IIT Bombay Mathematics Olympiad 2011

Question Paper for Level 1: CODE A

Please read the following instructions carefully before attempting the question paper:

• The examination is for two hours and carries 28 objective-type questions.

• Write the Question Paper code i.e. (A or B) in the space provided in the answer sheet.

• Write your relevant details on the answer sheet provided to you immediately after you

receive it and put your signature in the space provided. Do not write anything else

on the answer sheet.

• Read the questions carefully and mark your answers on the bubble sheet provided by

darkening the corresponding bubble of the option that you think is correct. Exactly one

answer is correct.

• Darken each bubble completely with a blue/black pen/pencil. Do not darken multiple

option bubbles for any question. It will be deemed as an incorrect answer.

• Each correct answer will be awarded +3 marks. Each incorrect answer will invite a

penalty of -1 mark. If you do not attempt a question, no marks will be awarded or

subtracted.

• You may guess an answer but guessing is discouraged. It may in fact, lower your score.

• Tie-breaker questions at the end may be attempted after you have completed solving

the objective-type questions. Tie-breaker questions will be evaluated if and only if there

is a tie.

• You may do the rough work on the question paper itself. No more extra sheet will

be provided.

• Calculators, cell phones and other such electronic devices are strictly banned in the exam-

ination center. Anyone found using them will be debarred from taking the examination

and he/she will have to forfeit his/her candidature.

• Please keep your cell phones switched off during the examination.

• Anyone caught copying or indulging in any other form of unfair means will have his/her

candidature forfeited and necessary disciplinary actions will be taken.

• Please submit your answer sheet along with question paper to the invigilator after

the examination.

1

1. 11+xb−a+xc−a + 1

1+xa−b+xc−b + 11+xb−c+xa−c =?

(a) 1

(b) 0

(c) xa−b−c

(d) None of these

2. The sum of first 20 terms of the series

−12 + 22 − 32 + 42 − 52 + 62 − · · ·

(a) 210

(b) 214

(c) 216

(d) 284

3. Find the next term of the sequence

3, 5, 11, 21, 43, . . .

(a) 124

(b) 85

(c) 126

(d) None of the above

4. A trader makes a profit equal to the selling price of 75 articles when he sold 100 of the

articles. What % profit did he make in the transaction?

(a) 300%

(b) 75%

(c) 33.33 %

(d) 150%

5. Four passengers in a compartment of the Mumbai-New Delhi Rajdhani Express discover

that they form an interesting group. Two are mathematicians and two are statisticians.

Two of them speak Bengali and the other two Hindi and no two of the same profession

speak the same language. Two are also Christians and two are Muslims. No two of the

same religion are of the same field of study, neither do they speak the same language.

Hindi-speaking statistician is a Christian. Which of the following statements must be

true?

2

(a) Bengali-speaking mathematician is a Muslim.

(b) Christian mathematician speaks Bengali.

(c) Bengali-speaking statistician is a Christian.

(d) None of the above

6. If n > 0 is a positive integer then what is the remainder when 91 + 92 + 93 + · · ·+ 92012

is divided by 6?

(a) 5

(b) 0

(c) 3

(d) 2

7. The marks scored by Neha in three subjects are in the ratio of 4 : 5 : 6. If she scored an

overall aggregate of 60% of the sum of the maximum marks and the maximum marks in

all three subjects is the same, in how many subjects did she score more than 60%?

(a) 0

(b) 1

(c) 2

(d) 3

8. When a number is divided by 36, it leaves a remainder of 19. What will be the remainder

when the number is divided by 12?

(a) 10

(b) 9

(c) 7

(d) None of the above

9. ABCD is a rectangle where AD = 10. P is a point on BC such that ∠APD = 90◦. If

DP = 8 then BP =?

(a) 6.4

(b) 5.2

(c) 4.8

(d) None of the above

10. If both 112 and 33 are factors of the number a× 43× 62× 1311, then what is the smallest

possible value of a?

3

(a) 123

(b) 363

(c) 3267

(d) 33

11. Three math classes: X, Y, and Z, take an algebra test.

• The average score in class X is 83.

• The average score in class Y is 76.

• The average score in class Z is 85.

• The average score of all students in classes X and Y together is 79.

• The average score of all students in classes Y and Z together is 81.

What is the average score for all the three classes, taken together?

(a) 80

(b) 81

(c) 81.5

(d) Cannot be determined

12. The calendar for the year 2007 will be same for the year

(a) 2014

(b) 2016

(c) 2017

(d) 2018

13. This question is based on the information given below:

• There is a cuboid whose dimensions are 4× 3× 3 cm.

• The opposite faces of dimensions 4× 3 are coloured yellow.

• The opposite faces of other dimensions 4× 3 are coloured red.

• The opposite faces of dimensions 3× 3 are coloured green.

Now the cuboid is cut into small cubes of side 1 cm. How many small cubes will have

only two faces coloured ?

(a) 12

(b) 24

4

(c) 20

(d) 16

14. A scalene triangle is a triangle that has three unequal sides. How many scalene triangles

exist whose sides a, b, and c are natural numbers less than 8?

(a) 18

(b) 14

(c) 13

(d) 16

15. Sayan and Debdip take a vacation at their grandparents’ house. During the vacation,

they do any activity together. They either played table tennis in the evening or practiced

Yoga in the morning, ensuring that they do not undertake both the activities on any single

day. There were some days when they did nothing. Out of the days that they stayed at

their grandparents’ house, they involved in one of the two activities on 22 days. However,

their grandmother while sending an end of vacation report to their parents stated that

they did not do anything on 24 mornings and they did nothing on 12 evenings. How long

was their vacation?

(a) 29 days

(b) 14 days

(c) 36 days

(d) Cannot be determined

16. If the price of petrol increases by 25%, by how much must an user cut down his consump-

tion so that expenditure on petrol remains constant?

(a) 33.33%

(b) 16.67%

(c) 25%

(d) 20%

17. What is the angle between the minute hand and the hour hand when the time is 15:40

hrs?

(a) 135◦

(b) 130◦

(c) 140◦

5

(d) 120◦

18. If one root of the quadratic equation 2x2 − 7x + q = 0 is 3 then the other root is

(a) 12

(b) -3

(c) −12

(d) 14

19. A tank can be filled by two taps - Tap 1 and Tap 2. The volume of the tank is 5000

litres. Tap 1 fills the tank at a rate of 1 litre/second. Tap 2 fills the tank at a rate of 3

litres in 2 seconds. On a particular day, Tap 2 is opened 331/3 minutes after the time at

which Tap 1 is opened. If after 45 minutes from the time when Tap 1 was opened, the

tank develops a hole which empties the tank at the rate of 2.5 litres/second, how full is

the tank in 2 hours from the time when Tap 1 opened?

(a) 34

(b) 23

(c) 14

(d) The tank will be full.

20. What is the value of M and N respectively if M39048458N is divisible by 8 and 11 where

M and N are single digit integers?

(a) 7, 8

(b) 8, 6

(c) 5, 4

(d) 6, 4

21. How many zeros are there at the end of 2011× 2010× 2009× · · · × 2× 1?

(a) 305

(b) 501

(c) 502

(d) 482

22. What is the last digit of 201714?

(a) 9

(b) 7

6

(c) 3

(d) 1

23. Find the number of triangles formed by drawing all possible lines between corners of a

hexagon?

(a) 20

(b) 12

(c) 18

(d) 22

24. Sanhita puts her timepiece on the table in such a way that at 6 P.M. hour hand points

to North. In which direction the minute hand will point at 9.15 P.M.?

(a) East

(b) South

(c) West

(d) North

25. A jogging park has two identical circular tracks touching each other and a rectangular

track enclosing the two circles. The edges of the rectangles are tangential to the circles.

Two friends, Virender and Gautam, start jogging simultaneously from the point where

one of the circular tracks touches the smaller side of the rectangular track. Virender jogs

along the rectangular track, while Gautam jogs along the two circular tracks in a figure

of eight. Approximately, how much faster than Virender does Gautam have to run, so

that they take the same time to return to their starting point?

(a) 4.72%

(b) 4.22%

(c) 3.88%

(d) 4.44%

26. For what value of n will the remainder of 351n and 352n be the same when divided by 7?

(a) 1

(b) 8

(c) 6

(d) None of the above

7

27. Let {an} be a sequence such that a1 = a2 = 1 and an+2 = 1an+1

+ an, n = 1, 2, . . .. Find

a2004.

(a) 12×3×4×···×2002×2003

(b) 3×5×···×2001×20032×4×···×2000×2002

(c) 2× 3× 4× · · · × 2002× 2003

(d) 1

28. How many prime divisors are there for the number 403 − 173 − 233?

(a) 5

(b) 4

(c) 3

(d) 7

8

Tie-breaker Questions( Attempting these questions is optional. Answers to these questions shall be evaluated only

in case of a tie. Also, you must show the method of calculation. )

1. Let all possible 6-digit numbers such that in each of which the digit occurs in

non-increasing order (from left to right, e.g. 6543300) are written as a sequence of

increasing order. Find the 1715th term of the sequence.

[Hint:- Try to count all the numbers whose digits occur in non-increasing order starting

from left most digit 1 to left most digit 7. Do not forget to write the elements in

increasing order.]

2. Show that if n is an integer, n2 + 11n + 2 is not divisible by 12769.

9