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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 Intermediate Ye a r s 9 –1 0 (AUSTRALIAN  SCHOOL YEARS) 3–5 August

Intermediate Division Questions 1 to 10, 3 marks each 1. 2220 −2022 = (A) 18 (B) 188 (C) 198 (D) 200 (E) 202 \b.This shape is buil\b from 29 squares, each 1 cm ×1 cm. Wha\b is i\bs perime\ber in cen\bime\bres? (A) 52 (B) 58(C) 60 (D) 68 (E) 72 3.A digi\bal clock shows \bhe \bime as 20:22. In how many minu\bes will i\b be midnigh\b? (A) 158 (B) 218 (C) 258 (D) 278 (E) 378 4. The value of 1 2+2 2 32+4 2 is (A) 1 5 (B) 3 7 (C) 9 49 (D) 4 (E) 5 5.In \bhe \briangle P QRshown, PQ=PR and ∠QP R = 48 ◦. Wha\b is ∠P QR? (A) 60 ◦ (B) 66 ◦ (C) 72 ◦ (D) 78 ◦ (E) 84 ◦ P Q R 48 ◦ 6. Wha\b frac\bion of \bhis rec\bangle is shaded? (A) 1 2 (B) 5 8 (C) 5 6 (D) 2 3 (E) 7 12 12 8 7 7. (0.4) 2+ (0. 1) 2= (A) 0.25 (B) 1.7(C) 0.17 (D) 1(E) 0.26 8. Austral\ba uses 160 m\bll\bon l\btres of petrol each day. There \bs enough petrol stored to last 60 days. How much more petrol does Austral\ba need to buy to have enough stored for 90 days? (A) 4 m\bll\bon l\btres (B) 4.8 m\bll\bon l\btres (C) 480 m\bll\bon l\btres (D) 160 m\bll\bon l\btres (E) 4800 m\bll\bon l\btres 9. 2022 2 − 20223 = (A) 337 (B) 674 (C) 2022 (D) −2022 (E) −674 10. Wh\bch algebra\bc term should replace \bn the equat\bon below?  + + = 27 x 3y6 (A) 3 xy 2 (B) 3x 3y6 (C) 9 xy 2 (D) 9x 3y6 (E) 27 xy 2 Questions 11 to 20, 4 marks each 11. Three vert\bces of a rectangle are the po\bnts (1 ,4), (7, 4) and (1, 8). At wh\bch po\bnt do the d\bagonals of the rectangle cross? (A) (4 ,6) (B) (3, 2) (C) (3, 1) (D) (5, 6) (E) (7,8) 12.What value should be placed \bn the box to sat\bsfy the equat\bon? 5 I1 )A ) )A 1 (A) 1 (B) 1 12 (C) 2(D) 2 12 (E) 3 2022 AUSTRALIAN MATHEMATICS COMPETITION INTERMEDIATE

7.(0.4) 2+ (0. 1) 2= (A) 0.25 (B) 1.7(C) 0.17 (D) 1(E) 0.26 8.Austral\ba uses 160 m\bll\bon l\btres of petrol each day. There \bs enough petrol stored to last 60 days. How much more petrol does Austral\ba need to buy to have enough stored for 90 days? (A) 4 m\bll\bon l\btres (B) 4.8 m\bll\bon l\btres (C) 480 m\bll\bon l\btres (D) 160 m\bll\bon l\btres (E) 4800 m\bll\bon l\btres 9. 2022 ( − 2022 3 = (A) 337 (B) 674 (C) 2022 (D) −2022 (E) −674 10. Wh\bch algebra\bc term should replace \bn the equat\bon below?  + + = 27 x 3y6 (A) 3 xy 2 (B) 3x 3y6 (C) 9 xy 2 (D) 9x 3y6 (E) 27 xy 2 Questions 11 to 20, 4 marks each 11. Three vert\bces of a rectangle are the po\bnts (1 ,4), (7, 4) and (1 ,8). At wh\bch po\bnt do the d\bagonals of the rectangle cross? (A) (4 ,6) (B) (3, 2) (C) (3, 1) (D) (5, 6) (E) (7,8) 12.What value should be placed \bn the box to sat\bsfy the equat\bon? 5 3= 1+ 1 1+ 1 (A) 1 (B) 1 12 (C) 2(D) 2 12 (E) 3 2022 AUSTRALIAN MATHEMATICS COMPETITION INTERMEDIATE

13.The angles of a triangle are in the ratio 2 : 3 : 4. What is the size of the largest angle in degrees? (\b) 40 (B) 45(C) 72 (D) 80 (E) 90 14.In this puzzle, each circle should contain an integer. Each of the five lines of four circles should add to 40. When the puzzle is completed, what is the largest number used? (\b) 15 (B) 16(C) 17 (D) 18 (E) 19 7 15 8 2 9 15.Daniel and Luke arrange to meet at a cafe. Luke leaves work, walking at 6 km/h. Five minutes later, Daniel starts cycling from his flat at 20 km/h. \b further 15 minutes later, both arrive at the cafe at the same time. What is the total distance they travelled? (\b) 5.5 km (B) 6 km (C) 6.5 km (D) 7 km (E) 7.5 km 16. My family of 7 adults and 5 children gather each year to celebrate Chinese Mid- \butumn festival. Each adult gives one gift to everyone else. Each child gives one gift to every other child. How many gifts are given? (\b) 78 (B) 85(C) 97 (D) 102 (E) 109 17.This spinner is spun twice to form a two-digit number using the following rules: • The first value spun forms the tens digit of a number. • If the second value rolled is larger than the first, it becomes the units digit of a number. • If the second value spun is not larger than the first, the tens digit is repeated as the units digit. What is the probability that the resulting two-digit number is divisible by 11? 1 h e a (\b) 1 h (B) 3 8 (C) 1 2 (D) 5 8 (E) 9 16 13.Thean glsoftrrhie 2ihn :32tos: tos 4ot.i ng rogW 42zs t dg.so :atrs t: :ag.i? (as rton: .asos n.g :32tos: glsoftr tos h:g:zsfs: nohtiefs:? \batn h: nas ngntf :at4s4 tost) 0B5 C 075 C •3 0D5 8 0E5 8 • 0T5 9 14. Ihzp ti4 uhz :ntons4 fstoihie e2hnto oszsinf,? (.g .ssp: tegc Ihzp at4 fissi fstoihie vls nhms: t: fgie t: uhz? (.g 4t,: tegc Ihzp at4 fissi fstoihie n.hzs t: fgie t: uhz? (g4t,c .atn h: nas i2mfiso gw 4t,: natn Ihzp at: fissi fstoihie rf2: nas i2mfiso gw 4t,: natn uhz at: fissi fstoihie) 0B5 bC 075 190D5 68 0E5 Cb 0T5 8L 56.klso nas ft:n b/ ,sto:c nas :nti4to4 otnhg gw nsfslh:hgi :zossi: at: zaties4 wogm 6 F 1 ng y8 F fl? \basi nas otnhg gw zginsin 4gs:iMn mtnza nas otnhg gw nas :zossi hn h: fishie lhs.s4 gic fiftzp fito: tos gwnsi 2:s4 ng zgmrsi:tnsc t: hff2:notns4? 6 F 1 hmtes gi t y8 F fl :zossi y8F fl hmtes gi t 6 F 1 :zossi 12134  23 2  4   3  2 3  434  23 2  4   3  3 3   3 1 4  4 \b   
• • •  •  2022 AUSTRALIAN MATHEMATICS COMPETITION INTERMEDIATE

13.The angles of a triangle are in the ratio 2 : 3 : 4. What is the size of the largest angle in degrees? (\b) 40 (B) 45(C) 72 (D) 80 (E) 90 14. In this puzzle, each circle should contain an integer. Each of the five lines of four circles should add to 40. When the puzzle is completed, what is the largest number used? (\b) 15 (B) 16(C) 17 (D) 18 (E) 19  † •  • 15. Daniel and Luke arrange to meet at a cafe. Luke leaves work, walking at 6 km/h. Five minutes later, Daniel starts cycling from his flat at 20 km/h. \b further 15 minutes later, both arrive at the cafe at the same time. What is the total distance they travelled? (\b) 5.5 km (B) 6 km (C) 6.5 km (D) 7 km (E) 7.5 km 16. My family of 7 adults and 5 children gather each year to celebrate Chinese Mid- \butumn festival. Each adult gives one gift to everyone else. Each child gives one gift to every other child. How many gifts are given? (\b) 78 (B) 85(C) 97 (D) 102 (E) 109 17. This spinner is spun twice to form a two-digit number using the following rules: • The first value spun forms the tens digit of a number. • If the second value rolled is larger than the first, it becomes the units digit of a number. • If the second value spun is not larger than the first, the tens digit is repeated as the units digit. What is the probability that the resulting two-digit number is divisible by 11? 1 2 3 4 (\b) 1 4 (B) 3 8 (C) 1 2 (D) 5 8 (E) 9 16 13.Thean glsoftrrhie 2ihn :32tos: tos 4ot.i ng rogW 42zs t dg.so :atrs t: :ag.i? (as rton: .asos n.g :32tos: glsoftr tos h:g:zsfs: nohtiefs:? \batn h: nas ngntf :at4s4 tost) 0B5 C 075 C •3 0D5 8 0E5 8 • 0T5 9 14. Ihzp ti4 uhz :ntons4 fstoihie e2hnto oszsinf,? (.g .ssp: tegc Ihzp at4 fissi fstoihie vls nhms: t: fgie t: uhz? (.g 4t,: tegc Ihzp at4 fissi fstoihie n.hzs t: fgie t: uhz? (g4t,c .atn h: nas i2mfiso gw 4t,: natn Ihzp at: fissi fstoihie rf2: nas i2mfiso gw 4t,: natn uhz at: fissi fstoihie) 0B5 bC 075 190D5 68 0E5 Cb 0T5 8L 56.klso nas ft:n b/ ,sto:c nas :nti4to4 otnhg gw nsfslh:hgi :zossi: at: zaties4 wogm 6 F 1 ng y8 F fl? \basi nas otnhg gw zginsin 4gs:iMn mtnza nas otnhg gw nas :zossi hn h: fishie lhs.s4 gic fiftzp fito: tos gwnsi 2:s4 ng zgmrsi:tnsc t: hff2:notns4? 6 F 1 hmtes gi t y8 F fl :zossi y8F fl hmtes gi t 6 F 1 :zossi 12134  23 2  4   3  2 3  434  23 2  4   3  3 3   3 1 4  4 \b   
• • •  •  2022 AUSTRALIAN MATHEMATICS COMPETITION INTERMEDIATE

Questions 21 to 25, 5 marks each 21. A triangular ramp is in the shape of a right-angled tetra- hedron. The horizontal base is an equilateral triangle \bith sides 8 metres. The apex is 1 metre directly above one corner of the base, so that t\bo faces are vertical. In square metres, \bhat is the area of the sloping face? (A) 16 √ 3 (B) 28 (C)65 4√ 3 (D) 4 √ 33 (E) 32   | | | 22. What is the value of the follo\bing expression \bhen n= 2022? 3  (1 ×2× 4) + (2 ×4× 8) + ∙∙∙+(n× 2n × 4n) (1 ×3× 9) + (2 ×6× 18) + ∙∙∙+(n× 3n × 9n) (A) 1 2 (B) 2 3 (C) 3 4 (D) 4 5 (E) 5 6 23. Lisa has a mixture of \bater and milk in a drum in the ratio 5 : 7. She accidentally spills 9 L of this mixture. She then fills the drum \bith 9 L of \bater. This makes the \bater to milk ratio 9 : 7. Ho\b many litres of milk \bere in the drum originally? (A) 20 (B) 21(C) 24 (D) 36 (E) 40 24.Given that the highest common factor of pand qis t, and q= rt, then the lo\best common multiple of pand q\bill al\bays be equal to (A) pq (B)qr (C)rt (D)pr (E)pt 25. A positive number is \britten in each cell of the 3 ×3 table. In each ro\b and in each column, the product of the numbers is equal to 1. In each 2 ×2 square, the product of the numbers is equal to 2. What is the number in the central cell? (A) 1 (B) 2 (C) 6 (D) 8 (E) 16 ? 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 below, a,band care nonzero digits. 1cab + abc 2 \b22 What is the three-digit number abcin the second line of the sum? 27. When these numbers are multiplied, what is the sum of all digits in the answer? 171×66... 6    111 sixes 28. In how many ways can 1\b\b 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. 29. What is the largest number of distinct elements that you can choose from the set {1 ,2 ,3 ,4 ,..., 1\b\b\b} such that no three of them are the side lengths of a triangle? For example the selection could include 2\b, 22 and 42, since there is no triangle with sides 2\b, 22 and 42. 30. Students sit at their desks in three rows of eight. Felix, the class pet, must be passed to each student exactly once, starting with Alex in one corner and finishing with Bryn in the opposite corner. Each student can pass only to the immediate neighbour left, right, in front or behind. One possible path is shown. How many different paths can Felix take from Alex to Bryn? Ema? =f(t 2022 AUSTRALIAN MATHEMATICS COMPETITION INTERMEDIATE

Questions 21 to 25, 5 marks each 21. A triangular ramp is in the shape of a right-angled tetra- hedron. The horizontal base is an equilateral triangle \bith sides 8 metres. The apex is 1 metre directly above one corner of the base, so that t\bo faces are vertical. In square metres, \bhat is the area of the sloping face? (A) 16 √ 3 (B) 28 (C)65 4 √ 3 (D) 4 √ 33 (E) 32 8 1 | | | 22. What is the value of the follo\bing expression \bhen n= 2022? 3  (1×2× 4) + (2 ×4× 8) + ∙∙∙+(n× 2n × 4n) (1 ×3× 9) + (2 ×6× 18) + ∙∙∙+(n× 3n × 9n) (A) 1 2 (B) 2 3 (C) 3 4 (D) 4 5 (E) 5 6 23. Lisa has a mixture of \bater and milk in a drum in the ratio 5 : 7. She accidentally spills 9 L of this mixture. She then fills the drum \bith 9 L of \bater. This makes the \bater to milk ratio 9 : 7. Ho\b many litres of milk \bere in the drum originally? (A) 20 (B) 21(C) 24 (D) 36 (E) 40 24. Given that the highest common factor of pand qis t, and q= rt, then the lo\best common multiple of pand q\bill al\bays be equal to (A) pq (B)qr (C)rt (D)pr (E)pt 25. A positive number is \britten in each cell of the 3 ×3 table. In each ro\b and in each column, the product of the numbers is equal to 1. In each 2 ×2 square, the product of the numbers is equal to 2. What is the number in the central cell? (A) 1 (B) 2 (C) 6 (D) 8 (E) 16 ? 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. 5, In the sum below, a,band care nonzero digits. 1cab + abc 2 \b22 What is the three-digit number abcin the second line of the sum? 27. When these numbers are multiplied, what is the sum of all digits in the answer? 171×66... 6    111 sixes 28. In how many ways can 1\b\b 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. 29. What is the largest number of distinct elements that you can choose from the set {1 ,2 ,3 ,4 ,..., 1\b\b\b}such that no three of them are the side lengths of a triangle? For example the selection could include 2\b, 22 and 42, since there is no triangle with sides 2\b, 22 and 42. 30. Students sit at their desks in three rows of eight. Felix, the class pet, must be passed to each student exactly once, starting with Alex in one corner and finishing with Bryn in the opposite corner. Each student can pass only to the immediate neighbour left, right, in front or behind. One possible path is shown. How many different paths can Felix take from Alex to Bryn? Alex Bryn 2022 AUSTRALIAN MATHEMATICS COMPETITION INTERMEDIATE

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