**1.** 10 people meet and shake hands. The maximum number of handshakes possible if there is to be no “cycle” of handshakes is (A cycle of handshakes is a sequence of k people a1, a2, ……, ak (k > 2) such that the pairs {a1, a2}, {a2, a3}, ……, {ak-1, ak}, {ak, a1} shake hands).

**A.** 7

**B.** 6

**C.** 9

**D.** 8

**2.**Alice and Bob play the following coins-on-a-stack game. 20 coins are stacked one above the other. One of them is a special (gold) coin and the rest are ordinary coins. The goal is to bring the gold coin to the top by repeatedly moving the topmost coin to another position in the stack.

Alice starts and the players take turns. A turn consists of moving the coin on the top to a position i below the top coin (0 = i = 20). We will call this an i-move (thus a 0-move implies doing nothing). The proviso is that an i-move cannot be repeated; for example once a player makes a 2-move, on subsequent turns neither player can make a 2-move.

If the gold coin happens to be on top when it’s a player’s turn then the player wins the game. Initially, the gold coin is the third coin from the top. Then

**A.** In order to win, Alice’s first move should be a 0-move.

**B.** In order to win, Alice’s first move should be a 1-move.

**C.** Alice has no winning strategy.

**D.** In order to win, Alice’s first move can be a 0-move or a 1-move.

**3. **After the typist writes 12 letters and addresses 12 envelopes, she inserts the letters randomly into the envelopes (1 letter per envelope). What is the probability that exactly 1 letter is inserted in an improper envelope?

**A.** 0

**B.** 12/212

**C.** 11/12

**D.** 1/12

**4. **Given 3 lines in the plane such that the points of intersection form a triangle with sides of length 20, 20 and 30, the number of points equidistant from all the 3 lines is

**A.** 4

**B.** 3

**C.** 0

**D.** 1

**5. **A hare and a tortoise have a race along a circle of 100 yards diameter. The tortoise goes in one direction and the hare in the other. The hare starts after the tortoise has covered 1/5 of its distance and that too leisurely. The hare and tortoise meet when the hare has covered only 1/8 of the distance. By what factor should the hare increase its speed so as to tie the race?

**A.** 40

**B.** 37.80

**C.** 8

**D.** 5

**6.** A circular dartboard of radius 1 foot is at a distance of 20 feet from you. You throw a dart at it and it hits the dartboard at some point Q in the circle. What is the probability that Q is closer to the center of the circle than the periphery?

**A.** 0.75

**B.** 1

**C.** 0.5

**D.** 0.25

**7.** On planet zorba, a solar blast has melted the ice caps on its equator. 8 years after the ice melts, tiny plantoids called echina start growing on the rocks. echina grows in the form of a circle and the relationship between the diameter of this circle and the age of echina is given by the formula d = 4 * v (t – 8 ) for t = 8where d represents the diameter in mm and t the number of years since the solar blast. Jagan recorded the radius of some echina at a particular spot as 8mm. How many years back did the solar blast occur?

**A.** 24

**B.** 12

**C.** 8

**D.** 16

**8. **For the FIFA world cup, Paul the octopus has been predicting the winner of each match with amazing success. It is rumored that in a match between 2 teams A and B, Paul picks A with the same probability as A’s chances of winning.

Let’s assume such rumors to be true and that in a match between Ghana and Bolivia, Ghana the stronger team has a probability of 2/3 of winning the game. What is the probability that Paul will correctly pick the winner of the Ghana-Bolivia game?

**A.** 4/9

**B.** 2/3

**C.** 1/9

**D.** 5/9

**9.** The citizens of planet nigiet are 8 fingered and have thus developed their decimal system in base 8. A certain street in nigiet contains 1000 (in base 8 buildings numbered 1 to 1000. How many 3s are used in numbering these buildings?

**A.** 256

**B.** 54

**C.** 192

**D.** 64

**10.** 36 people {a1, a2, …, a36} meet and shake hands in a circular fashion. In other words, there are totally 36 handshakes involving the pairs, {a1, a2}, {a2, a3}, …, {a35, a36}, {a36, a1}. Then size of the smallest set of people such that the rest have shaken hands with at least one person in the set is

**A.** 12

**B.** 13

**C.** 18

**D.** 11

**11.**There are two boxes, one containing 10 red balls and the other containing 10 green balls. You are allowed to move the balls between the boxes so that when you choose a box at random and a ball at random from the chosen box, the probability of getting a red ball is maximize**D.** This maximum probability is

**A.** 3/4

**B.** 14/19

**C.** 37/38

**D.** 1/2

**12.**A hollow cube of size 5 cm is taken, with a thickness of 1 cm. It is made of smaller cubes of size 1 cm. If 4 faces of the outer surface of the cube are painted, totally how many faces of the smaller cubes remain unpainted?

**A.** 900

**B.** 488

**C.** 500

**D.** 800

**13.**The IT giant Tirnop has recently crossed a head count of 150000 and earnings of $7 billion. As one of the forerunners in the technology front, Tirnop continues to lead the way in products and services in India. At Tirnop, all programmers are equal in every respect. They receive identical salaries ans also write code at the same rate.Suppose 12 such programmers take 12 minutes to write 12 lines of code in total. How many lines of code can be written by 72 programmers in 72 minutes?

**A.** 72

**B.** 432

**C.** 12

**D.** 6

**14.**The IT giant Tirnop has recently crossed a head count of 150000 and earnings of $7 billion. As one of the forerunners in the technology front, Tirnop continues to lead the way in products and services in India. At Tirnop, all programmers are equal in every respect. They receive identical salaries ans also write code at the same rate.Suppose 12 such programmers take 12 minutes to write 12 lines of code in total. How long will it take 72 programmers to write 72 lines of code in total?

**A.** 18

**B.** 72

**C.** 6

**D.** 12

**15.**Given a collection of points P in the plane, a 1-set is a point in P that can be separated from the rest by a line; i.e the point lies on one side of the line while the other lie on the other side. The number of 1-sets of P is denoted by n1(P).The maximum value of n1(P) over all configurations P of 11 points in the plane is

**A.** 10

**B.** 3

**C.** 5

**D.** 11

**16**.Given a collection of points P in the plane, a 1-set is a point in P that can be separated from the rest by a line; i.e. the point lies on one side of the line while the others lie on the other side. The number of 1-sets of P is denoted by n1(P). The maximum value of n1(P) over all configurations P of 19 points in the plane is

**A.** 18

**B.** 9

**C.** 3

**D.** 11

**17.**Alok and Bhanu play the following min-max game. Given the expression N = 9 + X + Y – Z where X, Y and Z are variables representing single digits (0 to 9), Alok would like to maximize N while Bhanu would like to minimize it. Towards this end, Alok chooses a single digit number and Bhanu substitutes this for a variable of her choice (X, Y or Z). Alok then chooses the next value and Bhanu, the variable to substitute the value. Finally Alok proposes the value for the remaining variable. Assuming both play to their optimal strategies, the value of N at the end of the game would be

**A.** 27

**B.** 18

**C.** 20

**D.** 0.0

**18.**Alok and Bhanu play the following min-max game. Given the expression N = X – Y – Z where X, Y and Z are variables representing single digits (0 to 9), Alok would like to maximize N while Bhanu would like to minimize it. Towards this end, Alok chooses a single digit number and Bhanu substitutes this for a variable of her choice (X, Y or Z). Alok then chooses the next value and Bhanu, the variable to substitute the value. Finally Alok proposes the value for the remaining variable. Assuming both play to their optimal strategies, the value of N at the end of the game would be

**A.** 2

**B.** 4

**C.** 9

**D.** -18

**19.**Alok and Bhanu play the following min-max game. Given the expression N = 38 + X*(Y – Z) where X, Y and Z are variables representing single digits (0 to 9), Alok would like to maximize N while Bhanu would like to minimize it. Towards this end, Alok chooses a single digit number and Bhanu substitutes this for a variable of her choice (X, Y or Z). Alok then chooses the next value and Bhanu, the variable to substitute the value. Finally Alok proposes the value for the remaining variable. Assuming both play to their optimal strategies, the value of N at the end of the game would be

**A.** 38

**B.** 119

**20. **10 suspects are rounded by the police and questioned about a bank robbery. Only one of them is guilty. The suspects are made to stand in a line and each person declares that the person next to him on his right is guilty. The rightmost person is not questione**D.** Which of the following possibilities are true?

**A.** All suspects are lying or the leftmost suspect is innocent.

**B.** All suspects are lying and the leftmost suspect is innocent .

**A.** A only

**B.** Neither A nor B

**C.** Both A and B

**D.** B only

**21.** A sheet of paper has statements numbered from 1 to 40. For all values of n from 1 to 40, statement n says: ‘Exactly n of the statements on this sheet are false.’ Which statements are true and which are false?

**A.** The even numbered statements are true and the odd numbered statements are false.

**B.** The 39th statement is true and the rest are false.

**C.** The odd numbered statements are true and the even numbered statements are false.

**D.** All the statements are false.

**22. **34 people attend a party. 4 men are single and the rest are there with their wives. There are no children in the party. In all 22 women are present. Then the number of married men at the party is

**A.** 12

**B.** 8

**C.** 16

**23. **30 teams enter a hockey tournament. A team is out of the tournament if it loses 2 games. What is the maximum number of games to be played to decide one winner?

**A.** 60

**B.** 59

**C.** 61

**D.** 30

**24.** A and B play a game of dice between them. The dice consist of colors on their faces (instead of numbers). When the dice are thrown, A wins if both show the same color; otherwise B wins. One die has 4 red face and 2 blue faces. How many red and blue faces should the other die have if the both players have the same chances of winning?

**A.** 3 red and 3 blue faces

**B.** 2 red and remaining blue

**C.** 6 red and 0 blue

**D.** 4 red and remaining blue

**25.** A and B play a game of dice between them. The dice consist of colors on their faces (instead of numbers). When the dice are thrown, A wins if both show the same color; otherwise B wins. One die has 3 red faces and 3 blue faces. How many red and blue faces should the other die have if the both players have the same chances of winning?

**A.** red and 1 blue faces

**B.** 1 red and 5 blue faces

**C.** 3 red and 3 blue faces

**D.** Any of the solutions given

**26. **There are two containers A and B. A is half filled with wine whereas B which is 3 times the size of A contains one quarter portion wine. If both containers are filled with water and the contents are poured into container C, what portion of container C is wine?

**A.** 30

**B.** 31

**C.** 42

**D.** 25

**27. **A sheet of paper has statements numbered from 1 to 45. For all values of n from 1 to 45, statement n says “At most n of the statements on this sheet are false”. Which statements are true and which are false?

**A.** The odd numbered statements are true and the even numbered are false.

**B.** The even numbered statements are true and the odd numbered are false.

**C.** All statements are false.

**D.** All statements are false

**28.** A sheet of paper has statements numbered from 1 to 25. For all values of n from 1 to 25, statement in says “At most n of the statements on this sheet are false”. Which statements are true and which are false?

**A.** The odd numbered statements are true and the even numbered are false.

**B.** All statements are false.

**C.** The even numbered statements are true and the odd numbered are false.

**D.** All statements are true .

**29.** Alice and Bob play the following chip-off-the-table game. Given a pile of 58 chips, Alice first picks at least one chip but not all the chips. In subsequent turns, a player picks at least one chip but no more than the number picked on the previous turn by the opponent. The player to pick the last chip wins. Which of the following is true?

**A.** In order to win, Alice should pick 14 chips on her first turn.

**B.** In order to win, Alice should pick two chips on her first turn.

**C.** In order to win, Alice should pick one chip on her first turn.

**30. **Suppose 19 monkeys take 19 minutes to eat 19 bananas. How many minutes would it take 8 monkeys to eat 8 bananas?

**A.** 152

**B.** 27

**C.** 19

**D.** 8

**31.** Suppose 12 monkeys take 12 minutes to eat 12 bananas. How many monkeys would it take to eat 72 bananas in 72 minutes?

**A.** 6

**B.** 72

**C.** 12

**D.** 18

**32.** A person drives with constant speed and after some time he sees a milestone with 2 digits. Then travels for 1 hours and sees the same 2 digits in reverse order. 1 hours later he sees that the milestone has the same 2 digits with a 0 between them. What is the speed of the car?

**A.** 54.00 mph

**B.** 45.00 mph

**C.** 27.00 mph

**D.** 36.00 mph

**33.** Fermat’s Last Theorem is a statement in number theory which states that it is impossible to separate any power higher than the second into two like powers, or, more precisely- If an integer n is greater than 2, then the equation a^n b^n = c^n has no solutions in non-zero integers a, b, and **C.** Now, if the difference of any two numbers is 9 and their product is 17, what is the sum of their squares?

**A.** 43

**B.** 45

**C.** 98

**D.** 115

**34. **Alchemy is an occult tradition that arose in the ancient Persian empire. Zosimos of Panopolis was an early alchemist. Zara, reads about Zosimos and decides to try some experiments. One day, she collects two buckets, the first containing one litre of ink and the second containing one litre of col a. Suppose she takes one cup of ink out of the first bucket and pours it into the second bucket. After mixing she takes one cup of the mixture from the second bucket and pours it back into the first bucket. Which one of the following statements holds now?

**A.** There is more cola in the first bucket than ink in the second bucket.

**B.** None of the statements holds true.

**C.** There is as much cola in the first bucket as there is ink in the second bucket.

**D.** There is less cola in the first bucket than ink in the second bucket.