ITMO Summer Camp Scholarship
Online Training Program, April 2015
Problem A. Bike Roads
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
bike.in
bike.out
2 seconds
64 megabytes
Andrew lives at countryside. The area he lives at has two bike roads, each of which has the form of a
circle with radius r. Roads have no common points.
Andrew’s house is located at one of the roads, and his school is located at the other one. Each day
Andrew rides his bike from home to school and back. He has noticed that riding along the road is easier
than riding by the ground. When riding along the road Andrew’s speed is u, and when riding by the
ground his speed is v. Now Andrew wonders what minimal time he needs to get from his house to school.
Let us introduce the coordinate system so that the center of the bike road where Andrew’s house is
located were (0, 0), and the center of the bike road where his school is located were (0, d). The radius of
each road is r. Andrew’s house is located at a point (x1 , y1 ), and his school is located at a point (x2 , y2 ).
His speed by the road is u, and his speed by the ground is v.
Input
Input ﬁle contains
point numbers: d, r, x1 , y1 , x2 , y2 , u and v (1 ≤ r ≤ 100, 2r < d ≤ 100,
√eight ﬂoating√
2
2
1 ≤ v < u ≤ 10, x1 + y1 = r, x22 + (y2 − d)2 = r). All equalities are up to 10−9 .
Output
Output one ﬂoating point number — the minimal time Andrew needs to get from his house to school.
Your answer must be accurate up to 10−6 .
Example
bike.in
20 5
5 0 5 20
2 1
20 5
-5 0 5 20
2 1
bike.out
16.5757337181
17.2040517249
Page 1 of 30
ITMO Summer Camp Scholarship
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Problem B. Cookies
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
cookies.in
cookies.out
2 seconds
64 megabytes
Santa Claus is planning to bring gifts to n children. He has m cookies and is planning to divide them to
n piles. However, as usually problems come unexpected. The child gets unhappy if somebody gets more
cookies than him.
Each child is characterized by his greediness, the greediness of the i-th child is gi . The unhappines of the
i-th child is equal to gi ai where ai is the number of children that get more cookies than him.
Now Santa wants to divide cookies in such a way that the total unhappiness is minimized. Each child
must get at least one cookie. Santa would like to give away all m cookies he has. Help him to do so.
Input
The ﬁrst line of the input ﬁle contains n and m (1 ≤ n ≤ 30, n ≤ m ≤ 5000). The second line contains
n integer numbers g1 , g2 , . . . , gn (1 ≤ gi ≤ 107 ).
Output
Print the minimal possible unhappiness at the ﬁrst line of the output ﬁle. The second line must contain n
integer numbers — the number of cookies the corresponding child must get. If there are several solutions,
output any one.
Example
cookies.in
3
1
4
2
20
2 3
9
1 5 8
cookies.out
2
2 9 9
7
2 1 3 3
Page 2 of 30
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Problem C. Diversion
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
diversion.in
diversion.out
2 seconds
64 megabytes
The kingdom of Farland has n cities connected by m bidirectional roads. Some of the roads are paved
with stone, and others are just country roads. The capital of the kingdom is the city number 1. The
roads are designed in such a way that it is possible to get from any city to any other using only roads
paved with stone, and the number of stone roads is minimal possible. The country roads were designed
in such a way that if any stone road is blocked or destroyed it is still possible to get from any city to any
other by roads.
Let us denote the number of stone roads needed to get from city u to city v as s(u, v). The roads were
created long ago and follow the strange rule: if two cities u and v are connected by a road (no matter,
stone or country), then either s(1, u) + s(u, v) = s(1, v) or s(1, v) + s(v, u) = s(1, u).
The king of Edgeland is planning to attack Farland. He is planning to start his operation by destroying
some roads. Calculations show that the resources he has are enough to destroy one stone road and one
country road. The king would like to destroy such roads that after it there were at least two cities in
Farland not connected by roads any more.
Now he asks his minister of defense to count the number of ways he can organize the diversion. But the
minister can only attack or defend, he cannot count. Help him!
Input
The ﬁrst line of the input ﬁle contains n and m — the number of cities and roads respectively
(3 ≤ n ≤ 20 000, m ≤ 100 000). The following m lines describe roads, each line contains three integer numbers — the numbers of cities connected by the corresponding road, and 1 for a stone road or 0
for a country road. No two cities are connected by more than one road, no road connects a city to itself.
Output
Output one integer number — the number of ways to organize the diversion.
Example
diversion.in
6
1
2
1
3
4
3
5
7
2
3
4
4
5
6
6
diversion.out
4
1
1
0
1
1
0
1
Page 3 of 30
ITMO Summer Camp Scholarship
Online Training Program, April 2015
Problem D. ePig
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
epig.in
epig.out
2 seconds
64 megabytes
Andrew and Ann are developing the new P2P software network ePig. The network is intended to be used
for sharing ﬁles. In this problem you will have to simulate the operation of the network when distributing
one large ﬁle.
Let there be n clients numbered from 1 to n. Initially the whole ﬁle is provided by client 1. All other clients
wish to get this ﬁle. The ﬁle is split into k chunks numbered from 1 to k. The transfer consists of a series
of rounds. Each round takes one minute and the bandwidth of the connection of each client is enough to
transform one chunk to the client and transform one chunk away from the client (simultaneously). After
a client gets some chunk it starts to provide it to other clients.
Before a round each client decides which chunk it will request. The client will request the chunk that is
provided by the smallest number of clients (except those chunks that it already has). If there are several
such chunks, it selects the one which has the smaller number.
After that the clients make chunk requests. Each client selects the client which has the chunk that it
decided to request, if there are several such clients, the client which provides the smallest number of
chunks is selected. If there are still several possible variants, the client which has the smallest number is
selected.
Each client considers all requests and satisﬁes one of them. The request satisﬁed by client X is the one
which comes from the most valued client. The value of the client is the number of chunks it allowed X
to be downloaded from him in the past. If there are several equally valued clients, X gives the chunk to
the one which has the smallest number of chunks available. If there are still several possible variants, the
chunk is provided to the client which has the smallest number.
After the requests that will be satisﬁed are selected the round begins. The clients whose requests were
rejected do not download anything this round, all other clients download the chunk they requested. After
that the new round starts.
Given n and k, you have to ﬁnd for each client, what is the number of rounds before it gets the whole
ﬁle.
Input
The ﬁrst line of the input ﬁle contains n and k (2 ≤ n ≤ 100, 1 ≤ k ≤ 200).
Output
Output n − 1 numbers — for each client except the ﬁrst one print the number of rounds before it gets
the whole ﬁle.
Example
epig.in
3 2
epig.out
3 3
The ﬁle distribution will proceed as follows. At the ﬁrst round clients 2 and 3 will request chunk 1 from
client 1. Request from client 2 will be satisﬁed. After that clients 2 and 3 will request chunk 2 from client
1. Client 3 will be satisﬁed. On the third round client 2 will request chunk 2 from client 3 and client 3
will request chunk 1 from client 2, both requests will be satisﬁed, and both will have the whole ﬁle.
Page 4 of 30
ITMO Summer Camp Scholarship
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Problem E. Geometry Problem
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
geometry.in
geometry.out
2 seconds
64 megabytes
Peter is studying in the third grade of elementary school. His teacher of geometry often gives him diﬃcult
home tasks.
At the last lesson the students were studying circles. They learned how to draw circles with compasses.
Peter has completed most of his homework and now he needs to solve the following problem. He is given
two segments. He needs to draw a circle which intersects interior of each segment exactly once.
The circle must intersect the interior of each segment, just touching or passing through the end of the
segment is not satisfactory.
Help Peter to complete his homework.
Input
The input ﬁle contains several test cases. Each test case consists of two lines.
The ﬁrst line of the test case contains four integer numbers x11 , y11 , x12 , y12 — the coordinates of the
ends of the ﬁrst segment. The second line contains x21 , y21 , x22 , y22 and describes the second segment in
the same way.
Input is followed by two lines each of which contains four zeroes these lines must not be processed.
All coordinates do not exceed 102 by absolute value.
Output
For each test case output three real numbers — the coordinates of the center and the radius of the
circle. All numbers in the output ﬁle must not exceed 1010 by their absolute values. The jury makes all
comparisons of real numbers with the precision of 10−4 .
Example
geometry.in
0
1
0
0
0
0
0
0
0
1
0
0
4
4
0
0
geometry.out
0.5 0 2
Page 5 of 30
ITMO Summer Camp Scholarship
Online Training Program, April 2015
Problem F. Hex
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
hex.in
hex.out
2 seconds
64 megabytes
The game of hex is played on a rhombic ﬁeld of size n × n divided to hexagonal cells. An example of the
ﬁeld for n = 6 is shown on the picture below.
Two players, one playing for noughts and one for crosses, make moves in turn. A move consists of putting
a player’s piece to one of the free cells. Two cells are considered adjacent if they have a common side.
The goal of noughts is to make a chain of adjacent cells from the left edge of the ﬁeld to the right edge
of the ﬁeld, the goal of crosses is to make such chain from the top edge to the bottom edge. A picture
below shows the game won by noughts.
The gameplay of hex strongly depends on so called templates. In this problem we will consider two simple
templates. The two-bridge template is the situation when two player’s pieces can be connected in two
ways in one move. Such pieces can be considered connected. A picture below shows the two-bridge for
crosses. Note that the cells marked by dots must not be occupied. If noughts play to one of these cells,
crosses reply to another cell connecting their pieces. The two cells marked by dots are called bridge cells.
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Another simple template is a similar construction at the edge of the board, called template II. A piece in
the second row at the player’s edge, with no pieces in the two cells at the edge adjacent to it, can always
be connected to the edge. A picture below shows template II for noughts.
The game is said to be won up to simple templates by one of the players if there is a sequence p1 , p2 , . . . , pk
of his pieces such that p1 and pk are at his diﬀerent edges or connected to them by template II, for all
i pieces pi and pi+1 are either adjacent or connected to each other by a two-bridge, and no two-bridges
share a bridge cell. A picture on the left below shows a game won by noughts up to simple templates. A
game on the right is not won by noughts up to simple templates yet, because a cell marked by a dot is
shared by two two-bridges.
Given the position of the ﬁeld, ﬁnd out whether the game is won up to simple templates by one of the
players.
Input
The ﬁrst line of the input ﬁle contains two integer numbers: n and m — the size of the ﬁeld and the
number of pieces on the ﬁeld (3 ≤ n ≤ 10 000, 1 ≤ m ≤ 30 000). The following m lines describe pieces.
Each piece is described by three integer numbers: x, y and s — the coordinates of the piece and the
player who owns the piece. Coordinates range from 1 to n, s is 0 for noughts and 1 for crosses. The
position need not be a valid game position, so the number of pieces of diﬀerent players can be diﬀerent.
No two pieces occupy the same cell. The coordinate system is shown on the picture below, each cell is
marked as x, y.
1,6 2,6 3,6 4,6 5,6 6,6
1,5 2,5 3,5 4,5 5,5 6,5
1,4 2,4 3,4 4,4 5,4 6,4
1,3 2,3 3,3 4,3 5,3 6,3
1,2 2,2 3,2 4,2 5,2 6,2
1,1 2,1 3,1 4,1 5,1 6,1
Page 7 of 30
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Output
Output “noughts” if the game is won by noughts up to simple templates, “crosses” if the game is won
by crosses up to simple templates and “none” if the game is not won up to simple templates yet.
Example
hex.in
6
2
4
5
3
4
5
6
2
3
4
6
3
4
4
5
6
4
3
4
5
2
1
8
4
2
3
2
4
5
6
1
hex.out
noughts
0
0
0
1
1
1
none
0
0
0
0
1
1
1
1
Page 8 of 30
ITMO Summer Camp Scholarship
Online Training Program, April 2015
Problem G. Important Roads
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
important.in
important.out
2 seconds
64 megabytes
The city where Georgie lives has n junctions some of which are connected by bidirectional roads.
Every day Georgie drives from his home to work and back. But the roads in the city where Georgie lives
are very bad, so they are very often closed for repair. Georgie noticed that when some roads are closed
he still can get from home to work in the same time as if all roads were available.
But there are such roads that if they are closed for repair the time Georgie needs to get from home to
work increases, and sometimes Georgie even cannot get to work by a car any more. Georgie calls such
roads important.
Help Georgie to ﬁnd all important roads in the city.
Input
The ﬁrst line of the input ﬁle contains n and m — the number of junctions and roads in the city where
Georgie lives, respectively (2 ≤ n ≤ 20 000, 1 ≤ m ≤ 100 000). Georgie lives at the junction 1 and works
at the junction n.
The following m lines contain information about roads. Each road is speciﬁed by the junctions it connects
and the time Georgie needs to drive along it. The time to drive along the road is positive and doesn’t
exceed 100 000. There can be several roads between a pair of junctions, but no road connects a junction
to itself.
It is guaranteed that if all roads are available, Georgie can get from home to work.
Output
Output l — the number of important roads — at the ﬁrst line of the output ﬁle. The second line must
contain l numbers, the numbers of important roads. Roads are numbered from 1 to m as they are given
in the input ﬁle.
Example
important.in
6
1
2
2
1
3
2
5
7
2
3
5
3
5
4
6
1
1
3
2
1
1
2
important.out
2
5 7
Page 9 of 30
ITMO Summer Camp Scholarship
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Problem H. Irreducible Young Diagrams
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
irreducible.in
irreducible.out
2 seconds
64 megabytes
Young diagram is a well known way to describe a partition of a positive integer number. A partition of a
number n is a representation as a sum of one or several integer numbers n = m1 + m2 + . . . + mk where
m1 ≥ m2 ≥ . . . ≥ mk .
A diagram consists of n boxes arranged in k rows, where k is the number of terms in the partition. A
row representing the number mi contains mi boxes. All rows are left-aligned, and sorted from longest to
shortest.
The diagram on the picture below corresponds to the partition 9 = 5 + 2 + 2.
Let us describe a way to transform the Young diagram. You are allowed to choose two adjacent boxes in
the diagram and remove them. The only restriction is that after the process the remaining diagram must
be a valid Young diagram. Also the resulting diagram must either be empty, or have a left-top corner at
the same position as the diagram that was transformed.
For example, removing the last boxes of the second and the third row from the diagram above, we get
the diagram for the partition 7 = 5 + 1 + 1.
Removing the last two boxes of the ﬁrst row from the same diagram we get the diagram for the partition
7 = 3 + 2 + 2.
There is one more way to transform the Young diagram in this example: remove the two boxes from the
last row.
The diagram that cannot be transformed in the above way is called irreducible. Clearly, an empty diagram
is irreducible.
Each diagram can be transformed until it is irreducible. Generally there may be several possible ways
of transforming the diagram. You are given a Young diagram. You have to ﬁnd all possible irreducible
diagrams that the given one can be transformed to.
Input
The ﬁrst line of the input ﬁle contains k — the number of rows in the diagram (1 ≤ k ≤ 100 000). The
second line contains k numbers: m1 , m2 , . . . , mk . The sum n = m1 + m2 + . . . + mk doesn’t exceed 108 .
Page 10 of 30
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Output
The ﬁrst line of the output ﬁle must contain one number l — the number of irreducible diagrams the
given diagram can be transformed to. The following l lines must describe these diagrams: each line must
contain t — the number of rows in the corresponding diagram, followed by t numbers — the number of
boxes in corresponding rows.
Example
irreducible.in
3
5 2 2
1
2
irreducible.out
1
1 1
1
0
Page 11 of 30
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Problem I. Nice Sequence
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
nice.in
nice.out
2 seconds
64 megabytes
Let us consider the sequence a1 , a2 , . . . , an of non-negative integer numbers. Denote as ci,j the number
of occurrences of the number i among a1 , a2 , . . . , aj . We call the sequence k-nice if for all i1 < i2 and for
all j the following condition is satisﬁed: ci1 ,j ≥ ci2 ,j − k.
Given the sequence a1 , a2 , . . . , an and the number k, ﬁnd its longest preﬁx that is k-nice.
Input
The ﬁrst line of the input ﬁle contains n and k (1 ≤ n ≤ 200 000, 0 ≤ k ≤ 200 000). The second line
contains n integer numbers ranging from 0 to n.
Output
Output the greatest l such that the sequence a1 , a2 , . . . , al is k-nice.
Example
nice.in
10 1
0 1 1 0 2 2 1 2 2 3
2 0
1 0
nice.out
8
0
Page 12 of 30
ITMO Summer Camp Scholarship
Online Training Program, April 2015
Problem J. Crazy Nim
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
nim.in
nim.out
2 seconds
64 megabytes
Alice and Bob like to play crazy nim. The game proceeds as follows. There are several stones arranged
in 3 piles that have a, b and c stones, respectively. Players make moves in turn, Alice moves ﬁrst.
Each turn a player can choose any pile and take any number of stones from it. There is one restriction:
it is not allowed to make two piles of equal positive size. The person who takes the last stone wins.
For example, if there are three piles with 1, 3 and 5 stones, the valid moves are:
•
•
•
•
•
•
take
take
take
take
take
take
1
1
3
1
3
5
stone from the ﬁrst pile;
stone from the second pile;
stones from the second pile;
stone from the third pile;
stones from the third pile;
stones from the third pile.
Given a, b and c, ﬁnd out who wins the game if both players play optimally.
Input
Input ﬁle contains several test cases. Each test case consists of three integer numbers a, b and c on a line
(1 ≤ a, b, c ≤ 109 , a ̸= b, a ̸= c, b ̸= c). The test cases are followed by a line that contains three zeroes.
This line must not be processed.
Output
For each line output who wins the game if both players play optimally. Adhere to the format of sample
output.
Example
nim.in
1 2 3
1 3 5
0 0 0
nim.out
Alice wins the game.
Bob wins the game.
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Problem K. Numbers
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
numbers.in
numbers.out
2 seconds
64 megabytes
Consider numbers from 1 to n.
You have to ﬁnd the smallest lexicographically number among them which is divisible by k.
Input
Input ﬁle contains several test cases. Each test case consists of two integer numbers n and k on a line
(1 ≤ n ≤ 1018 , 1 ≤ k ≤ n). The last test case is followed by a line that contains two zeroes. This line
must not be processed.
Output
For each test case output one integer number — the smallest lexicographically number not exceeding n
which is divisible by k.
Example
numbers.in
2000 17
2000 20
2000 22
0 0
numbers.out
1003
100
1012
Page 14 of 30
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Problem L. Integer Packing
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
packing.in
packing.out
2 seconds
64 megabytes
When transmitting data by network it is important to pack it eﬃciently since this allows to optimize
bandwidth usage. We will describe a decoding procedure in one eﬃcient way of packing integer numbers.
It uses variable length and has preﬁx property to ensure correct decoding. Your task will be to implement
the optimal encoding method.
The method is applicable for transmitting integer numbers from −263 to 263 − 1 (long type in Java).
Packed integer occupies from 1 to 9 bytes, depending on its value.
Decoding is performed in the following way. First the leading byte of the number is received. Looking
at its bits from higher to lower, ﬁnd the ﬁrst zero. Let q be the number of these leading ones (if the ﬁrst
byte is 0xff, q = 8). q means the number of bytes to follow. If q = 8, these eight bytes are a standard
big-endian (higher byte ﬁrst) representation of an integer number.
In the other case the following operation is performed: if the q + 2-nd bit of the ﬁrst byte is zero (bits are
numbered from 1, higher bits ﬁrst, if q = 7, the highest bit of the second byte is considered), leading ones
of the ﬁrst byte are replaced by zeroes, and the number is padded on the left by zero bytes to eight bytes.
In the other case, the q + 1-st bit (it is always zero) is replaced with one, and the number is padded on
the left with 0xff bytes to eight bytes. The result is a standard big-endian representation of an integer
number.
Given several integer number, you must encode each of them so that the length of the encoded number
were minimal possible and it decoded to the original number.
Input
The ﬁrst line of the input ﬁle contains n — the number of test cases (1 ≤ n ≤ 10 000). The following n
lines contain one number between −263 and 263 − 1, inclusive, each.
Output
Output n lines, each line must contain an encoded version of the corresponding number written as the
sequence of hexadecimal digits, higher bytes ﬁrst.
Example
packing.in
9
0
1
2
3
100
500
1000000
-1
-100
packing.out
00
01
02
03
8064
81f4
cf4240
7f
bf9c
Page 15 of 30
ITMO Summer Camp Scholarship
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Problem M. Permutation Reconstruction
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
permutation.in
permutation.out
2 seconds
64 megabytes
The famous Ulam Conjecture claims that if we take a graph G and consider a multiset Gmin of graphs
that are obtained from G by removing one of its vertices and all edges incident to it, then the graph
G can be reconstructed from Gmin . The Ulam conjecture is still not proven and no counterexample is
known.
In this problem we will consider a similar problem for permutations. Consider a permutation
a = ⟨a1 , a2 , . . . an ⟩ of numbers from 1 to n. Let us denote as a/i the permutation of n − 1 numbers
obtained from a by removing a number i and decreasing all numbers greater than i by one.
For example, if a = ⟨1, 3, 5, 2, 6, 4⟩ then a/2 = ⟨1, 2, 4, 5, 3⟩.
You are given a/1, a/2, . . . , a/n in some arbitrary order. You must restore the original permutation a.
Input
The ﬁrst line of the input ﬁle contains n — the order of the initial permutation (5 ≤ n ≤ 300). The
following n lines contain n − 1 numbers each and specify a/i for all i in some order.
Output
Output n integer numbers — the permutation a. It is guaranteed that such permutation exists.
Example
permutation.in
6
1
1
1
2
1
1
permutation.out
1 3 5 2 6 4
3
3
2
4
4
3
5
4
4
1
2
2
2
2
5
5
5
5
4
5
3
3
3
4
In this example the factors are given in the following order: a/6, a/4, a/2, a/1, a/3 and a/5.
Page 16 of 30
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Problem N. Rectangular Polygon
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
polygon.in
polygon.out
2 seconds
64 megabytes
A rectangular polygon is a polygon whose edges are all parallel to the coordinate axes. The polygon
must have a single, non-intersecting boundary. No two adjacent sides must be parallel.
Johnny has several sticks of various lengths. He would like to construct a rectangular polygon. He is
planning to use sticks as horizontal edges of the polygon, and draw vertical edges with a pen.
Now Johnny wonders, how many sticks he can use. Help him, ﬁnd the maximal number of sticks that
Johnny can use. He will use sticks only as horizontal edges.
Input
The ﬁrst line of the input ﬁle contains n — the number of sticks (1 ≤ n ≤ 100). The second line contains
n integer numbers — the lengths of the sticks he has. The length of each stick doesn’t exceed 200.
Output
Print l — the number of sticks Johnny can use at the ﬁrst line of the output ﬁle. The following 2l lines
must contain the vertices of the rectangular polygon Johnny can construct. Vertices must be listed in
order of traversal. The ﬁrst two vertices must be the ends of a horizontal edge. If there are several
solution, output any one. Vertex coordinates must not exceed 109 .
If no polygon can be constructed, output l = 0.
Example
polygon.in
4
1 2 3 5
4
1 2 4 8
4
1 1 1 1
polygon.out
3
0
1
1
3
3
0
0
4
0
1
1
2
2
1
1
0
0
0
1
1
2
2
0
0
1
1
-2
-2
-1
-1
In the ﬁrst example Johnny uses a stick of length 1 for (0, 0)−(1, 0) edge, a stick of length 2 for (1, 1)−(3, 1)
edge and a stick of length 3 for (3, 2) − (0, 2) edge. There is no way to use all four sticks.
Page 17 of 30
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Problem O. Decoding Prefix Codes
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
prefix.in
prefix.out
2 seconds
64 megabytes
A code is a mapping c : Σ → Γ∗ of characters of the given alphabet Σ to words of some another alphabet
Γ. In this problem Σ consists of lowercase letters of the English alphabet, and Γ = {0, 1}.
The code is called prefix-free or simply prefix, if no code word is a preﬁx of another word. For example,
the code c(‘a’) = 00, c(‘b’) = 01, c(‘c’) = 1 is preﬁx, but the code c(‘a’) = 0, c(‘b’) = 01 is not.
You are given a text and a string that is obtained from it by encoding it with some preﬁx code. You
must restore the code that was used to encode the text. If there are several possible variants, output any
one.
Input
The ﬁrst line of the input ﬁle contains the given text. Its length does not exceed 1000. It contains only
letters ‘a’–‘z’ of the English alphabet. There are at most 10 diﬀerent characters in the given text.
The second line contains the encoded version of the given text. It is guaranteed that it is obtained from
the given text by encoding by some preﬁx code. No word in the code used has length exceeding 10.
Output
Output the preﬁx code that could be used to encode the text to get the given encoded version. The
number of lines must be equal to the number of diﬀerent characters in the given text. Each line must
contain a character followed by a space and the code word for this character.
Example
prefix.in
hello
0100111000
prefix.out
e
h
l
o
001
01
1
000
Page 18 of 30
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Problem P. Hungry Queen
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
queen.in
queen.out
2 seconds
64 megabytes
Consider an inﬁnite chessboard with cells identiﬁed by pairs of integer numbers: (x, y). The black queen
is initially located at the cell (0, 0). The queen can move horizontally, vertically or diagonally, but cannot
move downwards. That is, after each turn the y-coordinate of the cell where the queen is after the turn
must be greater or equal to the y-coordinate of the cell where it was before the turn.
There are n white pawns on the board, they are located at cells (xi , yi ), where yi > 0.
The queen wants to take as many pawns as possible. White pawns do not move, and the queen can make
as many consecutive turns as needed. However, each turn the queen must take a pawn.
Find out what is the maximal number of pawns the queen can take, and which pawns it must take to
achieve this number.
Input
The ﬁrst line of the input ﬁle contains n (1 ≤ n ≤ 50 000). The following n lines contain two integer
numbers each — the coordinates (xi , yi ) of the pawns (|xi | ≤ 109 , 0 < yi ≤ 109 ). No two pawns occupy
the same position.
Output
The ﬁrst line of the output ﬁle must contain one integer number k — the number of pawns the queens
can take. The second line must contain k integer numbers — the numbers of the pawns the queen can
take in order she must do it. The pawns are numbered starting from 1 in order they are given in the
input ﬁle.
Example
queen.in
4
1 1
4 3
-1 3
4 2
queen.out
3
1 3 2
Page 19 of 30
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Problem Q. Boat Race
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
race.in
race.out
2 seconds
64 megabytes
Jerry is planning to take part in a boat race. The race will take place in a long narrow canal. The canal
runs from east to west, the banks of the canal have a form of a polyline.
Let us introduce the coordinate system in such a way that the western end of the canal has x-coordinate
equal to 0, the eastern end of the canal has x-coordinate equal to l. The polyline describing the southern
bank of the canal has vertices at points (x0 , y0 ), (x1 , y1 ), . . . , (xn , yn ) where 0 = x0 < x1 < . . . < xn = l.
The northern bank has the same form, but is w units to the north, so it is described by a polyline with
coordinates (x0 , y0 + w), (x1 , y1 + w), . . . , (xn , yn + w).
Jerry’s boat can start the race at any point of a start line (a segment (0, y0 ) − (0, y0 + w)) and end the
race at any point of the ﬁnish line (a segment (l, yn ) − (l, yn + w)). The boat is so small, that it can be
considered a point. When moving a boat can “touch” the banks of the canal, moving just along them.
To increase his chances of winning, Jerry wants to know what is the shortest path from the start line to
the ﬁnish line.
Input
The ﬁrst line of the input ﬁle contains n (1 ≤ n ≤ 100). The following n + 1 lines contain a pair of
integer numbers (xi , yi ) each and describe the southern bank of the canal (0 = x0 < x1 < . . . < xn ≤ 104 ,
|yi | ≤ 104 ). The last line of the input ﬁle contains integer number w (1 ≤ w ≤ 104 ).
Output
Output one ﬂoating point number — the length of the shortest path through the canal from the start
line to the ﬁnish line. Your answer must be accurate up to 10−6 .
Example
race.in
3
0
2
3
5
2
race.out
5.41421356237309505
0
2
-1
0
Page 20 of 30
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Problem R. SETI
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
seti.in
seti.out
2 seconds
64 megabytes
Amateur astronomers Tom and Bob try to ﬁnd radio broadcasts of extraterrestrial civilizations in the
air. Recently they received some strange signal and represented it as a word consisting of small letters
of the English alphabet. Now they wish to decode the signal. But they do not know what to start with.
They think that the extraterrestrial message consists of words, but they cannot identify them. Tom and
Bob call a subword of the message a potential word if it has at least two non-overlapping occurrences in
the message.
For example, if the message is “abacabacaba”, “abac” is a potential word, but “acaba” is not because
two of its occurrences overlap.
Given a message m help Tom and Bob to ﬁnd the number of potential words in it.
Input
Input ﬁle contains one string that consists of small letters of the English alphabet. The length of the
message doesn’t exceed 10 000.
Output
Output one integer number — the number of potential words in a message.
Example
seti.in
abacabacaba
seti.out
15
Page 21 of 30
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Problem S. Spam Filter
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
spam.in
spam.out
2 seconds
64 megabytes
Most spam ﬁlters use so called Bayesian ﬁlter to choose which message is spam and which is not. Let us
describe the idea of such ﬁlter.
The ﬁrst stage of using Bayesian ﬁlter is its training. During training the ﬁlter is shown a number of
messages for each of which it is speciﬁed whether it is spam or not. Denote the set of messages that are
spam as Spam and the set of other messages as Good.
Filter parses the messages to words and for each word w it calculates the probability that it is in a spam
message:
|{M |w ∈ M and M ∈ Spam}|
P (w ∈ M |M ∈ Spam) =
,
|Spam|
and the probability that it is not in a spam message
P (w ∈ M |M ∈ Good) =
|{M |w ∈ M and M ∈ Good}|
.
|Good|
After that for each word w in a message to classify the ﬁlter uses this information to calculate
P (M ∈ Spam|w ∈ M ) using Bayes’ formula:
P (M ∈ Spam|w ∈ M ) =
P (w ∈ M |M ∈ Spam)P (Spam)
.
P (w ∈ M |M ∈ Spam)P (Spam) + P (w ∈ M |M ∈ Good)P (Good)
Here P (Spam) = |Spam|/(|Spam| + |Good|) and P (Good) = |Good|/(|Spam| + |Good|). Here let 0/0 = 0
(this happens if the word did not occur in training messages).
After that the ratio of words that indicate that M ∈ Spam with probability of at least 1/2 is calculated.
If it reaches some threshold t the message is classiﬁed as spam. Note that each word is analyzed only
once even if it occurs several time in a message.
In this problem you will have to implement Bayesian spam ﬁlter. You will be given the set of messages
to train and after that the set of messages to classify. For each message to classify you will have to tell
whether it is spam or not.
Input
The ﬁrst line of the input ﬁle contains four integer numbers: s, g, n and t — the number of spam messages
to train on, the number of good messages to train on and the number of messages to classify, and the
threshold in percent (1 ≤ s ≤ 1000, 1 ≤ g ≤ 1000, 1 ≤ n ≤ 1000, 1 ≤ t ≤ 100).
The following s lines contain one message per line that are speciﬁed to be spam. A message is the
sequence of one or more words that consist of letters of the English alphabet and are separated and/or
surrounded by spaces, digits and punctuation marks (‘-’, ‘,’, ‘.’, ‘!’, ‘?’). All words are case insensitive.
The length of each message is at most 200 characters. Word length doesn’t exceed 20.
The following g lines contain good messages, one per line, and the last n lines contain messages to be
classiﬁed, one per line.
Output
For each message to be classiﬁed print “spam” if at least t% words indicate that it is spam, and “good”
in the other case.
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Example
spam.in
2 2 2 50
Buy our best computers!
You will find our computers best!
I have completed writing problems for trainings.
I have problems with my computer.
Computers? Solution!
I do not know what to buy. Need help.
spam.out
spam
good
Page 23 of 30
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Problem T. Fun with Squares
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
squares.in
squares.out
2 seconds
64 megabytes
Johnny has n squares with edges parallel to coordinate axes on the plane. Given two squares A and B
let us denote a ﬁgure that contains all points of A that do not belong to B as A − B.
−
=
−
=
−
= empty ﬁgure
Now Johnny wants to ﬁnd the number of collections of four distinct squares A, B, C and D such that
A − B and C − D are equal up to a parallel shift. Rotation is not allowed.
Input
The ﬁrst line of the input ﬁle contains n — the number of squares (4 ≤ n ≤ 400). The following n lines
describe squares. Each square is described by three integer numbers: x, y and l — the coordinates of its
left bottom corner and the length of its edge (−109 ≤ x, y ≤ 109 , 1 ≤ l ≤ 109 ).
Output
Output the number of sets of four distinct squares A, B, C and D such that A − B and C − D are
equal up to a parallel shift. Collections that contain the same set of squares but diﬀer by square roles
are considered diﬀerent.
Example
squares.in
4
0
1
2
1
4
0
1
2
4
squares.out
6
0
1
2
1
3
1
1
3
0
0
1
2
4
3
3
3
3
In the ﬁrst example the collections are: ⟨0, 1, 3, 2⟩, ⟨1, 0, 2, 3⟩, ⟨1, 3, 2, 0⟩, ⟨2, 0, 1, 3⟩, ⟨2, 3, 1, 0⟩, ⟨3, 2, 0, 1⟩.
Page 24 of 30
ITMO Summer Camp Scholarship
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Problem U. Longest Common Subpair
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
subpair.in
subpair.out
2 seconds
64 megabytes
A pair of strings (α, β) is called a subpair of a string γ if γ = γ1 αγ2 βγ3 for some (possibly empty) strings
γ1 , γ2 and γ3 . The length of the pair is the sum of lengths of its strings: |(α, β)| = |α| + |β|.
Given two strings ξ and η ﬁnd their longest common subpair, that is — such pair (α, β) that it is a
subpair of both ξ and η and its length is greatest possible.
Input
Input ﬁle contains two strings ξ and η, one on a line. Both strings contain only small letters of the
English alphabet. Both string are not empty. The length of each string doesn’t exceed 3000.
Output
Output α on the ﬁrst line of the output ﬁle and β on the second line.
Example
subpair.in
abacabadabacaba
acabacadacabaca
ab
bc
subpair.out
acaba
abaca
b
Page 25 of 30
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Problem V. Sum vs Product
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
sump.in
sump.out
2 seconds
64 megabytes
Peter has just learned mathematics. He learned how to add, and how to multiply. The fact that
2 + 2 = 2 × 2 has amazed him greatly. Now he wants ﬁnd more such examples.
Peters calls a collection of numbers beautiful if the product of the numbers in it is equal to their sum.
For example, the collections {2, 2}, {5}, {1, 2, 3} are beautiful, but {2, 3} is not.
Given n, Peter wants to ﬁnd the number of beautiful collections with n numbers. Help him!
Input
The ﬁrst line of the input ﬁle contains n (2 ≤ n ≤ 500).
Output
Output one number — the number of the beautiful collections with n numbers.
Example
sump.in
2
5
sump.out
1
3
The collections in the last example are: {1, 1, 1, 2, 5}, {1, 1, 1, 3, 3} and {1, 1, 2, 2, 2}.
Page 26 of 30
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Problem W. Superposition
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
superposition.in
superposition.out
2 seconds
64 megabytes
Tony is writing ﬁnancial software. One of the components he is writing now will have to deals with risk
and income graphs of various market instruments. Now he needs to plot the graph of a superposition of
two continuous piecewise linear functions.
Tony is given two functions: f (x) and g(x) and needs to ﬁnd the explicit representation of the function
g(f (x)).
Input
The input ﬁle consists of two blocks, one describes f and another describes g.
Each blocks starts with l — the number of intervals of linearity of a function (1 ≤ l ≤ 1000, l ̸= 2).
If l > 2, the following l − 1 lines contain the breaking points of the function. Each line contains two
integer numbers: xi , yi (i from 1 to l − 1, xi < xi+1 ). The function a(x) described by the block is deﬁned
as follows:


y1 , if x < x1 ;





y1 + k1 (x − x1 ), if x1 ≤ x < x2 ;
a(x) = . . .


yl−2 + kl−2 (x − xl−2 ), if xl−2 ≤ x < xl−1 ;




yl−1 , if x ≥ xl−1 .
Here ki = (yi+1 − yi )/(xi+1 − xi ).
If l = 1 the following line contains one integer number y, the function described is a(x) = y.
All coordinates in the input ﬁle do not exceed 109 by their absolute values.
Output
Output the function g(f (x)) in the same format as the functions in the input ﬁle. The number of intervals
of linearity in the description must be minimal possible. It is guaranteed that the number of intervals of
linearity of the resulting function will not exceed 100 000. Output ﬂoating point numbers with absolute
or relative precision of at least 10−8 .
Example
superposition.in
3
0
2
4
0
1
2
0
3
superposition.out
4
0 0
0.666666666667 3
1.333333333333 0
0
3
0
Page 27 of 30
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Problem X. High Speed Trains
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
trains.in
trains.out
2 seconds
64 megabytes
The kingdom of Flatland has n cities. Recently the king of Flatland visited Japan and was amazed by
high speed trains Shinkansens going all around the country. Therefore he decided to build the system of
high speed trains in Flatland.
Each high speed train line will be bidirectional and connect exactly two diﬀerent cities of Flatland.
Although there is actually no need of high speed trains in Flatland, the king ordered that there must be
at least one high speed train line from each city of Flatland.
The minister of transportation told the king that there are several train system satisfying his requirements.
The king was amazed by the fact and asked the minister to count the number of possible systems.
Help the minister to calculate the number of train systems.
Input
The input ﬁle contains one integer number n (2 ≤ n ≤ 100).
Output
Output one integer number — the number of diﬀerent train systems that can be arranged in Flatland.
Example
trains.in
4
trains.out
41
Page 28 of 30
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Problem Y. New Year Tree Transportation
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
tree.in
tree.out
2 seconds
64 megabytes
People of Byteland celebrate New Year. Unlike people of most other countries, they do not decorate
ﬁr-trees for the New Year celebration. Instead they decorate binary trees. A binary tree is a rooted tree
such that every node has at most two children.
A nice binary tree with n nodes was prepared to be set up on the main square of Byteland capital. However
ﬁrst it must be transported from the place where it was grown up to the capital. The transportation will
be arranged by the railroad. But it turned out that the standard railroad car can carry the tree only if
it has at most k nodes.
So it was decided to cut several edges of the tree so that each of the remaining connected parts had at
most k nodes. After the tree is transported to the capital it would be reassembled and set up. Due to
security reasons each car must carry only one tree part.
Of course the department of transportation of Byteland would like to use as few cars as possible to
transport the tree. However, minimizing the number of cars seemed to be too diﬃcult problem. Therefore
the minister of transportation ordered to cut the tree in such a way that the number of cars needed at
least did not exceed ⌈2n/k⌉.
But the people who are transporting the tree couldn’t solve even this problem. Help them! Given a
binary tree ﬁnd the way to cut some of its edges in such a way that each of the remaining connected
parts had at most k nodes and the number of parts didn’t exceed ⌈2n/k⌉.
Input
The ﬁrst line of the input ﬁle contains n — the number of nodes of the tree, and k — the maximal
capacity of the car (1 ≤ n ≤ 100 000, 1 ≤ k ≤ n). The following n − 1 lines describe the edges of the tree.
Each edge is described by two integer numbers: the parent node and the child node. The given tree is
guaranteed to be a binary tree. Nodes are numbered from 1 to n, root has number 1.
Output
The ﬁrst line of the output ﬁle must contain l — the number of edges that must be cut. The second line
must contain l integer numbers — the edges to be cut. Edges are numbered from 1 to n − 1 as they are
listed in the input ﬁle.
If the tree cannot be cut in the described way, output l = −1.
Example
tree.in
5
1
1
5
5
2
2
5
3
4
tree.out
2
2 4
Page 29 of 30
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Problem Z. TV Show
Input ﬁle:
Output ﬁle:
Time limit:
Memory limit:
tvshow.in
tvshow.out
2 seconds
64 megabytes
Charlie is going to take part in one famous TV Show. The show is a single player game. Initially the
player has 100 dollars. The player is asked n questions, one after another. If the player answers the
question correctly, the sum he has is doubled. If the answer is incorrect, the player gets nothing and
leaves the show.
Before each question the player can choose to leave the show and take away the prize he already has.
Also once in the game the player can buy insurance. Insurance costs c dollars which are subtracted from
the sum the player has before the question. Insurance has the following eﬀect: if the player answers
the question correctly his prize is doubled as usually, if the player answers incorrectly the prize is not
doubled, but the game continues. The player must have more than c dollars to buy insurance.
Charlie’s friend Jerry works on TV so he managed to steal the topics of the questions Charlie will be
asked. Therefore for each question i Charlie knows pi — the probability that he will answer this question
correctly. Now Charlie would like to develop the optimal strategy to maximize his expected prize. Help
him.
Input
The ﬁrst line of the input ﬁle contains two integer numbers n and c (1 ≤ n ≤ 50, 1 ≤ c ≤ 109 ). The
second line contains n integer numbers ranging from 0 to 100 — the probabilities that Charlie will answer
questions correctly, in percent.
Output
Output one real number — the expected prize of Charlie if he follows the optimal strategy. Your answer
must have relative or absolute error within 10−8 .
Example
tvshow.in
2 100
50 50
2 50
50 50
2 50
60 0
tvshow.out
100
112.5
120
The optimal strategy in the second example is to take insurance for the second question. In this case the
expected prize is 1/2 × 0 + 1/2 × (1/2 × 150 + 1/2 × 300).
In the third example it is better to leave the show after the ﬁrst question, because there is no reason to
try to answer the second one.
Page 30 of 30