Final ICMC Paper (Senior).docx

7/27/2019 Final ICMC Paper (Senior).docx

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2012 Intra-College Mathematics Competition

Organised by NJC Mathematics Society Page 1 of 6

National Junior College Mathematics Society 2012

INTRA-COLLEGE MATHEMATICS COMPETITION

SENIOR SECTIONTuesday, 14th February 2012

60 minutes

QUESTION PAPER

Instructions:

1. Please record your answers clearly on the answer sheet provided.

2. Scratch paper, graph paper, ruler, compass, and protractor are permitted, but are notessential. No other aids are allowed.

3. No steps are required to justify your answer.

4. Calculators and other electronic devices are not allowed.

5. Figures are not necessarily drawn to scale.

6. You will have 60 MINUTES to complete the competition.

7. There are 6 printed pages for this paper (including this cover page).


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Section A

(There are 10 multiple-choice questions in this section. 3 marks are awarded for a correctanswer and 0 marks for no response or a wrong answer.)

1. The day after tomorrow is two days before Tuesday. What day is today?A. WednesdayB. ThursdayC. FridayD. Saturday

2. Jack has three cats. One day Tom asked him: Are there any male cats among them? andJack replied yes. Assuming that Jack is telling the truth, what is the probability that all of

Jacks cats are males?

A. 1/2B. 1/4C. 1/8D. 1/7

3. Let , and be three real numbers such that| + 1| + (2 1) + 2 + 4 = 0

Find the value of + + .A. 3/2B. 1C. 1/2D. 0

4. Alice and Bob run on a 100 m circular track. They start at the same time from the samepoint - Alice runs clockwise at 10 m/s while Bob runs counter clockwise at 8 m/s. After

100 seconds, how many times have they met (including the starting point)?

A. 10B. 15C. 19D. 20

5. If a worker paints a house in 5 days, how many workers does it take to paint 100 housesin 2 days?

A. 150B. 200C. 250D. 300


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6. If = 1 + 2, = 1 2, and ( 1 4 + )(3 ) = , then the valueof isA. -5B. 5C. -9D. 9

7. How many squares are there in the figure below?(Note: This diagram is drawn to scale)

A. 25B. 41C. 50D. 55

8. There are 26 students in Mr. Tans class. What is the probability that at least 3 of themhave their birthdays in the same month?

A. 0%B. 33%C. 67%D. 100%

9. What is the last digit of 442012 + 332012 + 222012 + 112012?A. 7B.

2

C. 4D. 3

10.If m is a prime number and n is an odd number, where m 2 + n = 2015, what is the value of(nm)?

A. 2007B. 2009C. 2011D. 2012


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Section B

(There are 10 multiple-choice questions in this section. 5 marks are awarded for a correctanswer, 1 markfor no response and 0 marks for a wrong answer.)

11.How many integers with four different digits are there between 1000 and 9999 such thatthe absolute value of the difference between the first and the last digit is 2?

A. 448B. 840C. 896D. 1500

12. John uses the same escalator every day to go to his office. When he stands still on theescalator he reaches the top in 30s. If John walks up the escalator at speed v he reaches

the top in 15s. How long does it take John to reach the top if he walks up at speed 2 v?A. 0sB. 7.5sC. 10sD. 12s

13.Let , , be 3 integers all larger than 1 and let be a positive integer such that = 2 4, = 4 0, and = 1 2. Find the value of .A. 48B. 60C. 72D. 84

14.On planet Mysterio, the clocks have 10 hours each, and each hour has 100 minutes. Now,if the time on the planet is 5:50, what will be the angle between the minute-hand and

hour-hand of the clock?

A. 0oB. 15oC. 18oD. 30o

15.If 2012 = + and both , are non-negative integers, how many pairs ofsolutions (,) are there?A. 1B. 2C. 3D. 4


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16.Given ,, are positive integers where , which of the following cannot be aperfect square?

A. + + B. + + C.

+ +

D. + + 17.How many numbers greater that ten thousand can be formed with the digits 0,1,2,2,3

without repetition? (Note that the digit 2 appears exactly twice in each number formed.)

A. 48B. 60C. 96D. 120

18.How many real solutions are there to this equation?( + 1 1 )

= 1

A. 2B. 4C. 5D. 6

19.Amos the goat is tied by a rope to a corner of a rectangular shed. The dimensions of theshed are 9 x 7 metres and the rope is 10 metres long. The shed is surrounded by grass.

The area, in square metres, that the goat can graze upon is

A. 155/2B. 229/4C. 75D. 160 + 5/2

20.Mr. Ooi has just replaced the four tyres on his car with brand new ones. Knowing that atyre lasts 40000 km when placed on a front wheel and lasts 60000 km when placed on arear wheel, what is the optimal distance the car can travel before any tires wear out?

A. 40000 kmB. 48000 kmC. 50000 kmD. 52000 km


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Section C

(There are 5 short-answer questions in this section. 8 marks are awarded for a correct answer,1 markfor no response and 0 marks for a wrong answer.)

21.In the following diagram, O is a point in rectangle ABCD such that the area of triangleOAB is 42 cm2, and the area of triangle OBC is 36 cm2. If the area of triangle OCD is 15%

of the area of rectangle ABCD, find the area of triangle ODA in cm2.

22.If 1 + 1 + 1 = 0, then what is the value of4?

23.What is the largest positive integer nfor which n3

+ 100 is divisible by n + 10?

24.In a SG$1 coin, there is an inscribed regular octagon. The area of the octagon is 2422mm2. What is the diameter of the SG$1 coin in mm?

25.If and are positive integers such that

, then what is the smallest possible

value of?

END OF PAPER

AB

C D

O

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Final ICMC Paper (Senior).docx