61. 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.
62 A chord PQ is drawn on a circle. If the length of PQ is equal to
the radius of the circle, then what is the probability that any line
drawn at random from point P through the circle is smaller than PQ A.
1/2 B. 1/3 C. 2/3 D. 3/5.
63 Between 2 station the first, second and the third class fares were
fixed in the ratio 8:6:3, but afterwards the first class were reduced
by 1/6 and second class by 1/12. In a year no. of first second and
third class passenger were in ratio 9:12:26 and the money at the
booking counter was 1326. How much was paid by the first class
passenger? A. 320 B. 390 C.420 D. None. Of these.
64 . A magician has one hundred cards numbered 1 to 100. He puts them
into three boxes, a red one, a white one, and a blue one, so that
each box contains at least one card. A member of the audience selects
two of the three boxes, chooses one card from each, and announces the
sum of the numbers on the chosen cards. Given this sum, the magician
identifies the box from which no card has been chosen. How many ways
are there to put all the cards into the boxes so that this trick
always works? (Two ways are considered different if at least one card
is put into a different box.)
65. On a circle are marked 999 points. How many ways are there to
assign to each point one of the letters A, B, or C, so that on the
arc between any two points marked with the same letter, there are an
even number of letters differing from these two?
66. On a train are riding 175 passengers and 2 conductors. Each
passenger buys a ticket only after the third time she is asked to do
so. The conductors take turns asking a passenger who does not already
have a ticket to buy one, doing so until all passengers have bought
tickets. How many tickets can the conductor who goes first be sure to
sell?
67. Around a table are seated representatives of n countries (n
greater than or equal to 2), such that if two representatives are
from the same country, their neighbors on the right are from two
different countries. Determine, for each n, the maximum number of
representatives.
68. 98 points are given on a circle. Maria and Joe take turns drawing
a segment between two of the points which have not yet been joined by
a segment. The game ends when each point has been used as the
endpoint of a segment at least once. The winner is the player who
draws the last segment. If Joe goes first, who has a winning
strategy?
69. A computer screen shows a 98 * 98 chessboard, colored in the
usual way. One can select with a mouse any rectangle with sides on
the lines of the chessboard and click the mouse button: therefore,
the colors in the selected rectangle switch (black becomes white,
white becomes black). Find, the minimum number of mouse clicks needed
to make the chessboard all one color.
70. A farmer has four straight pieces of fencing: 1, 2, 3, and 4
yards in length. What is the maximum area he can enclose by
connecting the pieces?
71. Somewhere in Northern Asia, groups of 20 man were planning a
special group suicide this year. Each of the them will be placed in a
random position along a thin, 100 meter long plank of wood which is
floating in the sea. Each man is equally likely to be facing either
end of the plank. At time t=0, all of them walk forward at a slow
speed of 1 meter per minute. If a man bumps into another man, the two
both reverse directions. If a man falls off the plank, he drowns.
What is the longest time that must elapse till all the man have
drowned?
72. I have chosen a number from 1 to 144, inclusive. You may pick a
subset (1, 2, .. 144), i.e. U can frame any subset based on AP, GP,
or any other thing and then ask me whether my number is in the
subset. An answer of “yes" will cost you 2 dollars, an answer
of “no" only 1-dollar. What is the smallest amount of money
you will need to be sure to find my number?
73. The side lengths of a triangle and the diameter of its incircle
are four consecutive integers in an arithmetic progression. Find all
such triangles.
74. A 10-digit number is said to be interesting if its digits are all
distinct and it is a multiple of 11111. How many interesting integers
are there?
75. Two matching decks have 36 cards each; one is shuffled and put on
top of the second. For each card of the top deck, we count the number
of cards between it and the corresponding card of the second deck.
What is the sum of these numbers?
76. A cube of side length n is divided into unit cubes by partitions
(each partition separates a pair of adjacent unit cubes). What is the
smallest number of partitions that can be removed so that from each
cube, one can reach the surface of the cube without passing through a
partition?
77. A room has its dimension as 4 ,5 and 6 mt. An ant wants to go
from one corner of the room to the diagonally opposite corner. The
minimum distance it has to travel is
A.10.1 B. 10.8 C. 10.9 D. 11.2
78. A rumor began to spread one day in a town with a population of
100,000. Within a week, 10,000 people had heard this rumor. Assuming
that the rate of increase of the number of people who have heard the
rumor is proportional to the number who have NOT yet heard it, how
long will it take until half of the population of the town has heard
the rumor? A. 44 B.46 C.45 D. none of these
79. A camel must travel 15 miles across a desert to the nearest city.
It has 45 bananas but can only carry 15at a time. For every mile
camel walks, it needs to eat a banana. What is the maximum number of
bananas that can transport to the city? A. 10 B.8 C. 15 D. none of
these
80. So far this basketball season, all of Siddhart points have come
from two- point and three point field goals. He had scored 43 points.
He has made one more than twice as many three pointers as two
pointers. How many of each kind of field goal has Siddhart made?
81. Let g (t)=9t^4+6t^2+2. For which value of t is this function a
minimum?
82. Consider any two-digit number whose digits are not zero and are
not the same. What is the greatest integer that divides evenly the
difference between the square of the number and the square of the
reverse?
ANS Largest difference = 91^2 - 19^2 = 7920 Hence, the greatest
integer is 7920.
83. What is the maximum number of Friday the thirteenths that can
occur in any given year?
84. What letter is in the 150th entry of the pattern
ABBCCCDDDD......?
85. Tanisha and Richa had lunch at the mall. Tanisha ordered three
slices of pizza and two colas. Richa ordered two slices of pizza and
three colas. Tanisha's bill was Rs60.0, and Rachel's bill was Rs52.5.
What was the price of one slice of pizza? What was the price of one
colon?
86. An architect plans a room to be 14m x 30m. He wishes to increase
both dimensions by the same amount to obtain a room with 141 sq.
metres more area. By how much must he increase each dimension?
Ans 3/2, -47/2
87. A geometry teacher drew some quadrilaterals on the chalkboard.
There were 5 trapezoids, 12 rectangles, 5 squares, and 8 rhombuses.
What is the least number of figures the teacher could have drawn?
Ans 20
88. Given an n x n square board, with n even. Two distinct squares of
the board are said to be adjacent if they share a common side, but a
square is not adjacent to itself. Find the minimum number of squares
that can be marked so that every square (marked or not) is adjacent
to at least one marked square.
89. A painted wooden cube, such as a child's block, is cut into
twenty-seven equal pieces. First the saw takes two parallel and
vertical cuts through the cube, dividing it into equal thirds; then
it takes two additional vertical cuts at 90 degrees to the first
ones, dividing the cube into equal ninths. Finally, it takes two
parallels and horizontal cuts through the cube, dividing it into
twenty-seven cubes. How many of these small cubes are painted on
three sides? On two sides? On one side? How many cubes are unpainted?
90. It was claimed that the shepherd was the shepherd of 2000 sheep.
The shepherd exclaimed, "I am not the shepherd of two thousand
sheep!" Pointing to his flock, he added, "If I had that
many sheep plus another flock as large as that, then again half as
many as I have out there, I would be the shepherd of two thousand
sheep." How many sheep were in the shepherd's flock?
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