31. The side lengths of a triangle and the diameter of its incircle
are 4 consecutive integers in an arithmetic progression. Find all
such triangles.

32. We have a 102 * 102 sheet of graph paper and a connected figure
of unknown shape consisting of 101 squares. What is the smallest
number of copies of the figure, which can be cut out of the square?

33. Between points A and B there are two railroad tracks. One of them
is straight and is 4 miles long. The other one is the arc of a circle
and is 5 miles long. What is the radius of curvature of the curved
track?

34. Number of ordered pair of integer x^2+6x+y^2 = 4 is. A.2 b.4 c.6
d.8.

35. 100 cards are numbered 1 to 100 (each card different) and placed
in 3 boxes (at least one card in each box). How many ways can this be
done so that if two boxes are selected and a card is taken from each,
then the knowledge of their sum alone is always sufficient to
identify the third box?

36. A cube is divided into 27 pairwise congruent smaller cubes. Find
the maximum number of small cubes that can be stabbed by a straight
line.

37. Suppose that you have a table with 8 rows and 8 columns and that
the numbers from 1 to 64 are placed in the table in such a way that
all the rows and columns add up to the same number. What is this
number?

38 A certain shade of gray paint is obtained by mixing 3 parts of
white paint with 5 parts of black paint. If 2 gallons of the mixture
is needed and the individual colors can be purchased only in
one-gallon or half- gallon cans, what is the least amount of paint,
in gallons, that must be purchased in order to measure out the
portions needed for the mixture?

(A) 2 (B) 2 .5 (C) 3 (D) 3.5 (E) 4

39. What is the least number of digits (including repetitions) needed
to express 10^100 in decimal notation?

(A) 4 (B) 100 (C) 101 (D) 1,000 (E) 1,001

40 You are at a track day at your local racecourse in your new
Porsche. Because it's a crowded day at the track, you are only
allowed to do two laps. You haven't driven your car at the track yet,
so you took the first lap easy, at 30 miles per hour. But you do want
to see what your ridiculous sports car can do. How fast do you have
to go on the second lap to end the day with an average speed of 60
miles per hour?

41. A farmer sold 10 sheep at a certain price and 5 others at Rs. 50
less per head : the sum he received for each lot was expressed in
rupee by the same 2 digits: find the prices per sheep.

42. If A tell B , that he has thought of a 3 digit no. , in such a
way that the sum of all of its digit happens to be a prime no. If
based on this information only B tries to find the no. , what is the
probabilities that she will find the prime no. in the first guess .

43. The area of intersection of 2 circular discs each of radius r and
with the boundary of each disc passing through the center of the
other is.

44. An acrobat thief enters an ancient temple, and finds the
following scenario:

1. The roof of the temple is 100 meters high.

2. In the roof there are two holes, separated by 1 meter.

3. Through each hole passes a single gold rope, each going all the
way to the floor.

4. There is nothing else in the room.

The thief would like to cut and steal as much of the ropes as he can.
However, he knows that if he falls from height that is greater than
10 meters, he will die. The only thing in his possession is a knife.

How much length of rope can the acrobat thief get?

45. A 25 meter long wound cable is cut into 2 and 3 meter long
pieces. How many different ways can this be done if also the order of
pieces of different lengths is taken into account?.

46. A gang of 17 pirates steal a sack of gold coins. When the start
dividing the loot among them there was 3 coin left. They start
fighting for the 3 left over coins and a pirates was killed. They
decided to re-divided the coins, but again there was 10 coin left
over. Again they fight and another pirates was killed. However they
were third time lucky as this time loot was equally distributed. Now
tell the least no. of coin satisfying above condition.

47. The coefficient of x^43 in the product (x-2)(x-5)(x- …….(x-131)
is

A. 3087 B.4462 C.5084 D.2926

48. Find all a, b ,c such that the roots of x^3+ax^2+bx-8=0

49. A can hit a target 4 times in 5 shots; B 3 times in 4 shots ; and
C 2 times in 3 shots. They all fire shot , find the probabilitiy that
2 shots at least hit.?

Ans. 6/13

50. ABC is an isosceles triangle with AC = BC . The median AD and BE
are perp. To each other and intersect at G . If GD = a unit , find
the area of the quadrilateral CDGE.

51. A regular octagon with sides 1 unit long is inscribed in a circle
. Find the radius of the circle.

52 The pages of a report are numbered consecutively from 1 to 10. If
the sum of the page numbers up to and including page number x of the
report is equal to one more than the sum of the page numbers
following page number x, then x =

(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

53. Two urns contain the same total numbers of balls, some blacks and
some whites in each. From each urn are drawn n balls with
replacement, where n >= 3. Find the number of drawings and the
composition of the two urns so that the probability that all white
balls are drawn from the first urn is equal to the probability that
the drawing from the second is either all whites or all blacks.

54. Two people agree to meet at a given place between noon and 1 p.m.
By agreement, the first one to arrive will wait 15 min for the
second, after which he will leave. What is the probability that the
meeting actually take place if each of them selects his moment of
arrival at random during 12 noon to 1 p.m?

55. A positive no n, not exceeding 100 is chosen in such a way that
if n is less than or equal to 50, then probabilities of choosing n is
10 and if n > 50 then it is 3p. What is the probability of
choosing a square?

56. What is the greatest area of a rectangle the sum of whose 3 sides
is equal = 100.

Ans. 1250 .

57. A plane figure consists of a rectangle and an equilateral
triangle constructed on one side of the rectangle . How large is the
area of that figure in the term of perimeter.

Ans. 6+ sqrt(3)/132 p^2.

58. A reservoir is shaped like a regular triangle . A ferry boat goes
from point A to point B both located on the bank . A cyclist who has
to get form A to B can use the ferry boat or to ride along the bank .
hat is the least ratio between the speed of cyclist and the speed of
the boat at which the use of the boat does not give any gain in time,
the location of the point A and B being arbitrary ?

Ans. 2.

59. Find the greatest volume of the cylinder inscribed in the given
cone.

Ans. V(R/3) = 4/27 * pie H*R*R

60. In a party with 1982 persons, among any group of four there is at
least one person who knows each of the other three. What is the
minimum no. of people in the party who know everyone else.

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