• 沒有找到結果。

2015IMAS First Round Junior Division Problems

N/A
N/A
Protected

Academic year: 2021

Share "2015IMAS First Round Junior Division Problems"

Copied!
9
0
0

加載中.... (立即查看全文)

全文

(1)

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

ccmp@seed.net.tw

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

ccmp@seed.net.tw

(2)

I

I

n

n

t

t

e

e

r

r

n

n

a

a

t

t

i

i

o

o

n

n

a

a

l

l

M

M

a

a

t

t

h

h

e

e

m

m

a

a

t

t

i

i

c

c

s

s

A

A

s

s

s

s

e

e

s

s

s

s

m

m

e

e

n

n

t

t

s

s

f

f

o

o

r

r

S

S

c

c

h

h

o

o

o

o

l

l

s

s

2015 JUNIOR DIVISION FIRST ROUND PAPER

Time allowed:75 minutes

INSTRUCTION AND INFORMATION

GENERAL

1. Do not open the booklet until told to do so by your teacher.

2. No calculators, slide rules, log tables, math 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 20 multiple-choice questions, each with 5 choices. Choose the most reasonable answer. The last 5 questions require whole number answers between 000 and 999 inclusive. The questions generally get harder as you work through the paper. There is no penalty for an incorrect response.

5. This is a mathematics assessment, not a test; do not expect to answer all questions. 6. Read the instructions on the answer sheet carefully. Ensure your name, school

name and school year are filled in. It is your responsibility that the Answer Sheet is correctly coded.

7. When your teacher gives the signal, begin working on the problems.

THE ANSWER SHEET

1. Use only pencils.

2. Record your answers on the reverse side of the Answer Sheet (not on the question paper) by FULLY filling in the circles which correspond to your choices.

3. Your Answer Sheet will be read by a machine. The machine will see 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 IMAS reserves the right to re-examine students before deciding whether to grant official status to their scores.

(3)
(4)

2015 JUNIOR DIVISION FIRST ROUND PAPER

Questions 1-10, 3 marks each

1. What is the value of ( 2) 8   (1 2 22 22015 0)  16?

(A)0 (B)32 (C)33

(D) 2016

2 1 (E)22016 31

2. Someone set the alarm clock for 1:30 pm and fell asleep at 12:35 pm. When he was awaken by the alarm clock, for how long had this person been sleeping? (A)1hour and 5 minutes (B)55 minutes (C)95 minutes (D)105 minutes (E)11 hours and 5 minutes

3. In a quadrilateral ABCD, AB DC// ,

//

BC ED, ADAE and  C 110.

What is the measure of A?

(A)20° (B)35° (C)40° (D)55° (E)70° 4. In a sale, each dress is reduced to 49% of its price and if two dresses are

purchased at the same time, both are reduced to 45% of their prices. Lily buys two dresses together and pays 90 dollars for both. By doing this instead of buying them separately, how many dollars has she saved?

(A)10 (B)8 (C)6 (D)4 (E)3.6 5. Sixteen points are arranged in a 3 cm by 3 cm formation.

Four of them are removed, leaving behind twelve points as shown in the diagram. If we choose three of these twelve points as vertices of a triangle, what is the largest possible area of this triangle, in cm2?

(A)9 (B)9 2 (C)3 (D)2 (E) 3 2 E D C B A

(5)

6. Class A has 17 students more than Class B, which has 15 students less than Class C. Of the following five numbers, which can be the total number of students in these three classes?

(A)150 (B)151 (C)152 (D)153 (E)154 7. From 0, 1, 2, 3, 4 and 5, we choose two different numbers x and y. What is the

largest possible value of 2(xy)2  (x y)2?

(A)75 (B)163 (C)175 (D)187 (E)200 8. Divide the rectangle ABCD into four isosceles right triangles and one square, as

in the diagram below. If the area of square EFGH is 100 cm2, what is the area of rectangle ABCD, in cm2?

(A)750 (B)1000 (C)1100 (D)1200 (E)1600 9. A group of students are staying in a hotel. If five of them share a room, then there

is no room for six of them. If six of them share a room, there are just enough rooms, one of which has less than six students. Of the following five numbers, which cannot be the number of students?

(A)46 (B)51 (C)56 (D)61 (E)66

10. In a pentagon, one angle is48. The second angle is three times as large as the first. The third angle is30less than the second. The fourth angle is10less than the fifth. What is the measure of the fourth angle, in degrees?

(A)112 (B)122 (C)132 (D)142 (E)152

Questions 11-20, 4 marks each

11. There are three shirts, three pairs of trousers and three pairs of shoes. Of each type, one is red, one is black and one is white. In how many different ways can we choose one of each type so that something white is chosen?

(A)8 (B)9 (C)18 (D)19 (E)27 J 2 E D C B A H G F

(6)

12. In triangle ABC, AB is perpendicular to BC. D and E are points on BC such that

BAD DAE EAC

     and ADC   C 56 . What is the measure of

BAC  ? (A)42° (B)45° (C)51° (D)60° (E)84° 13. If a a 1 b   and 1 b a

a   , what is the value of

2 2 ( 1) b a ? (A)1 (B)2 (C)3 (D)4 (E)5

14. C and D are points on AB such that AC : CD : DB = 1 : 2 : 3. Semicircles are drawn on the same side of AB with respective diameters AB, AC, CD and DB. What fraction of the area of the largest semicircle is the total area of the other three semicircles? (A)1 4 (B) 1 3 (C) 13 36 (D) 7 12 (E) 7 18 15. Each coin is worth either 1 dollar, 5 dollars or 10 dollars. Their total worth is 60

dollars. They may be divided into three, four or five piles of equal worth. What is the minimum number of coins?

(A)6 (B)11 (C)15 (D)16 (E)20 J 3 E D C B A D C B A

(7)

16. From a cube of side length 10 cm, a cylinder with diameter 6 cm and depth 8 cm is hollowed out. What is the volume, in cm3, of the remaining part of the cube? Take =3.14.

(A)426.08 (B)517.46 (C)573.94 (D)717.46 (E)773.92 17. If a, b and c are all positive integers, which of the following numbers can be the

value of (a b c a)(  b c a)(  b c)(  a b c)?

(A)44 (B)46 (C)48 (D)50 (E)52

18. A three-layer structure consists of 14 unit cubes. The bottom layer consists of 9 cubes in a 3 by 3 configuration. The middle layer consists of 4 cubes in a 2 by 2 configuration. The top layer consists of a single cube. The exposed surface area of this structure is painted, including the bottom. What is the total area of the unpainted surface of the individual cubes?

(A)20 (B)31 (C)42 (D)53 (E)64

19. In an election between four candidates, they are supported respectively by 11, 12, 13 and 14 of the first 50 voters. Six more votes are to be cast, each for one of the four candidates. In how many ways can the candidate currently with 13 supporters become the uncontested winner?

(A)16 (B)17 (C)18 (D)19 (E)20

20. Let x, y and z be distinct positive prime numbers such that x y z and

2 2 2

xyz are also prime numbers. What is the minimum value of x y z?

(A)17 (B)19 (C)23 (D)29 (E)31 J 4 8 cm 10 cm 10 cm 10 cm 6 cm

(8)

Questions 21-25, 6 marks each

21. ABCDEF is a regular hexagon. G is the midpoint of AB and H is the point on AF such that FH=2AH. If the area of triangle AHG is 1 cm2. What is the area, in cm2, of ABCDEF?

22. What is the value of abc where a, b and c are positive real numbers such that ( ) 48

a b c , b c( a)70and (c a b) 88?

23. What is the value of ba where a and b are real numbers such that

2

6 9 9

baa   b b ?

24. What is the maximum value of a if a2 | (10 11 12   19)?

25. For any permutation of 1, 2, 3, 4, 5, 6, 7 and 8, add the second number to the first, multiply the sum by the third number, add the fourth number to the product, multiply the sum by the fifth number, and so on. What is the minimum value of the final sum?

*** J 5 E D C B A H G F

(9)

參考文獻

相關文件

172, Zhongzheng Rd., Luzhou Dist., New Taipei City (5F International Conference Room, Teaching Building, National Open University)... 172, Zhongzheng Rd., Luzhou Dist., New

Reading Task 6: Genre Structure and Language Features. • Now let’s look at how language features (e.g. sentence patterns) are connected to the structure

Now, nearly all of the current flows through wire S since it has a much lower resistance than the light bulb. The light bulb does not glow because the current flowing through it

I certify that I have audited the financial statements of the Subsidized Schools Provident Fund set out on pages 24 to 45, which comprise the balance sheet as at

Corollary 13.3. For, if C is simple and lies in D, the function f is analytic at each point interior to and on C; so we apply the Cauchy-Goursat theorem directly. On the other hand,

Corollary 13.3. For, if C is simple and lies in D, the function f is analytic at each point interior to and on C; so we apply the Cauchy-Goursat theorem directly. On the other hand,

Unless prior permission in writing is given by the Commissioner of Police, you may not use the materials other than for your personal learning and in the course of your official

Unless prior permission in writing is given by the Commissioner of Police, you may not use the materials other than for your personal learning and in the course of your official