[PDF] 2020 Australian Mathematics Competition AMC Intermediate Years 9 and 10.pdf | Plain Text

Copyright © 2020 Australian Mathematics TrustACN 083 950 341 2020 AMC AUSTRALIAN MATHEMATICS COMPETITION Intermediate Years 9–10 (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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I4 18. Two sides of a regular hexagon are extended to create a small triangle. Inside this triangle, a smaller regular hexagon is drawn, as shown. In area, how man\b times bigger is the larger hexagon than the smaller hexagon? (A) 4 (B) 6(C) 8 (D) 9 (E) 12 19.The numb er 1 × 2× 3× 4× 5× 6× 7× 8× 9× 10 n is a perfect square. What is the smallest possible value of n? (A) 7 (B) 14 (C) 21 (D) 35 (E) 70 20.In the triangle AB Cshown, Dis the midpoint of AC,E is the midpoint of BD andFis the midpoint of AE. If the area of triangle BEFis 5, what is the area of triangle AB C? (A) 30 (B) 35 (C) 40 (D) 45 (E) 50 B C A D E F Questions 21 to 25, 5 marks each 21. A scientist measured the amount of bacteria in a Petri dish over several weeks and also recorded the temperature and humidit\b for the same time period. The results are summarised in the following graphs. Week Temperature A B C D E Humidity Bacteria Temperature Humidity During which week was the bacteria population highest? (A) week A (B) week B (C) week C (D) week D (E) week E 2020 Australian Mathematics Competition — Intermediate

I5 22. Five friends read a total of 40 books between them over the holidays\b Everyone read at least one book but no-one read the same book as anyone else\b Asilata read twice as many books as Eammon\b Dane read twice as many as Bettina\b Collette read as many as Dane and Eammon put together\b Who read exactly eight books? (A) Asilata (B) Bettina (C) Colette (D) Dane (E) Eammon 23. There are 5 sticks of length 2 cm, 3 cm, 4 cm, 5 cm and 8 cm\b Three sticks are chosen randomly\b What is the probability that a triangle can be formed with the chosen sticks? (A) 0\b25 (B) 0\b3(C) 0\b4 (D) 0\b5 (E) 0\b6 24.Five squares of unit area are circumscribed by a circle as shown\b What is the radius of the circle? (A) 3 5 (B) 2 √ 5 I (C) √10 5 (D) √13 5 (E) √185 e 25. 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 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. Ifnis a positive integer, n! is found by multiplying the integers from 1 to n\b Fo r example, 4! = 4 ×3× 2× 1 = 24\b What are the three rightmost digits of the sum 1! + 2! + 3! + ∙∙∙+ 2020! ? 2020 Australian Mathematics Competition — Intermediate

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2020 AMC — INTERMEDIATE