30. A country has 1998 airports connected by some direct flights. For
any three airports, some two are not connected by a direct flight.
What is the maximum number of direct flights that can be offered?
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. Remainder when 3^37 is divided by 79. ? A.78 b.1 c.2 d.35
34. Number of ordered pair of integer x^2+6x+y^2 = 4 is. A.2 b.4 c.6
d.8.
35. Remainder when 3^2002 + 7^2002 + 2002 is divided by 29.
36. a1, ... , a8 are reals, not all zero. Let cn = a1n + a2n + ... +
a8n for n = 1, 2, 3, ... . Given that an infinite number of cn are
zero, find all n for which cn is zero.
37. In how many ways can Rs. 18.75 be paid in exactly 85 coins ,
consists of 50 paise, 25 p and 10p coins.
38. In a certain community consisting of p persons , a % can read and
write : of the males alone b % , and of the females alone c % can
write : find the no. of males and females in the community (in a, b,
c and p terms) .
39. A no. of 3 digit in base 7 when expressed in base 9 has it’s
digits reversed in order : find the no.
40 A man has to take a plot of land fronting a street : the plot is
to be rectangular , and 3 times it’s frontage added to 2 it’s
depth is to be 26 metres. What is the greatest no of square metres he
can take.
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. Find 3 consecutive integer each divisible by a square greater
than 1.
45.Find the 3 consecutive no , the first of which is divisible by a
square , the second by a cube and third by a fourth power .
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 the 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. Find all prime for which the quotient (2^p-1 - 1)/p is a square.
48. find all a, b ,c such that the roots of x^3+ax^2+bx-8=0
49. Find all pairs of integer (x,) such that
(x-1)^2=(x+1)^2+(y+1)^2.
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.
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