2021 澳洲 AMC E-Senior 真题 答案 详解

---

中英双语真题

2021 AUSTRALIAN MATHEMATICS COMPETITION
SENIOR

Senior Division

1-10 Questions, 3 marks each
Questions 1 to 10, 3 marks each

1. Each small triangle is the same size. What fraction of the largest triangle is shaded?
   (A) ( \frac{1}{2} )
   (B) ( \frac{1}{3} )
   (C) ( \frac{1}{4} )
   (D) ( \frac{2}{5} )
   (E) ( \frac{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\frac{11}{12})
   (D) 11
   (E) (11\frac{1}{12})

4. In this diagram, what is the size of the angles marked θ?
   (A) 70°
   (B) 75°
   (C) 80°
   (D) 85°
   (E) 90°

5. The value of ( \frac{(20 \times 21)+21}{21} ) is
   (A) 20
   (B) 21
   (C) 22
   (D) 41
   (E) 42

英文真题

1. The parallelogram shown has an area of 48 square units. The value of ( a ) is
   (A) 7
   (B) 8
   (C) 9
   (D) 10
   (E) 11

2. Mervin is allowed to paint the four walls and the ceiling of his rectangular bedroom as he wishes, subject 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, have a denominator less than 6:
[ \frac{1}{5}, \frac{1}{3}, \frac{2}{5}, \frac{2}{3}, \frac{3}{4}, \frac{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) (\frac{21}{20})
(D) (\frac{27}{20})
(E) (\frac{29}{20})

1. In autumn, Tilly's meadow changes rapidly with the weather.
   The number of flowering plants starts at 150000 but they are dying off so each week the number halves. 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

2. A formula in physics is given as:
   [ r = \frac{mV}{qB} ]
   If ( q ) was trebled, ( m ) was halved and ( r ) and ( B ) remained the same, then ( V ) would
   (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

真题文字详解(英文)

66 2021 AMC Senior Solutions Senior Solutions 1. Of the 16 small triangles, 6 are shaded. As a fraction, $$\frac{6}{16} = \frac{3}{8}$$ hence (E). 2. 2021 ÷ 7 = 288 remainder 5, hence (D). 3. 12 - $$\frac{1}{12}$$ - 1 = 11 - $$\frac{1}{12}$$ = 10$$\frac{11}{12}$$, hence (C). 4. The angles in a quadrilateral sum to 360°, so that 3θ + 150 = 360. Then 3θ = 210 so θ = 70, hence (A). 5. $$\frac{(20 × 21) + 21}{21} = \frac{21 × 21}{21} = 21$$, hence (B). 6. The time Henry took in minutes was 3$$\frac{3}{4}$$ = $$\frac{15}{4}$$. In hours, this is $$\frac{15}{4} × \frac{1}{60} = \frac{1}{16}$$. His average speed in kilometres per hour was $$\frac{3}{2} ÷ \frac{1}{16} = \frac{3}{2} × \frac{16}{1} = 24$$, hence (C). 7. $$\frac{8^3 × 3^6}{6^5} = \frac{(2^3)^3 × 3^6}{(2 × 3)^5} = \frac{2^9 × 3^6}{2^5 × 3^5} = 2^{9-5} × 3^{6-5} = 2^4 × 3 = 48$$ hence (B). 8. The product of $$\frac{1}{3}$$ = 0.3 and $$\frac{2}{3}$$ = 0.6 is $$\frac{1}{3} × \frac{2}{3} = \frac{2}{9}$$. As a decimal this is 0.2, hence (B). 9. The area of the parallelogram is the base times the height perpendicular to the base. From the coordinates given, the base is parallel to the x-axis, of length (a − 1). The height is then parallel to the y-axis, of length 8 − 2 = 6. Hence 6(a − 1) = 48, so a − 1 = 8 and a = 9, hence (C). 10. Let the four walls be north, south, east and west. There are three choices for the ceiling colour. Then the north wall must be a second colour, so there are two choices for the north wall. Then there is no further choice: the east and west walls will be the third colour, and the south wall the second colour. In all there are 3 × 2 = 6 ways that the room can be painted, hence (C). www.amt.edu.au