51. A regular octagon with sides 1 unit long is inscribed in a circle
. Find the radius of the circle.
52 A regular octagon with sides 1 unit long is circumscribed in a
circle . Find the radius of the circle.
53. In a recent exam, your teacher asked 2 difficult questions :
"Eugenia ,which is further north, Venice or Vladivoski?"
and "What is the latitude and longitude of the north pole?".Of
the class, 33 1\3% of the students were wrong on the latitude
question , but only 25% missed the other one. 20% answered both
questions incorrectly. 37 students answered both questions correctly
. How many students took the examination?
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 th esum of whose 3 sides
is equal = 100.
Ans. 1250 .
57. A plane figure consists of a rectangle and an equaliteral
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 inscribred 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.
Ans. ??(not have)
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.
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