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Romania ACM Programming Contest – Bucharest & Iasi Subregions
May 15, 2010
A - Odd-Even Sum
Given a natural number N, compute the value S(N)=1-2+3-4+5-6+...N (i.e. the sum of
all the odd numbers less than or equal to N minus the sum of all the even numbers less than or
equal to N).
Input Data
The first line contains the number T of test cases. Each of the next T lines contains a natural
number N (0≤N≤2.000.000.000).
Output Data
You should print T lines. The ith line of output should contain the value S(N), where N is
the number from the ith test case (i.e. the number on the (i+1)st line of the input).
Example
Standard Input
4
0
10
11
12
Standard Output
0
-5
6
-6
B - Domino Cover
Consider a rectangular board composed of HxN squares (1≤H≤7
;
1≤N≤1.000.000.000). We want to fully cover the board by placing non-overlapping
dominoes. A domino is a 2x1 or a 1x2 rectangular piece (i.e. it covers two squares having a
common side). Compute the total number of possibilities in which we can cover the entire
board with non-overlapping dominoes.
Input Data
The input data consists of 7 sets of test cases, one for each possible value of H. Each set of
test cases contains the number T of test cases on the first line. Each of the next T lines
contains a value of N (a test case is thus described by the value H corresponding to the set of
test cases and by the value N on the corresponding line). The sets of test cases are given in
increasing order of H and are separated by a blank line.
Output Data
For each test case print the total number of possibilities to cover the board, modulo 30047.
Example
Standard Input
Standard Output
Explanation
2
1
2
0
1
3
5
0
The values printed to the standard output
correspond to the following test cases (in order):
H=1, N=1
H=1, N=2
H=2, N=3
2
Romania ACM Programming Contest – Bucharest & Iasi Subregions
May 15, 2010
3
4
2
5
6
2
7
8
41
781
2245
0
5639
28993
5567
0
23808
H=2,
H=3,
H=3,
H=4,
H=4,
H=5,
H=5,
H=6,
H=6,
H=7,
H=7,
N=4
N=5
N=6
N=7
N=8
N=9
N=10
N=11
N=12
N=13
N=14
2
9
10
2
11
12
2
13
14
C - Valleys & Mountains
Let’s consider a permutation h(1), ..., h(N) of the numbers 1, ..., N, where each
position i (1≤i≤N) also has an associated value v(i).
A valley is a contiguous subsequence h(i), h(i+1), ..., h(j), such that:
1. j-i+1≥3
2. the subsequence is decreasing from h(i) to some number h(p), and then increasing
from h(p) up to h(j)
3. i+1≤p≤j-1 (i.e. p must not be the first or the last position of the sequence).
A mountain is a contiguous subsequence h(i), h(i+1), ..., h(j), such that:
4. j-i+1≥3
5. the subsequence is increasing from h(i) to some number h(p), and then decreasing
from h(p) to h(j)
6. i+1≤p≤j-1 (i.e. p must not be the first or the last position of the sequence).
The value of a contiguous subsequence h(i), h(i+1), …, h(j) is equal to the sum of
the values of the positions contained in the subsequence (i.e. it is equal to
v(i)+v(i+1)+…+v(j)).
Find the maximum value of a valley and the maximum value of a mountain.
Input Data
The first line of input contains the number T of test cases which are described next. The first
line of each test case contains the number N (3≤N≤50.000). The ith line of the next N lines
contains two integer values separated by one blank: h(i) and v(i) (-20.000≤
v(i)≤+20.000).
Romania ACM Programming Contest – Bucharest & Iasi Subregions
May 15, 2010
Output Data
For each test case you should print one line containing two numbers MV and MM, separated by
one blank: the maximum value of a valley and the maximum value of a mountain for the
corresponding test case. If no valley exists, you should print the string „no valley”
(without quotes) instead of the number MV. If no mountain exists, you should print the string
„no mountain” (without quotes) instead of the number MM.
Example
Standard Input
Standard Output
2
10
1 -1
3 -2
5 3
7 -4
6 5
4 6
2 -7
8 10
9 -9
10 10
4
4 10
3 -1
1 -10
2 -9
15 10
-10 no mountain
D - Minimum Enclosing Diamond
A diamond is a square rotated by 45 degrees (i.e. its two diagonals are parallel to the
coordinate axes). The size of a diamond is the length of its diagonals.
Given a set of N points in the plane, find the minimum size of a diamond containing all the
points inside of it or on its borders.
Input Data
The first line of input contains the number T of test cases which are described next. The first
line of a test case contains the number N (1≤N≤50.000) of points. The next N lines contain
two integer numbers each, separated by a blank: the x and y coordinates of a point.
Output Data
For each test case print a line containing the size of the minimum enclosing diamond. Some
careful analysis will reveal that this number is an integer.
Example
Standard Input
Standard Output
2
3
1 2
3 6
0 4
6
19
Romania ACM Programming Contest – Bucharest & Iasi Subregions
May 15, 2010
4
-1 3
-2 -5
-10 6
4 4
E - Dual-Processor Scheduling
There are N jobs (numbered from 1 to N) which have to be executed on a dual-processor
machine with non-identical processors. Each job i must be executed t1[i] contiguous
seconds on processor 1 and t2[i] contiguous seconds on processor 2. At each time moment,
at most one job can be executed on the same processor. Moreover, the two time intervals
when a job i is executed on processor 1 and on processor 2 must not overlap (but it is
allowed to start the execution on the other processor as soon as the execution on one of the
processors ends). There are no ordering constraints regarding the execution of each job on the
two processors (i.e. each job i may be executed first on any of the two processors and later on
the other one) or regarding the execution of the jobs on the same processor. This also implies
that the jobs may be executed in different orders on the two processors.
Considering that the execution of the jobs starts at the time moment 0, find a schedule of the
N jobs on the two processors, such that the earliest time moment TE (expressed in seconds)
when the execution of all the jobs on both processors ends is minimum.
Input Data
The first line of input contains the number T of test cases which are described next. The first
line of a test case contains the number N (1≤N≤100.000) of jobs. The ith line of the next N
lines contain two integer numbers each, separated by a blank: the values t1[i] and t2[i],
in this order (1≤t1[i],t2[i]≤20.000).
Output Data
For each test case print a line containing the minimum value of TE.
Example
Standard Input
Standard Output
2
3
3
4
6
4
1
2
4
7
13
15
5
1
6
8
4
2
1
F - Multiversion Stack
We consider a stack which can have multiple versions. The only initial version is numbered
with 0 and is an empty stack. We will perform a sequence of PUSH and POP operations on
this stack.
Romania ACM Programming Contest – Bucharest & Iasi Subregions
May 15, 2010
A PUSH operation has the form PUSH(v) and performs the following actions:
1. let C be the index of the operation in the sequence of operations (C is indexed starting
from 1; both types of operations are indexed)
2. a new version vnew is created, which contains all the elements of the version v;
vnew is the smallest natural number which does not exist as a version
3. the element C+v is added at the top of the version vnew
4. print at the standard output a line containing the number of elements in the stack
version vnew
A POP operation has the form POP(v) and performs the following actions:
1. a new version vnew is created, which contains all the elements of the version v;
vnew is the smallest natural number which does not exist as a version
2. the topmost element of the stack version vnew is removed from the version vnew and
printed to the standard output on a single line ; if the version vnew does not contain
any element, then -1 is printed to the standard output on a single line
Input Data
The first line of input contains the number NOP of operations in the sequence of operations.
Each of the next NOP lines describes one operation. An operation is described by two integer
numbers, separated by a blank: op and v (v≥0). If op=1 then the operation to be performed
is PUSH(v). If op=2 then the operation to be performed is POP(v). The parameter v will
always denote an existing version.
This problem contains only one test case.
Output Data
Print the output of the sequence of operations.
Example
Standard Input
Standard Output
Explanation
9
1
1
1
1
2
2
2
2
1
1
2
3
1
5
3
1
-1
3
The sequence of operations is:
Push(0), Push(1), Push(2), Push(0),
Pop(3), Pop(2), Pop(1), Pop(7).
The stacks corresponding to each version are the
following (left-to-right corresponds to the bottom-to-top
order):
version 0: empty
version 1: 1
version 2: 1, 3
version 3: 1, 3, 5
version 4: 4
version 5: 1, 3
version 6: 1
version 7: empty
version 8: empty
version 9: 1, 3, 11
0
1
2
0
3
2
1
7
2
Romania ACM Programming Contest – Bucharest & Iasi Subregions
May 15, 2010
G - Valleys & Mountains 2
Romania is well-known for its beautiful and unique landscape. In particular, it is known that
Romania has V valleys and M mountains. During a trip to Romania, George took a photo of
Romania’s landscape. The photo is represented as a sequence of the characters / and \ .
There are 2*N characters in the sequence, N characters of type / and N characters of type \. A
valley is represented by two consecutive characters \/ (\ followed by /). A mountain is
represented by two consecutive characters /\ (/ followed by \). The map starts and ends at
the sea level (altitude 0) and:
1) for every / character the altitude increases by 1
2) for every \ character the altitude decreases by 1
3) the altitude never decreases below 0
In order to decide if Romania’s landscape is indeed unique, George would like to count the
total number of possible photos containing N characters of type /, N characters of type \,
having V valleys and M mountains and which obey all the specified altitude constraints.
Input Data
This problem has no input.
Output Data
Let C(N,V,M) be the number of sequences with N characters of type /, N characters of type
\, V valleys and M mountains and which obey all the specified altitude constraints. Print all
the non-zero values C(N,V,M) of all the tuples (N,V,M) in increasing lexicographic order
of the tuples, for all the values of N from 1 to 30.
Example
Standard Input
Standard Output
Explanation
1
1
1
1
3
1
1
6
6
1
1
10
20
10
1
...
The values printed to the standard output correspond,
in order, to the following tuples:
(1,0,1)
(2,0,1)
(2,1,2)
(3,0,1)
(3,1,2)
(3,2,3)
(4,0,1)
(4,1,2)
(4,2,3)
(4,3,4)
(5,0,1)
(5,1,2)
(5,2,3)
(5,3,4)
(5,4,5)
...
The 3 sequences for the tuple (3,1,2) are:
1) //\\/\
2) /\//\\
3) //\/\\
Romania ACM Programming Contest – Bucharest & Iasi Subregions
May 15, 2010
H - Minimum Steiner Tree
Let’s consider an undirected complete graph with N nodes numbered from 1 to N. We denote
by C[i][j] the cost of the edge (i,j) (we have C[i][j]=C[j][i]≥1 and
C[i][i]=0). Some of the nodes of the graph are marked as being special. We want to select
a subset S of graph edges such that there is a path from every special node to every other
special node using the edges from S. The cost of the subset S is the sum of the costs of the
edges contained in it. We are interested in finding the minimum cost of such a subset S.
Input Data
The first line of input contains the number T of test cases which are described next. The first
line of a test case contains two integer numbers, separated by a blank: N (1≤N≤19) and Q
(1≤Q≤100.000). The next N lines contain N numbers each, separated by blanks: the jth
number on the ith of these lines represents the value C[i][j]
(1≤C[i][j]≤100.000.000). Each of the next Q lines contains a query sequence SQ of
N characters (plus the terminating newline character). The ith character in SQ is 1 if the node
i is marked as being special, and 0 otherwise.
Output Data
For each query sequence SQ from the input print a line containing the minimum cost of a
subset of edges S such that there is a path from every node marked as special in SQ to every
other special node using only the edges from S. When computing the answer for a query
sequence you must consider its corresponding graph. Moreover, the other query sequences
from the same test case do not have any influence on which node is marked as special or not
(only the current query sequence SQ specifies this).
Example
Standard Input
Standard Output
1
6 15
0 5 100000000 5 5 8
5 0 9 7 7 8
100000000 9 0 5 5 100000000
5 7 5 0 5 5
5 7 5 5 0 5
8 8 100000000 5 5 0
001110
010000
100000
100000
010010
100100
000100
101111
001010
010010
111011
010101
001001
001111
011001
10
0
0
0
7
5
0
20
5
7
20
12
10
15
17
Romania ACM Programming Contest – Bucharest & Iasi Subregions
May 15, 2010
I - Minimum Sum of Averages Division
Consider a sequence of N positive natural numbers: v1, ..., vN. We want to divide this
sequence into K contiguous subsequences such that:
1. the K subsequences are non-overlapping
2. each position i (1≤i≤N) of the sequence belongs to a subsequence
3. the total cost of the subsequences is minimum
The cost of a contiguous subsequence vi, ..., vj is equal to:
(vi+vi+1+...+vj-1+vj) div (j-i+1).
where (vi+vi+1+...+vj-1+vj) denotes the sum of all the numbers in the subsequence.
We denote by A div B the quotient of the division of A at B (i.e. the integer division of A at
B). For instance, 19 div 3=6 and 20 div 4=5.
Input Data
The first line of input contains the number T of test cases which are described next. The first
line of a test case contains two integer numbers, separated by a blank: N (1≤N≤300) and K
(1≤K≤N). The second line of each test case contains the values v1, ..., vN, separated by
blanks. We have 1≤vi≤30.000 (1≤i≤N).
Output Data
For each test case print a line containing the minimum total cost of dividing the sequence into
the given number of contiguous subsequences.
Example
Standard Input
Standard Output
Explanation
2
4 2
10 20 30 40
6 2
10 100 1000 100 10 1
40
245
In the first test case, the two subsequences are:
10 => cost =10/1=10
20 30 40 => cost =(20+30+40)/3=30
The total cost is 10+30=40.
In the 2nd test case, the two subsequences are:
10 100 1000 100 10 => cost=1220/5=244
1 => cost=1/1=1
The total cost is 244+1=245.