注意：
允許學生個人、非營利性的圖書館或公立學校合理使用 本基金會網站所提供之各項試題及其解答。可直接下載 而不須申請。
重版、系統地複製或大量重製這些資料的任何部分，必 須獲得財團法人臺北市九章數學教育基金會的授權許 可。
申請此項授權請電郵
[email protected]
Notice:
Individual students, nonprofit libraries, or schools are permitted to make fair use of the papers and its
solutions. Republication, systematic copying, or multiple reproduction of any part of this material is permitted only under license from the Chiuchang Mathematics Foundation.
Requests for such permission should be made by
e-mailing Mr. Wen-Hsien SUN
[email protected]Questions 1 to 10, 3 marks each 1. The value of 2011 − 1102 is
(A) 1111 (B) 1191 (C) 1001 (D) 989 (E) 909
2. In the diagram, the value of x is
...
...
127 ◦ x ◦
(A) 143 (B) 127 (C) 90 (D) 153 (E) 37
3. The value of 14 ÷ 0.4 is
(A) 3.5 (B) 35 (C) 5.6 (D) 350 (E) 0.14
4. Which of the following could be the graph of y = 2x + 1?
✲
✻ x
(A) y
...
✲
✻ x
(B) y
...
✲
✻ x
(C) y
...
✲
✻ x
(D) y
...
...
...
✲
✻ x
(E) y
...
...
...
...
5. The expression 8x − 4y − 3x + 2y equals
(A) 4x − y (B) 5x − 2y (C) 5x − 6y (D) 11x − 2y (E) 11x − 6y
I 2
6. By what number must 1
3 be divided to obtain 4 as a result?
(A) 1
12 (B) 6 (C) 1 1
3 (D) 1
4 (E) 12
7. Which one of the following is not equal to 3 9 ?
(A) (3 3 ) 3 (B) 3 3 × 3 3 × 3 3 (C) 27 3 (D) 9 3 × 27 (E) 9 4
8. The numbers represented by points R and P on the number line below are multi- plied. Which point would best represent the product of these two numbers?
... ... ...
0 1 2
M . S R P . . . T . N .
(A) M (B) N (C) P (D) S (E) T
9. P QRS is a trapezium in which P Q = 2 units and RS = 3 units.
What fraction of the trapezium is shaded?
(A) 1
5 (B) 1
4 (C) 1
3 (D) 2
5 (E) 1
2
...
...
...
...
...
...
...
P Q
R S
2
3
... .. . .. .... .. . .. .. .. .. .. .. . .. .. . .. .. . .. .. . .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .
.. .. .. .
.. .. .. ..
.. .. .. ..
.. .. .. .. .
.. .. .. ..
.. .. .. .. .
.. .. .. .. .
.. .. .. .. ..
.. .. .. .. ..
.. .. .. .. ..
.. .. .. .. ..
.. .. .. .. .. .
.. .. .. .. .. .
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. .. .
.. .. .. .. .. .. .
.. .. .. .. .. .. ..
.. .. .. .. .. .. ..
.. .. .. .. .. .. ..
.. .. .. .. .. .. ..
.. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .
.. .. .. .. .. .. ..
.. .. .. .. .. .. .
.. .. .. .. .. .. .
.. .. .. .. .. .. .
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. ..
.. .. .. .. ..
.. .. .. .. ..
.. .. .. .. ..
.. .. .. .. .
.. .. .. .. .
.. .. .. .. .
.. .. .. .
.. .. .. .
.. .. .. .
.. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. . .. . ..
...
10. An 8 × 8 × 8 hollow cube is constructed from 1 × 1 × 1 cubes so that its six walls are 1 cube thick. The number of 1 × 1 × 1 cubes needed to make the hollow cube is
(A) 169 (B) 296 (C) 298 (D) 384 (E) 512
Questions 11 to 20, 4 marks each
11. In my neighbourhood, 90% of the properties are houses and 10% are shops. 10%
of the houses are for sale and 30% of the shops are for sale. What percentage of the properties for sale are houses?
(A) 9% (B) 80% (C) 33 1
3 % (D) 75% (E) 25%
12. P QRS is a square. T U V W is a smaller square placed inside as shown with P R = 2T V . The ratio of the shaded area to the area of the square P QRS is
(A) 2 : 3 (B) 3 : 4 (C) 1 : 3 (D) 1 : 2 (E) 2 : 5
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
P Q
...
...
...
...
...
S R
T U
W V
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. .. ..
.. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. ..
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
13. The numbers on the six faces of this cube are consecutive even numbers.
If the sums of the numbers on each of the three pairs of opposite faces are equal, find the sum of all six numbers on this cube.
(A) 196 (B) 188 (C) 210 (D) 186 (E) 198
14. The positive integers are arranged in a zigzag fashion across five rows as follows:
A 1 9 17
B 2 8 10 16 18
C 3 7 11 15 19
D 4 6 12 14 .
E 5 13 .
In which row will 2011 appear?
(A) A (B) B (C) C (D) D (E) E
15. Two tourists are walking 12 km apart along a flat track at a constant speed of 4 km/h. When each tourist reaches the slope of a mountain, she begins to climb with a constant speed of 3 km/h.
✛ 12 km ✲ ✡ ✡ ✑ ✡ ✣ ✑ ✸
✑ ✑
✰ ? km
What is the distance, in kilometres, between the two tourists during the climb?
(A) 16 (B) 12 (C) 10 (D) 9 (E) 8
I 4
16. The six faces of a dice are numbered − 3, − 2, − 1, 0, 1, 2. If the dice is rolled twice and the two numbers are multiplied together, what is the probability that the result is negative?
(A) 1
2 (B) 1
4 (C) 11
36 (D) 13
36 (E) 1
3
17. A 36 cm by 24 cm rectangle is drawn on 1 cm grid paper such that the 36 cm side contains 37 grid points and the 24 cm side contains 25 grid points. A diagonal of the rectangle is drawn. How many grid points lie on that diagonal?
(A) 10 (B) 12 (C) 13 (D) 15 (E) 21
18. Three people play a game with a total of 24 counters where the result is always that one person loses and two people win. The loser must then double the number of counters that each of the other players has at that time.
At the end of three games, each player has lost one game and each person has 8 counters. At the beginning, Holly had more counters than either of the others.
How many did she have at the start?
(A) 9 (B) 11 (C) 13 (D) 16 (E) 24
19. Mary has 62 square blue tiles and a number of square red tiles. All tiles are the same size. She makes a rectangle with red tiles inside and blue tiles on the perimeter. What is the largest number of red tiles she could have used?
(A) 62 (B) 182 (C) 210 (D) 224 (E) 240
20. An isosceles triangle has a horizontal base of length 12 centimetres. It is divided into four equal areas by three parallel lines as shown.
...
. ...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
✛ x cm ✲
What is the value of x?
(A) 3 √
2 (B) 4 (C) 4.5 (D) 3 (E) 3 √
3
Questions 21 to 25, 5 marks each
21. Of the staff in an office, 15 rode a pushbike to work on Monday, 12 rode on Tuesday and 9 rode on Wednesday.
If 22 staff rode a pushbike to work at least once during these three days, what is the maximum number of staff who could have ridden a pushbike to work on all three days?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8
22. I drive a distance of 200 km around the city and my car’s average speed is 25 km/h.
How far do I then need to drive at an average speed of 100 km/h to raise my car’s average speed for the whole time to 40 km/h?
(A) 400 km (B) 200 km (C) 150 km (D) 120 km (E) 100 km
23. How many 3-digit numbers can be written as the sum of three (not necessarily different) 2-digit numbers?
(A) 194 (B) 198 (C) 204 (D) 287 (E) 296
24. A circle of radius 90 units and a circle of radius 40 units are tangent to each other and tangent to two lines as shown in the diagram below.
...
..
..
..
..
..
..
....
..
..
..
....
..
....
..
....
....
....
....
....
....
..
....
..
....
...
...
...
...
...
...
...
...
...
...
... ...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
... ...
. .
90 40
X Y
What is the distance XY ?
(A) 120 (B) 180 (C) 216 (D) 234 (E) 260
I 6
25. An arrangement of numbers has different differences when the differences between neighbours are all different. For example, the numbers
1 4 2 3 have differences 3, 2 and 1 − all different.
If the numbers from 1 to 6 are arranged with different differences, and with 3 in the third position,
3 what is the sum of the last three digits?
(A) 12 (B) 13 (C) 14 (D) 15 (E) 16
For questions 26 to 30, shade the answer as an integer from 0 to 999 in the space provided on the answer sheet.
Question 26 is 6 marks, question 27 is 7 marks, question 28 is 8 marks, question 29 is 9 marks and question 30 is 10 marks.
26. The first digit of a six-digit number is 1. This digit 1 is now moved from the first digit position to the end, so it becomes the last digit. The new six-digit number is now 3 times larger than the original number. What are the last three digits of the original number?
27. The diagram shows the net of a cube. On each face there is an integer: 1, w, 2011, x, y and z.
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
....
x y 2011 z
w
1
If each of the numbers w, x, y and z equals the average of the numbers written on
the four faces of the cube adjacent to it, find the value of x.
28. Two beetles sit at the vertices A and H of a cube ABCDEF GH with edge length 40 √
110 units. The beetles start moving simultaneously along AC and HF with the speed of the first beetle twice that of the other one.
....
....
..
....
....
....
....
....
....
..
....
....
....
....
....
....
..
....
....
....
..
....
....
....
....
....
....
..
....
....
....
....
....
....
..
....
....
....
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
A D
B C
E
F G
H
✈s
s ✈
...
....
....
....
..
....
....
....
....
....
....
....
....
....
....
....
....
....
..
....
....
....
....
..
... ... ...
... ... ...
... ... ...
... ...
... ... ...
....
....
..
... ....
..
...
....
..
....
... ..
....
..
... ....
..
...
....
..
....
... ..
....
..
... ....
....
... ..
....
..
..
What will be the shortest distance between the beetles?
29. In the diagram, 4 P QR has an area of 960 square units. The points S, T and U are the midpoints of the sides QR, RP and P Q, respectively, and the lines P S, QT and RU intersect at W .
...
. ...
...
....
....
..
....
....
..
....
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
....
..
....
...
P Q
R
S
U
T N
L M W ✓✓
✼
...
