LITHUANIAN OLYMPIADS INFORMATICS

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50If inputs of network are equal to x 1=5, x 2=9, x 3=4, x 4=2, then outputs will beequal to: y 1=4, y 2=2, y 3=5, y 4=9.Task. Parallel sorting network is presented in the picture 234.5. The sequencebeing sorted is known too: x 1=6, x 2=5, x 3=4, x 4=3, x 5=2, x 6=1. Calculate the values ofy 1, y 2, y 3, y 4, y 5, y 6.235. Roman Numerals. Latin letters I, V, X, L, C, D, M, meaning numbers 1, 5, 10,50, 100, 500, 1000 accordingly, are used for writing Roman numerals. Value of a Romannumber is calculated in the following way: if the larger digit is before the smallerone or if two digits are the same, then they are added, and if the smaller digit is beforethe larger one, then the smaller one is subtracted from the larger digit. For example,MCXLIV == 1000 + 100 + 50 – 10 + 5 – 1 == 1144.These rules are not enough if one wants to find out how is the Roman numberbeing formed, but they can be used for convert Roman number into Arabic unambiguously.Assume, that the largest Roman number is MMMCMXCIX (3999).Task. Write a program to convert the given Roman number into Arabic.236. Three-digit Numbers. Given a sequence of digits. There is minus one (-1) atthe end of it. It does not belong to the sequence (it is not a digit), it just indicates theend of it. The sequence has not less than three members.Task. Write a program to calculate how many different three-number digits it ispossible to make each time taking 3 digits standing next to each other. Digits can notbe not exchanged; the first digit of number cannot be equal to 0.For example, if the input data are: 2 4 5 5 4 5 5 2 -1, the solution should be5, because five different three-digit numbers can be formed (245, 455, 554, 545, 552).STAGE IISENIOR DIVISION237. Parallel sorting (theoretical task). Sequences of numbers often have to besorted in ascending or descending order. During the process of sorting, elements ofsequences are compared and swapped. Sometimes these actions can be performed inparallel. Consider the sequence on pict. 234.1.Two pairs of elements can be compared and swapped at the same moment: thesecond element with the fourth one and the fifth with the sixth one – the sorted sequenceis obtained.Comparator is used in algorithms of parallel sorting. It is a circuit having twoinputs and two outputs (picture 234.2: inputs are on the left, outputs – on the right).For example, when x 1=3, x 2=1, we obtain situation in picture 234.3.Comparators are joined with each other and so sorting networks are obtained.They sort members of sequence in ascending order. Note that inputs and outputs of
any comparator can be only theadjacent elements of the sequencebeing sorted. Only one comparatorcan work with the same elementin parallel. Undoubtedly, not everysuch network sort sequencecompletely Consider the example:If inputs of network on the leftare equal to x 1=5, x 2=9, x 3=4, x 4=2,x 5=6, then outputs will be equal to:y 1=5, y 2=2, y 3=4, y 4=6, y 5=9.Let’s call all the comparations,performed in parallel (i.e. at theX 11 2 3 4žings. step žings. step žings. step žings. stepPicture 237.151y 1yX 22X 3y 3X 4X 5y 5same time) a step. Network is assumed to be more effective if number of steps is less.Task. Create a parallel sorting network to sort the sequence of four members completely.Number of steps must be the least.Represent this network in scheme, as it is shown in picture 237.1.238. Skyscraper. There are n offices, numbered from 1 to n in a new skyscraper.The table (-es) with the number of the office should be fastened to the door of eachoffice. Only one digit is written on one table. If the number of the office is not onedigitnumber, several tables should be fastened to each door.Task. Write a program to calculate the numbers of tables with each digit separatelyneeded to fasten numbers to the doors of all the offices.For example, if n==6, the output should be 0 1 1 1 1 1 1 0 0 0; here thefirst number means the needed amount of tables with digit 0, the second one meansthe amount of tables with digit 1 and etc. The last number means the amount of tableswith digit 9.239. Yang’s Table. Yang’s table consists of different natural numbers groupedinto the rows of non-increasing length, aligned left; members of each row increasefrom left to right, and members of each column increase downwards.There exists an algorithm to update Yang’s table a new element (number). Considerthe example in picture 239.1.The Yang’s table above must be updated with the new element 8.We do the following: write 8 to theplace of number 9 being in the first row,because 9 is the smallest number of thisrow greater than 8. Now number 9 is“pushed” to the second row, where itreplaces number 10. This one “pushes”13 from the third row to the fourth one.1 3 5 9 12 162 6 10 154 13 141117Picture 239.1y 4
Page 1 and 2: Valentina DagienėJūratė Skūpien
Page 3 and 4: PrefaceLithuanian Informatics Olymp
Page 5 and 6: ) There are n (n ≥ 2) coins. Writ
Page 7 and 8: STAGE II7SENIOR DIVISION182. Code o
Page 9 and 10: 9186. Amusement Park. Two carousels
Page 11 and 12: to crossing c, then number of cross
Page 13 and 14: The message was hidden in the squar
Page 15 and 16: Below is sample of map of the Kingd
Page 17 and 18: One instruction can involve 1, 2 or
Page 19 and 20: Andrius has not quite decided how m
Page 21 and 22: Output. Output must be written to t
Page 23 and 24: station) ts (1 ≤ ts ≤ 150). In
Page 25 and 26: We will analyze the following compu
Page 27 and 28: OLYMPIAD XII27STAGE IJUNIOR DIVISIO
Page 29 and 30: 29Rank of tomatoes originallyRank o
Page 31 and 32: Task. Given the first member of seq
Page 33 and 34: wards). There are no spaces between
Page 35 and 36: actions until there are no more dig
Page 37 and 38: written from the first position of
Page 39 and 40: Task. Write a program to find the l
Page 41 and 42: Input data are presented in the inp
Page 43 and 44: 43Input Output Explanation3 4 342 4
Page 45 and 46: small Latin letters, multiplication
Page 47 and 48: 230. Interest. A man went to borrow
Page 49: 8730 - 0378 == 8352; n 5;8532 - 235
Page 53 and 54: of the boat and of the bicycle as w
Page 55 and 56: Output. N integers (numbers of jobs
Page 57 and 58: 1 2 3 4 5 6 7 8 9 10 11 12 13 141 g
Page 59 and 60: 248. Boxes. There are N boxes of va
Page 61 and 62: Output. The output file must contai

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