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Task A. Alchemist
Our hero Stancho is not only a programmer but an alchemist also. He succeeds to find the
philosophical stone, with using of which he is transforming some combinations of chemical
elements to another combinations of chemical elements. Each of the elements enters in the
alchemist reaction with mass 1 gram and each obtained element is also with mass 1 gram (it is clear
that to escape the law for conservation of the masses is not a big problem for Stancho). Stancho is
starting with 1 gram of lead, then he can include the obtained elements in new reaction, then in new,
and so on, trying to obtain as much as possible gold. Write a program that for given list of reactions
is finding the maximal quantity of gold that Stancho is able to obtain.
Input
On the first row of the standard input the number of the test cases will be given. Each test case will
start with the number K of different elements that could participate in reactions (2K6). The
second line is a list of the elements, separated by spaces (lead and gold are always in the list).
Names of the elements are not longer than 10 letters. In the third line the number L of possible
reactions will be given (1L100). Each two consecutive of the last 2L lines contain the description
of one alchemist reaction: the list of input elements is given on the first, the output elements – on
the second. All names in the lists are separated by spaces.
Output
For each test case the program has to print the maximal quantity of gold that Stancho is able to
obtain or the text AS MANY AS NECESSARY if it is possible to obtain any quantity of gold.
Sample input
1
5
lead gold sulfur mercury copper
5
lead
gold sulfur mercury
lead
gold copper mercury
mercury gold
sulfur
mercury copper
mercury sulfur gold
sulfur mercury
gold
Sample output
3
Task B. Beer
In the programming Stancho is not very strong. But in drinking beer – yes, he is. That is why, for
his birthday he decided to surprise his friends by proposing to each of them a collection of beers as
follows. In the pub where the birthday will be celebrated many different kind of beers are proposed.
Stancho has N friends. To each of them Stancho would like to propose the same number of bottles.
The collection of each friend has to be different but with the same prize. It is possible that for some
kinds of beer more than one bottle is included in the collection. For example, 2 bottles of Tuborg
and 1 bottle of Guiness is different collection from 1 bottle of Tuborg and 2 bottles of Guiness. The
pub is promising that from each kind of beer there is (theoretically) unlimited quantity of bottles.
Write a program which composes a different collection for each of Stancho’s friends. If more than
one solution exists then the program has to find one of them.
Input
On the first row of the standard input the number of the test cases will be given. Each test case will
start with the number N of friends (1N100). In the second line the number K of different kinds of
beer will be given (1K100). Each of the last K lines will contain the name of the beer (string of
length less then or equal 10) and the price of the beer — an integer less then or equal 32767 GBU
(generalized beer units).
Output
For each of the test cases the program has to print N lines, one line for each collection. The
description of the collection is a list of couples name – quantity. All names and numbers in the list
have to be separated by spaces. If there is no solution, then the file has to contain only one line for
this test case with the word NO.
Sample input
1
2
3
kamenitza 1
tuborg 2
guiness 3
Sample output
guiness 1 kamenitza 2
tuborg 2 kamenitza 1
Task C. Chemical reactions
There is one cup with a mixture of chemical compounds. Many chemical reactions can happen with
this mixture changing the chemical compounds in the cup. Each possible mixture of chemical
compounds has its energy level. All mixtures have different energy levels. All possible mixtures are
numbered with numbers from 1 to N. For each mixture there are several possible reactions that
convert it to another mixture. In all cases a reaction that converts the current mixture to a mixture
with minimal energy takes place.
There is only one mixture that has energy 0 and there are no possible reactions for it. Write a
program that determines all mixtures that lead to the mixture with energy 0.
Input
The first line of the standard input contains T – the number of test cases. Each case starts with the
number of mixtures – N (N < 10000). Next N lines describe one mixture. Each line starts with two
integers – the energy, and Mi – the number of reactions available with this mixture. Next Mi
numbers on the line are the numbers of the mixtures that are product of the possible reactions.
Output
For each test case output “Test case <number of case>:” followed by the numbers of the
mixtures that lead to the mixture with energy 0.
Sample input
1
6
5
0
9
7
8
6
1
0
2
2
1
1
2
1 6
6 3
4
5
Sample output
Test case 1: 1 2 3
Task D. Dog Task
Hunter Bob often walks with his dog Ralph. Bob walks with a constant speed and his route is a polygonal line (possibly
self-intersecting) whose vertices are specified by N pairs of integers (Xi, Yi) – their Cartesian coordinates.
Ralph walks on his own way but always meets his master at the specified N points. The dog starts his journey
simultaneously with Bob at the point (X1, Y1) and finishes it also simultaneously with Bob at the point (XN, YN).
Ralph can travel at a speed that is up to two times greater than his master's speed. While Bob travels in a straight line
from one point to another the cheerful dog seeks trees, bushes, hummocks and all other kinds of interesting places of
the local landscape which are specified by M pairs of integers ( X j , Y j ). However, after leaving his master at the point
(Xi, Yi) (where 1  i  N ) the dog visits at most one interesting place before meeting his master again at the point (Xi+1,
Yi+1).
Your task is to find the dog's route, which meets the above requirements and allows him to visit the maximal possible
number of interesting places. The answer should be presented as a polygonal line that represents Ralph's route. The
vertices of this route should be all points (Xi, Yi) and the maximal number of interesting places ( X j , Y j ). The latter
should be visited (i.e. listed in the route description) at most once.
An example of Bob's route (solid line), a set of interesting places (dots) and one of the best Ralph's routes (dotted line)
are presented in the following picture:
Y
2
4
1
3
X
Input
The first line of the standard input contains the number of test cases. Each test case starts with two integers N and M,
separated by a space ( 2  N  100 , 0  M  100 ) on a single line. The second line of each test case contains N pairs of
integers X1, Y1, ..., XN, YN, separated by spaces, that represent Bob's route. The third line of each test case contains M
pairs of integers X 1 , Y1 , ..., X M , YM , separated by spaces, that represent interesting places.
All points in the input file are different and their coordinates are integers not greater than 1000 by the absolute value.
Output
For each test case output two lines to the standard output. The first line for each test case should contain the single
integer K – the number of vertices of the best dog's route. The second line should contain K pairs of coordinates X 1 ,
Y1 , ..., X K , YK , separated by spaces, that represent this route. If there are several such routes, then you may write any
of them.
Sample input
4 5
1 4 5 7 5 2 -2 4
-4 -2 3 9 1 2 -1 3 8 -3
Sample output
6
1 4 3 9 5 7 5 2 1 2 -2 4
Task E. Exponentials
Stancho is building a tower. But it is not an ordinary tower, it is a power tower! Actually he has
been challenged by Pancho to prove his mathematics skills. Stancho and Pancho are now playing
the following game:
 Fancho (the master of fair-play) tells Stancho and Pancho an integer N (N ≥ 2).
 Stancho writes on a paper a sequence of integers a1, a2, …, ak such that they are all greater
than 1 and their product is not greater than N.
 Pancho does the same.

The one whose sequence is valid and the power tower
a1
a2
a3
ak
has greater
value wins. In case of a tie, the one whose sequence is lexicographically smaller wins. In
case of a tie again, the player whose name begins with 'S' wins.
It is not hard to see that Stancho has a winning strategy. Stancho has even proved it in the article
"Why Stancho always beats Pancho in the game Power Tower." published by Mineral Water
Journal this year. However, Stancho's brain is not as fast as his 32 bit CPU, so he wants to cheat a
little... Just a little... And you have to write a program which plays the game for him. Otherwise,
you will have 7 years of unhappiness.
Input
On the standard input, several game descriptions will be given. Each one consists of a single
number - the value of N (1 < N ≤ 2000000000).
Output
For each game description, write to the standard output a line with the winning sequence of
numbers separated by space.
Sample input
17
130
Sample output
2 2 2 2
2 2 2 2 2 2 2
* Since I’m not sure how readable it will be printed, the power tower is a1^a2^...^ak, where ^
means exponentiation.
Task F. Fractions
A simple continuous fraction has the form as shown on
the right, where the ai’s are integer numbers. It can be
denoted as [a1, a2, ..., an]. It is not difficult to show that
any rational number p/q, with integers p > q > 0, can be
represented in a unique way by a simple continuous
fraction with n terms, such that p/q = [a1, a2, ..., an−1, 1],
where n and the ai’s are positive natural numbers.
Your task is to find and print the simple continuous
fraction that corresponds to a given rational number.
Input
To the standard input a series of cases will be given, each one in a line. A line describing a case
contains p and q, two integer numbers separated by a space, with 1019 > p > q > 0. The end of the
input is indicated by a line containing “0 0”.
Output
Cases must be analyzed in the order that are read from the input and the result must be written to the
standard output. Output for each case will consist of several lines. The first line indicates the case
number, starting at 1, using the format: Case i: replacing i by the corresponding case number.
The second line displays the input data in the form p / q. The remaining lines must contain the
continuous fraction corresponding to the rational number, p/q , specified in the given input line. The
continuous fraction must be printed accordingly to the following rules:
 Horizontal bars are formed by sequences of dashes ‘-’.
 The width of each horizontal bar is exactly equal to the width of the denominator under it.
 Blank characters should be printed using periods ‘.’.
 The number on a fraction numerator must be printed center justified. That is, the number of
spaces at either side must be same, if possible; in other case, one more space must be added
at the right side.
Sample Input
75 34
65 60
0 0
Sample Output:
Case 1:
75 / 34
..........1......
2.+.------------............1....
....4.+.--------..............1..
........1.+.----................1
............5.+.................1
Case 2:
65 / 60
......1...
1.+.-----.........1
....11.+..........1
Task G. Gancho
Stancho is becoming very good in programming. He can write in many programming languages and
his programs are always optimal. Because he is very good programmer he always writes his sources
on one line. Of course this is not problem for the compilers because usually ‘;’ is used as separator.
Stancho is so proud of his idea for writing code on one line that he decided to name this source code
– Stancho code. Everything would have been perfect if some parts of the source didn't start to miss.
Stancho thinks that Gancho deletes some of his sources. Fortunately he keeps old copies of his
sources.
However Stancho is interested in understanding what parts of his code were removed. He knows
that some substrings of his original code were removed and their number is minimal. Write a
program that finds where these substrings are. The program must also print these removed
substrings in sorted order. If there are multiple ways to remove substrings with same number of
deletions, select this way that has first removed substring with minimum start point, if there are
again multiple possibilities choose this one with minimum end point of the first removed substring.
If there are again multiple possibilities choose this one with minimum start point of the second
removed substring and so on...
Input
The first line of the standard input contains T – the number of test cases. Each case consists of two
lines – the first line is the Stancho code with some parts removed, the second one is the original
Stancho code. The first line is up to 1000 characters long. The second one is up to 2000 characters
long. All lines may contain ASCII characters with codes between 32 (space) and 127 (including 32
and 127).
Output
For each test case output “Test case <number of case>:” followed by the number of
substrings N. Next 2N numbers must describe each substring – the start and the end position in the
string. For additional information see the sample output.
Sample input
2
abalen
nabaxelenpe
int m()
int main()
Sample output
Test case 1: 3 1 1 5 6 10 11
Test case 2: 1 6 8
Task H. Mystic
Stancho discovered that for some sequences there is a number of balance and peace, of happiness
and joy, of freedom and prosperity – the mystic number. It is a number from the sequence such that
the numbers which are smaller than it are exactly as much as the numbers which are greater than it.
Since he has many sequences, he needs a program which finds the mystic number of each sequence.
Input
Several sequences will be given on the standard input. Each of them is described in two lines - the
first contains N (0 < N < 100) the number of elements in it. N is always odd number. On the second
there are exactly N different numbers separated by spaces. The input ends with a line with number
0.
Output
For each of the input sequence, write to the standard output a line with its mystic number.
Sample input
3
1 2 3
5
5 6 3 4 2
0
Sample output
2
4
Task I. Trinversions
Many of you have heard about the research Stancho made about the complexity of the insert sort
algorithm. In fact, he discovered that many algorithms depend on the number of inversions from the
given sequence (inversion in the sequence A is a pair of indices i, j such that i < j and A[i] >
A[j]). He also discovered that this area is so well studied, that it is hard to add something
significant to it. That’s why he is trying new and completely different approach. First of all, he will
not sort all kinds of sequences any more – he is specializing on sorting digits of a decimal numbers.
Second, his algorithms will not be based on the traditional swap primitive (which swaps two
values) but a new triswap primitive, which is built in all quantum computers. The triswap
instruction works as follows – give 3 values to it which are currently sorted in descending order and
it will reorder them, such that they will now be sorted in ascending order. The great optimization
comes from the fact that it is actually exchanging only 2 values (the first and last) but is sorting 3
elements! Of course, there is a drawback – if the triplet given to it is not a trinversion, its behavior is
undefined, and the whole quantum CPU may collapse, destroying many parallel universes. A
trinversion for a sequence A is a triplet of indices i, j, and k such that i < j < k and A[i] >
A[j] > A[k].
Stancho has already implemented many triswap based sorting algorithms. However it is very
dangerous to test them. Because of this, he wants to see how they will perform on some numbers.
He wants you to write a program which counts the trinversions in the sequence of digits in some
decimal numbers.
Input
Several decimal integers will be given on the standard input. Each of them is greater than 0 and less
than 101000000.
Output
For each of the given numbers, write to the standard output a single line with the number of
trinversions of its digits.
Sample input
12341651
64271241
Sample output
1
13
Task J. Magics
Stancho just read something about graphs. As he understands, graphs are some circles with some
arrows between them. The number of circles is always N (no other letter can be used) and the
number of arcs is exactly M. Each circle has an unique identifier between 0 and N-1, and each arc
starts from one circle and ends in another (or the same). Stancho has drawn one beatuful graph on
the wall of his room. One of the toys of Stancho – Munio the Magic Toy is a magic toy. Every night
after Stancho fall asleep, Munio wakes up, jumps on some circle of the graph, and jumps exactly K1 times to other circles. No more than K-1 times he jumps, and no less! He may jump from one
circle to another only if there is an arrow from the first to the second. He remembers the identifiers
of the circles he has jumped on, and their sequence he calls a magic path. Munio always succeeds to
make a magic path and never makes a magic path he has already made.
Input
On the standard input, descriptions of the Stancho's graph in several parallel universes will be
given. Each description begins with 3 integers – N, M and K (1 ≤ N ≤ 500, 1 ≤ K ≤ 20). Then M
lines follow describing each arrow by two numbers between 0 and N-1 – the identifiers of the
starting circle and the ending circle.
Output
For each of the of the parallel universes, write to the standard output a line with one number – how
many nights Munio the Magic Toy can jump satisfying the above conditions.
Sample input
3
0
1
5
0
0
0
0
1
1
1
2
2
3
2 2
1
2
10 3
1
2
3
4
2
3
4
3
4
4
Sample output
2
10