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University of Montenegro
Institute of Foreign Languages
________________________
(Name, Ind.No.)
MOCK EXAM
I Circle one of the given options to best complete the text: (12)
Pigeonhole Principle
The Pigeonhole Principle 1. _________________ that if n+1 pigeons fly to n holes, there 2. _________________
be a pigeonhole containing at least two pigeons. This apparently trivial principle is very 3. _________________.
The pigeonhole principle is an example of a counting argument which can be 4. _________________ to many
formal problems, including ones involving infinite sets that cannot be put into one-to-one correspondence.
The first statement of the principle is believed to have been made 5. _________________ Dirichlet in 1834 under
the name Schubfachprinzip ("drawer principle" or "shelf principle"). In Italian too, the original name "principio dei
cassetti" is still in use; in some other languages (for example, Russian) this principle 6. _________________ the
Dirichlet principle (not to be confused with the minimum principle for harmonic functions of the same name).
Let us 7. _________________ some examples.
Example (Putnam 1978) 8. _________________ A be any set of twenty integers 9. _________________ from the
arithmetic progression 1,4, . . . ,100. Prove that there must be two distinct integers in A whose sum is 104.
10. _________________: We partition the thirty four elements of this progression 11. _________________
nineteen groups {1},{52}, {4,100} , {7,97}, {10,94},. . . {49,55}. Since we are choosing twenty integers and we
have nineteen sets, by the Pigeonhole Principle there must be two integers that belong 12. _________________
one of the pairs, which add to 104.
1. a) states
2. a) may
3. a) power
4. a) application
5. a) by
6. a) calls
7. a) to see
8. a) let
9. a) choose
10. a) solve
11. a) on
12. a) on
1
b) is stated
b) might
b) powerful
b) apply
b) from
b) call
b) seeing
b) must
b) chose
b) salvation
b) into
b) to
2
3
4
c) is stating
c) ought to
c) powerless
c) applied
c) to
c) is called
c) see
c) make
c) is chosen
c) solution
c) from
c) about
5
6
7
d) statement
d) must
d) powered
d) applicated
d) on
d) is calling
d) have seen
d) take
d) chosen
d) solving
d) and
d) at
8
9
10
11
12
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II Complete the text with the appropriate words. (5)
theorem
celebrated
order
states
integer
set
random
proved
special
either
Ramsey theory is a ________________ part of extremal combinatorics. It ______________ that any sufficiently
large ________________ configuration will contain some sort of ________________.
Frank Ramsey ________________ that for every integer k there is an ________________ n, such that every graph
on n vertices ________________ contains a clique or an independent ________________ of size k. This is a
________________ case of Ramsey’s ________________.
III Supply the missing articles where needed. (5)
_______ mathematics (colloquially, maths or math) is _______ body of knowledge centered on such concepts as
quantity, structure, _______ space, and change, and also _______ academic discipline that studies them. Benjamin
Peirce called it "_______ science that draws necessary conclusions".[2] Other practitioners of mathematics
maintain that mathematics is _______ science of pattern, and that _______ mathematicians seek out _______
patterns whether found in _______ numbers, space, science, computers, ________ imaginary abstractions, or
elsewhere.
IV Turn the direct into indirect speech: (5)
1. ‘I think that this theory cannot be proven,’ Gauss said.
______________________________________________________________________________
2. ‘What is applied mathematics?’ the student asked.
______________________________________________________________________________
3. ‘I’ll show you how to solve this problem,’ he said.
______________________________________________________________________________
4. ‘I expect you to hand in you homework by tomorrow,’ the teacher said.
______________________________________________________________________________
5. ‘I gave an interesting lecture,’ the professor boasted.
______________________________________________________________________________
V Complete the text using the correct form of the verbs in brackets: (8)
Srinivasa Ramanujan was a mathematical prodigy. "I remember once ______________ (go) to see him when he
was lying ill at Putney," the mathematician G. H. Hardy once ________________ (remember). "I ______________
(take) the taxicab number 1729, and remarked that the number seemed to me rather a dull one, and that I hoped it
was not an unfavorable omen. "'No,' he replied, 'it _______________ (be) a very interesting number; it is the
smallest number expressible as the sum of two cubes in two different ways.'"
50-50 Proposition
While lecturing on probability at Warwick University one day in October 1972, Jeffrey Hamilton, demonstrating
the effect of chance, _______________ (take) a coin from his pocket and casually tossed it in the air. The
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probability that the coin would land face up (heads) was exactly the same as the probability that it
_________________ (land) face down (tails); it was, Hamilton explained, a 50-50 proposition. Hamilton and the
assembled students then ___________________ (watch) as it hit the floor, bounced, rolled, spun around - and
came to rest on its edge. After a stunned silence, a wild applause ______________ (hear) in the room.
VI Translate the following text: (10)
Q. Can you give any examples of how your mathematics are seen in nature?
A.“The Fibonacci Sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. The formula basically is a guide to adding the
previous two numbers in the Hindu-Arabic system to get a new number ad infinitam. Interestingly enough, this is
found everywhere in living things because of the way things grow exponentially in nature. Also, did you notice
how much art and music have to do with the sequence? If you look at piano keys or famous works of art you will
always see recurring patterns obviously of the "Golden numbers.” We take the ratio of two successive numbers in
Fibonacci's series, (1, 1, 2, 3, 5, 8, 13) and we divide each by the number before it, we will find the following series
of numbers: = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538... The ratio seems to be
settling down to a particular value, which we call the golden ratio or the golden number. It has a value of
approximately 1.61804.”
VII Find synonyms in the text: (5)
How many golf balls can fit in a school bus?
Job: Product Manager
Answer: This is one of those questions Google asks just to see if the applicant can explain the key challenge to
solving the problem. Reader Matt Beuchamp came up with a dandy answer, writing: I figure a standard school bus
is about 8ft wide by 6ft high by 20 feet long - this is just a guess based on the thousands of hours I have been
trapped behind school buses while traffic in all directions is stopped.
That means 960 cubic feet and since there are 1728 cubic inches in a cubic foot, that means about 1.6 million
cubic inches.
I calculate the volume of a golf ball to be about 2.5 cubic inches (4/3 * pi * .85) as .85 inches is the radius of a golf
ball.
Divide that 2.5 cubic inches into 1.6 million and you come up with 660,000 golf balls. However, since there are
seats and crap in there taking up space and also since the spherical shape of a golf ball means there will be
considerable empty space between them when stacked, I'll round down to 500,000 golf balls.
Which sounds ludicrous. I would have guessed no more than 100k. But I stand by my math.
Of course, if we are talking about the kind of bus that George Bush went to school on or Barney Frank rides to
work every day, it would be half that....or 250,000 golf balls.
Very good _________________
Form an opinion ________________
to prevent someone from escaping from somewhere ___________________
things that are useless or unimportant ___________________
fairly large, especially large enough to have an effect or be important _____________________
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