First Round Problems
IOI V
PROBLEM 1
You have a necklace of n beads (n <100) some of which are red, others blue and
others white, arranged at random. Let's see two examples for n = 29:
1 2 1 2
o x x o x o o x
o x x x
o o x o
o o @ o
x o @ @
x x o o
x x x x
x x o x
o o x o
x o o o
x o o o
o o o x
o x o o o @
Figure a Figure b
o red bead
x blue bead
@ white bead
(The beads considered first and second in the text that follows have been marked
in the picture).
The configuration in Fig. a) may be represented as a string of b's and r's, where b
represents a blue bead and r a red one, as follows:brbrrrbbbrrrrrbrrbbrbbbbrrrrb
Suppose you are to break the necklace, lay it out straight, and then collect beads
of the same colour from one end until you reach a bead of a different colour, and
do the same for the other end (which may not be of the same colour as the beads
collected before this).
Determine the point where the necklace should be broken so that the most
number of beads can be collected.
For example, for the necklace in Fig. a), 8 beads can be collected, with the
breaking point either between bead 9 and bead 10, or between bead 24 and bead
25.
In some necklaces, white beads had been included as shown in Figure b) above.
When collecting beads, a white bead that is encountered may be treated as either
red or blue, and painted with the desired colour. The string that represents this
configuration will include the symbols: r, b and w.
Write a program to do the following:
1. Read a configuration from an ASCII input file, NECKLACE.DAT,
with each configuration on one line. Write this data into an ASCII output
file, NECKLACE.SOL. An example of an input file would be:
Example:
NECKLACE.DAT
brbrrrbbbrrrrrbrrbbrbbbbrrrrb
bbwbrrrwbrbrrrrrb
2. For each configuration, determine the maximum number, M, of beads
collectable, along with a breaking point.
3. Write to the outfile, NECKLACE.SOL, the number M and the breaking
point. The solutions for different configurations should be separated with
a blank record.
Example of a possible solution:
NECKLACE.SOL
brbrrrbbbrrrrrbrrbbrbbbbrrrrb
8 between 9 and 10
bbwbrrrwbrbrrrrrb
10 between 16 and 17
PROBLEM 2
Some companies are partial owners of other companies because they have
acquired part of their total shares. For example, Ford owns 12% of Mazda. It is
said that a company A controls company B if, at least, one of the following
conditions is satisfied:
a) A = B
b) A owns more than 50% of B
c) A controls k (k > 1) companies,
C(1), ..., C(k), so that:
C(i) owns x(i)% of B for 1 < i < k and x(1) + .... + x(k) > 50.
The problem to solve is:
Given a list of triples (i,j,p) which means that the company i owns p% of
company j, calculate all the pairs (h,s) so that company h controls company s.
There are at most 100 companies.
Write a program to:
1 Read from an ASCII input file, COMPANY.DAT, the list of
triples, (i,j,p), to be considered for each case (that is, each data set),
where i, j and p are positive integers. Different cases (data sets) will
be separated with a blank record.
2 Find all the pairs (h,s) so that company h controls company s.
3 Write to an ASCII output file, COMPANY.SOL, all the pairs (h,s)
found, with h different from s. The pairs (h,s) must be written in
consecutive records and in increasing order of h. The solutions for
different cases must be separated with a blank record.
Example:
COMPANY.DAT COMPANY.SOL
2 3 25 4 2
1 4 36 4 3
4 5 63 4 5
2 1 48
3 4 30
4 2 52
5 3 30
1 2 30 2 3
2 3 52 2 4
3 4 51 2 5
4 5 70 3 4
5 4 20 3 5
4 3 20 4 5
PROBLEM 3
N rectangles of different colours are superposed on a white sheet of paper. The
sheet's sizes are: a cm wide and b cm long. The rectangles are put with their
sides parallel to the sheet's borders. All rectangles fall within the borders of the
sheet. As result, different figures of different colours will be seen. Two regions
of the same colour are considered to be part of the same figure if they have at
least one point in common; otherwise, they are considered different figures. The
problem is to calculate the area of each of these figures. a, b are even positive
integers not greater than 30.
The coordinate system considered has origin at the sheet's center and the axes
parallel to the sheet's borders:
Different data sets are written in an ASCII input file, RECTANG.DAT:
a, b and N will be in the first line of each data set, separated by a blank
space.
• In each of the next N lines you will find:
• the integer coordinates of the position where the left lower vertex of
the rectangle was put.
• followed by the integer coordinates of the position where the upper
right vertex of the rectangle was put
• and, then, the rectangle's colour represented by an integer between 1
and 64. White colour will be represented by 1.
The order of the records corresponds to the order used to put the rectangles on the
sheet. Different data sets will be separated with a blank record.
Write a program to:
1. Read the next data set from RECTANG.DAT
2. Calculate the area of each coloured figure
3. Write in an ASCII output file, RECTANG.SOL, the colour and the area
of each coloured figure as shown in the example below. These records will
be written in increasing order of colour. The solutions to different data sets
will be separated by a blank record.
Example:
RECTANG.DAT RECTANG.SOL
20 12 5 1 172
-7 -5 -3 -1 4 2 47
-5 -3 5 3 2 4 12
-4 -2 -2 2 4 4 8
2 -2 3 -1 12 12 1
3 1 7 5 1
30 30 2 1 630
0 0 5 14 2 2 70
-10 -7 0 13 15 15 200
You have won a contest sponsored by an airline. The prize is a ticket to travel
around Canada, beginning in the most western point served by this airline, then
traveling only from west to east until you reach the most eastern point served,
and then coming back only from east to west until you reach the starting city.
No city may be visited more than once, except for the starting city, which must
be visited exactly twice (at the beginning and the end of the trip). You are not
allowed to use any other airline or any other means of transportation. Given a
list of cities served by the airline, and a list of direct flights between pairs of
cities, find an itinerary which visits as many cities as possible and satisfies the
above conditions beginning with the first city and visiting the last city on the
list and returning to the first city.
Different data sets are written in an ASCII input file, C:\IOI\ITIN.DAT. Each data
set consists of:
• in the first line: the number N of cities served by the airline and the number V of
direct flights that will be listed. N will be a positive integer not larger than 100.
V is any positive integer.
• in each of the next N lines: a name of a city served by the airline. The names are
ordered from west to east in the input file. That is, the i-th city is west of the j-th
city if and only if i < j (There are no two cities in the same meridian). The name of
each city is a string of, at most, 15 digits and/or characters of the Latin alphabet,
for example: AGR34 or BEL4
(There are no spaces in the name of a city)
• in each of the next V lines: two names of cities, taken from the list of cities,
separated by a blank space. If the pair city1 city2 appears in a line, it indicates
that there exists a direct flight from city1 to city2 and also a direct flight from
city2 to city1.
Different data sets will be separated by an empty record (that is, a line containing
only the end of line character). There will be no empty record after the last data set.
The following example is stored in file
C:\IOI\ITIN.DAT.
8 9 5 5
Vancouver C1
Yellowknife C2
Edmonton C3
Calgary C4
Winnipeg C5
Toronto C5 C4
Montreal C2 C3
Halifax C3 C1
Vancouver Edmonton C4 C1
Vancouver Calgary C5 C2
Calgary Winnipeg
Winnipeg Toronto
Toronto Halifax
Montreal Halifax
Edmonton Montreal
Edmonton Yellowknife
Edmonton Calgary
The input may be assumed correct. No checking is necessary.
The solution found for each data set must be written to an ASCII output file,
C:\IOI\ITIN.SOL: in the first line, the total number of cities in the input data set;
in the next line, the number M of different cities visited in the itinerary, and in the
next M+1 lines the names of the cities, one per line, in the order in which they are
visited. Note the first city visited must be the same as the last. Only one solution is
required. If no solution is found for a data set, only two records for this data set must
be written in ITIN.SOL, the first one giving the total number of cities, and the second
one saying: "NO SOLUTION".
A possible solution for the above example:
ITIN.SOL
8 5
7 NO SOLUTION
Vancouver
Edmonton
Montreal
Halifax
Toronto
Winnipeg
Calgary
Vancouver
Put your program solution into an ASCII file named C:\IOI\DDD.xxx. Extension .xxx
is .BAS for Qbasic, .LCN for LOGO, .C for C, .PAS for PASCAL.