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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 1 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 2 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 _____________________ 3