[PDF] 2020 Australian Mathematics Competition AMC Senior Years 11 and 12.pdf | Plain Text

Copyright © 2020 Australian Mathematics TrustACN 083 950 341 2020 AMC AUSTRALIAN MATHEMATICS COMPETITION Senior Years 11–12 (Australian school years) THURSDAY 30 JULY 2020 NAME TIME ALLOWED: 75 MINUTES 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 different 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 official status to their score. Reminder: You may sit this competition once, in one division only, or risk no score.

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S4 18.The rectangle OAB Cis drawn in the quadrant of a circle ODE, so that AD= 2 and CE= \b. What is the radius of the circle? (A) 11 (B) 13(C) 15 (D) 17 (E) 20 O D B E A C 2 \b 19. The minimum value of the function f(x)=2 x2−2x−3 is (A) 1 (B) 1 2 (C) 1 4 (D) 1 8 (E) 1 16 20.Two sides of a regular pentagon are extended to create a triangle. Inside this triangle, a smaller regular pentagon is drawn, as shown. In area, how many times bigger is the larger pentagon than the smaller pentagon? (A) 4 (B) 2√ 5 (C) 5 (D) √5+3 2 (E) √ 5 Questions 21 to 25, 5 marks each 21. Fo rn≥ 1, s nis defined to be the number consisting of nconsecutive ones, so s 1= 1, s 2= 11, s 3= 111, and so on. Which one of the following numbers is divisible by 7? (A) s 902 (B)s 903 (C) s 904 (D) s 905 (E)s 906 22. A circle is inscribed in the quadrilateral AB C D so that it touches all four sides, as shown. Sides AB and DC are parallel with lengths 2 cm and 4 cm, respectively, and sides ADand BChave equal length. What, in centimetres, is the length of AC? (A) √ 17 (B) 2 √ 5 (C)√ 13 (D) 5 (E) 3√ 2 D 4 cm> C — B 2 cm > A — 2020 Australian Mathematics Competition — Senior

S5 23.A rectangular sheet of paper that is three times as tall as it is wide is folded along one diagonal, making the pentagon shown\b What is the ratio of the area of this pentagon to the area of the original rectangle? (A) 13 : 18 (B) 3 : 4(C) 7 : 12 (D) 2 : 3 (E)√ 10 : 4 24. Alex writes down the value of the following sum, where the final term is the number consisting of 2020 consecutive nines: 9 + 99 + 999 + 9999 + ∙∙∙+ 99 ...9    2019 nines + 99 ...9    2020 nines How many times does the digit 1 appear in the answer? (A) 0 (B) 2016 (C) 2018 (D) 2020 (E) 2021 25. Three real numbers a,band care such that a + b+ c= 4 and 1 a + b+ 1 b + c+ 1 c + a=5 Then, c a + b+ a b + c+ b c + ais equal to (A) 3 2 (B) 4 5 (C) 2 (D) 20 (E) 17 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 different integer from 1 to 10 is placed on each of the faces of a cube\b Each vertex is then assigned a number which is the sum of the numbers on the three faces which touch that vertex\b Only the vertex numbers are shown here\b What is the product of the 4 smallest face numb ers? 19 14 14 9 21 16 21 26 2020 Australian Mathematics Competition — Senior

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