2015 澳洲 AMC E-Senior 真题 答案 详解

中英双语真题

Advanced Paper
1-10 questions, 3 points each
1. What is the value of (21 \times 2015) ?
(A) 45231 (B) 54321 (C) 42315 (D) 14325 (E) 23514

1. Given (K = L + R^2), (L = 4), (K = 85), and (R) is a positive number, what is the value of (R) ?
   (A) 5 (B) 7 (C) 9 (D) 4 (E) 8

2. Xiaoying and Xiaohong worked on a farm during their school vacation picking fruit. Xiaoying worked for 5 days, and Xiaohong worked for 3 days. The farm paid them $1000 together. If they were paid according to the ratio of the number of days they worked, how much can Xiaohong get?
   (A) $325 (B) $300 (C) $250 (D) $375 (E) $500

3. A regular triangle, a square, and a regular pentagon are connected by their adjacent sides as shown in the figure. What is the size of angle (x) ?
   (A) 108° (B) 105° (C) 90° (D) 120° (E) 102°

4. Calculate (3^{-2} - 2^{-3}).
   (A) -1 (B) 0 (C) (-\frac{1}{72}) (D) (\frac{1}{72}) (E) (\frac{17}{72})

5. Xiaoying measured three sides of a rectangle and found that their sum was 80 cm. Xiaofang measured three sides of another rectangle and found that their sum was 88 cm. What is the perimeter of this rectangle?
   (A) 112 cm (B) 132 cm (C) 96 cm (D) 168 cm (E) 156 cm

6. Toss two standard dice, then multiply the two numbers shown to obtain a product. What is the probability of getting a product that is a multiple of 6?
   (A) (\frac{1}{6}) (B) (\frac{5}{12}) (C) (\frac{1}{4}) (D) (\frac{1}{3}) (E) (\frac{1}{2})

英文真题

Senior Division
Questions 1 to 10, 3 marks each

1. What is the value of (21 \times 2015)?
2. (A) 45231
3. (B) 54321
4. (C) 42315
5. (D) 14325

6. (E) 23514

7. If (K = L + R^2), (L = 4), (K = 85) and (R) is positive, then (R) equals

8. (A) 5
9. (B) 7
10. (C) 9
11. (D) 4

12. (E) 8

13. On their school holidays, Xenia and Ngoc worked for a farmer picking fruit. Xenia worked for 5 days and Ngoc for 3 days. The farmer paid them $1000, which they shared in the same ratio as the days they worked. Ngoc's share was

14. (A) $325
15. (B) $300
16. (C) $250
17. (D) $375

18. (E) $500

19. An equilateral triangle, a square and a regular pentagon are joined as in the diagram. What is the size of angle (x)?

20. (A) (108^\circ)
21. (B) (105^\circ)
22. (C) (90^\circ)
23. (D) (120^\circ)

24. (E) (102^\circ)

25. (3^{-2} - 2^{-3}) equals

26. (A) -1
27. (B) 0
28. (C) (-\frac{1}{72})
29. (D) (\frac{1}{72})
30. (E) (\frac{17}{72})

真题文字详解(英文)

Solutions - Senior Division 1. $21 \times 2015 = 42315$, hence (C). 2. The first equation is $85=4+R^2$, which simplifies to $R^2=81$ and has positive solution $R=\sqrt{81}=9$, hence (C). 3. Ngoc's share is $1000\times\frac{3}{8}=\$375$, hence (D). 4. The angles in the regular polygons are $60^\circ$, $90^\circ$ and $108^\circ$, a total of $258^\circ$. The remaining angle is $x=360^\circ-258^\circ=102^\circ$, hence (E). 5. $3^{-2}-2^{-3}=\frac{1}{9}-\frac{1}{8}=\frac{8}{72}-\frac{9}{72}=-\frac{1}{72}$, hence (C). 6. (Also J17, I11) Alternative 1 Jenna must leave out a longer side and Dylan a shorter side, where the longer side is $8cm$ longer than the shorter side. So the sides are $x cm$ and $(x + 8) cm$. Then Jenna's measurement is $2x+x+8=80$, so that $3x=72$ and $x=24$. The rectangle is $24cm$ by $32cm$. The perimeter is then $2\times 24+2\times 32=112cm$, hence (A). Alternative 2 Suppose the rectangle has width $w$ and height $h$. Dylan's and Jenna's measurements are $2w+h$ and $2h+w$. Adding these, $3w+3h=80+88=168$ and so $w+h=168\div 3=56$. Then the perimeter is $2(w+h)=112cm$, hence (A). 7. (Also J22, I14) The score is a multiple of $6$ if one of the dice is $6$ or if one of the dice is even ($2$, $4$ or $6$) and the other is $3$. We tabulate these possibilities amongst the $36$ equally likely rolls. Second dice 1 2 3 4 5 6 First dice 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Then the probability that the score is a multiple of $6$ is $\frac{15}{36}=\frac{5}{12}$, hence (B). 2015 AMC - Senior Solutions 59