of xin the equation 3 x+3 x+1 +3 x+2 = 13 √ 3 is (A) 0 (B) 1 2 (C) 13 √ 3 − 9 9 (D) √ 3 (E) 13√ 3 − 3 16. Four numbers are equally spaced on the number line, in the given order 1 †‰, 1 22,X,Y What is the value of Y? (A) 1 26 (B) 2 21 (C) 3 44 (D) 2 55 (E) 3 26 17. A small equilateral triangle sits inside a larger equi- lateral triangle as shown\b What is the ratio of the areas of the smaller and larger equilateral triangles? (A)1:3 (B) 7 : 16 (C) 9 : 16 (D) 37 : 64 (E) 21 : 32 626 2 6 2 18. The square pyramid shown is divided into 3 pieces, P, Q and R, by two planes that are parallel to the square base\b Each of the 3 pieces has the same height\b The volume of the piece Pis 5 cm 3\b What is the volume of piece R, in cm 3? (A) 15 (B) 25 (C) 40 (D) 95 (E) 125 P Q R 19. A sequence of values a 1,a 2, \b\b\b, a 100 is calculated as follows: a 1=1 ,a 2=2,a 3=a 2+1 a 1 ,...a n= a n −1 +1 a n −2 ,...a 100 = a 99 +1 a 98 What is a 100 ? (A) 1 (B) 2(C) 3 (D) 5 (E) 50 20. A triangular ramp is in the shape of a right-angled tetra- hedron\b The horizontal base is an equilateral triangle with sides 8 metres\b The apex is 1 metre directly above one corner of the base, so that two faces are vertical\b In square metres, what is the area of the sloping face? (A) 16 √ 3 (B) 28 (C)65 4 √ 3 (D) 4 √ 33 (E) 32 8 1 | | | Questions 21 to 25, 5 marks each 21. A cuboctahedron is a solid formed by joining the midpoints of the edges of a cube as shown\b What is the volume of a cuboctahedron of side length 2? (A) 10 √ 2 3 (B) 40 √ 2 3 (C) 8 √ 2 (D) 6 √ 2+8 √ 3 (E) 8 √ 2+ 8 3√ 3 2 2 2 2 2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR
13.The four integers 3, 4, 8, 11 have their mean and range calculated\b A fifth integer is then included that is different from the other four\b This doesn’t change the range, but the mean is now an integer\b What is this new mean? (A) 4 (B) 5(C) 6 (D) 7 (E) 8 14. Two triangles AB C and BCD are both right angled as shown\b Lines ABand CD are parallel\b Also AC= 1 and AB= 2\b What is the length of CD? (A) √ 2 (B) 3 2 (C) √ 3 (D) 16 9 (E) 5 3 A C B D > 2 > 1 15. The value of xin the equation 3 x+3 x+1 +3 x+2 = 13 √ 3 is (A) 0 (B) 1 2 (C) 13 √ 3− 9 9 (D) √ 3 (E) 13 √ 3− 3 9 16. Four numbers are equally spaced on the number line, in the given order 1 20 , 1 22 ,X,Y What is the value of Y? (A) 1 26 (B) 2 21 (C) 3 44 (D) 2 55 (E) 3 26 17. A small equilateral triangle sits inside a larger equi- lateral triangle as shown\b What is the ratio of the areas of the smaller and larger equilateral triangles? (A)1:3 (B) 7 : 16 (C) 9 : 16 (D) 37 : 64 (E) 21 : 32 626 2 6 2 18. The square pyramid shown is divided into 3 pieces, P, Q and R, by two planes that are parallel to the square base\b Each of the 3 pieces has the same height\b The volume of the piece Pis 5 cm 3\b What is the volume of piece R, in cm 3? (A) 15 (B) 25 (C) 40 (D) 95 (E) 125 P 3 . 19. A sequence of values a 1,a 2, \b\b\b, a 100 is calculated as follows: a 1=1 ,a 2=2,a 3=a 2+1 a1 ,...a n=a n −1 +1 an−2 ,...a 100 = a 99 +1 a98 What is a 100 ? (A) 1 (B) 2(C) 3 (D) 5 (E) 50 20.A triangular ramp is in the shape of a right-angled tetra- hedron\b The horizontal base is an equilateral triangle with sides 8 metres\b The apex is 1 metre directly above one corner of the base, so that two faces are vertical\b In square metres, what is the area of the sloping face? (A) 16 √ 3 (B) 28 (C)65 4√ 3 (D) 4 √ 33 (E) 32 ˆ • | | | Questions 21 to 25, 5 marks each 21. A cuboctahedron is a solid formed by joining the midpoints of the edges of a cube as shown\b What is the volume of a cuboctahedron of side length 2? (A) 10 √ 2 (B) 40√ 2 (C) 8 √ 2 (D) 6 √ 2+8√ 3 (E) 8 √ 2+ 8 3 √3 • • • • 2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR
22.In the two equations ax−b= cand dy+e= f, each of the letters a,b,c, d, e, and f is replaced by a di\berent digit from 1 to 9. When the two equations are solved for xand y, the lowest possible value of x+ yis (A) less than −7 (B) between −7 and −5 (C) between −5 and −3 (D) between −3 and −1 (E) greater than−1 23. A semicircle is inscribed in a right-angled isosceles triangle and a square is inscribed in the semicircle as shown. What is the ratio of the area of the square to the area of the triangle? (A)1:3 (B)1: √ 2 (C) 2 : 5 (D) 1 : 2 √ 2 (E) 3 : 8 24. In the grid shown, the numbers 1 to 8 are placed so that when joined in ascending order they make a trail. The trail moves from one square to an adjacent square but does not move diagonally. In how many ways can the numbers 1 to 8 be placed in the grid to give such a trail? 1 4 5 2 3 8 7 6 (A) 12 (B) 20(C) 24 (D) 28 (E) 36 25.When I cycled around the lake yesterday, my children Sally and Wally decided to ride the same route in the opposite direction. We all set o\b at the same time, from the same point, and finished at that same spot. We each rode at our own steady speed. It took me 77 minutes. Sally and I passed each other, waving, exactly 42 minutes after we started. Precisely 2 minutes later, Wally and I passed each other, puffing. To the nearest minute, how much longer did Wally take than Sally to ride around the lake? (A) 2 (B) 4(C) 6 (D) 8 (E) 10 2.3 45. . 3 3 3. . \b \b3. . 3 5. 3 .3 • 3• 3\b•• Inthne woq o tuaisot wu=oane nd otuo +,c fnt uolw lwnrlu nd hwtuu puthrluq nd hwu wu=oaneb wu ytrhuq \bnye hwu otuo nd hwu htroeasu yrhw hwuqu hwtuu puthrluqc gwoh rq hwu qim nd hwu 1, otuoq hwoh Inthne ytrhuq \bnye9 axb −√ √ √ √ √ √ √ \b \b \b \b √ acb √ √ √√ √ √ √ \b √ √ \b √ • √ ••• adb • \b •√ • √ \b •• • √ √ √ - √ • √ •€‚ƒ„ •√ √ • √ √ √ • \b … √ •€‚ƒ„ √ \b - •√ \b \b √ …√ √ √ √ √ † √ ‡ −\b •√ √ √ ˆ - √ … •€‚ƒ„ √ yeb − ‰ € √ • √ Š √ √ ‹ − √ √ ‰ € √ • √ Š Œ √ … • √ √ √ √ √ Œ \b √√ √ √ • \b √ √√ √ √ • \b Ž √ √\b√ √ √ √√ √ √ 2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR
22.In the two equations ax−b= cand dy+e= f, each of the letters a,b,c, d, e, and f is replaced by a di\berent digit from 1 to 9. When the two equations are solved for xand y, the lowest possible value of x+ yis (A) less than −7 (B) between −7 and −5 (C) between −5 and −3 (D) between −3 and −1 (E) greater than−1 23. A semicircle is inscribed in a right-angled isosceles triangle and a square is inscribed in the semicircle as shown. What is the ratio of the area of the square to the area of the triangle? (A)1:3 (B)1: √ 2 (C) 2 : 5 (D) 1 : 2 √ 2 (E) 3 : 8 24. In the grid shown, the numbers 1 to 8 are placed so that when joined in ascending order they make a trail. The trail moves from one square to an adjacent square but does not move diagonally. In how many ways can the numbers 1 to 8 be placed in the grid to give such a trail? 1 4 5 2 3 8 7 6 (A) 12 (B) 20(C) 24 (D) 28 (E) 36 25.When I cycled around the lake yesterday, my children Sally and Wally decided to ride the same route in the opposite direction. We all set o\b at the same time, from the same point, and finished at that same spot. We each rode at our own steady speed. It took me 77 minutes. Sally and I passed each other, waving, exactly 42 minutes after we started. Precisely 2 minutes later, Wally and I passed each other, puffing. To the nearest minute, how much longer did Wally take than Sally to ride around the lake? (A) 2 (B) 4(C) 6 (D) 8 (E) 10 2.3 45. . 3 3 3. . \b \b3. . 3 5. 3 .3 • 3• 3\b•• Inthne woq o tuaisot wu=oane nd otuo +,c fnt uolw lwnrlu nd hwtuu puthrluq nd hwu wu=oaneb wu ytrhuq \bnye hwu otuo nd hwu htroeasu yrhw hwuqu hwtuu puthrluqc gwoh rq hwu qim nd hwu 1, otuoq hwoh Inthne ytrhuq \bnye9 axb −√ √ √ √ √ √ √ \b \b \b \b √ acb √ √ √√ √ √ √ \b √ √ \b √ • √ ••• adb • \b •√ • √ \b •• • √ √ √ - √ • √ •€‚ƒ„ •√ √ • √ √ √ • \b … √ •€‚ƒ„ √ \b - •√ \b \b √ …√ √ √ √ √ † √ ‡ −\b •√ √ √ ˆ - √ … •€‚ƒ„ √ yeb − ‰ € √ • √ Š √ √ ‹ − √ √ ‰ € √ • √ Š Œ √ … • √ √ √ √ √ Œ \b √√ √ √ • \b √ √√ √ √ • \b Ž √ √\b√ √ √ √√ √ √ 2022 AUSTRALIAN MATHEMATICS COMPETITION SENIOR
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 Senior Ye a r s 1 1 –1 2 (AUSTRALIAN SCHOOL YEARS)