Problem A. Sudoku Checker ≡

Statement

The puzzle game of Sudoku is played on a board of N² × N² cells. The cells are grouped in N × N squares of N × N cells each. Each cell is either empty or contains a number between 1 and N².

The sudoku position is correct when numbers in each row, each column and each square are different. The goal of the game is, starting from some correct position, fill all empty cells so that the final position is still correct.

This game is fairly popular in the Internet, and there are many sites which allow visitors to solve puzzles online. Such sites always have a subroutine to determine a correctness of a given position.

You are to write such a routine.

Input file format

Output file format

Constraints

Sample tests

2
0 0 0 0
0 0 0 0
0 0 2 0
0 0 0 1

CORRECT

2
2 1 3 0
3 2 4 0
1 3 2 4
0 0 0 1

INCORRECT

Problem B. Going to be a programmer? ≡

Statement

Igor is a computer science freshman student. As opposed to most of his classmates, he actually knows a little about computer programming. Because of this, other students often ask him to write their homework assignments for the "Introductory Programming" course.

He noticed that, although each assignment is unique, they usually follow a simple pattern. In particular, one assignment states:

"Count the number of integers in the array which are [greater than|less than|equal to|not equal to] [value]",

where parts in brackets differ from student to student. Assignments also contain sample input and output to encourage students to test their programs before submission.

Igor have many classmates, so to save time he decided to write a homework generator. He started with the following template:

  count := 0;
  for i := 1 to N do
    if a[i] #cond# #value# then
      Inc(count);

Your program will be given sample input — an array of N integer values, and the corresponding output R. It must find such substitution strings for #cond# and #value# that after execution of the code snippet above count will be equal to R, or determine that it is impossible.

Input file format

Output file format

The first line of output must contain one of the strings '<', '>', '=', '<>' — substitution for #cond#. The second line must contain an integer — substitution for #value#.

If there is no solution, output file must contain a single character '?' (ASCII 63). If there is more then one solution, output any of them.

Constraints

Sample tests

3
10 20 30
2

>
15

3
1 1 1
2

?

Problem C. How many iterations? ≡

Statement

Young programmer Vasya is learning the beginnings of the algorithm complexity theory. For starters, he wants to be able to quickly estimate the number of iterations of nested loops. As a first example, he wrote the following code:

long long f(int n)
{
   long long ans = 0;
   for (int i = 1; i <= n; i++) 
      for (int j = i+1; j <= n; j++)
         for (int k = j+1; k <= n; k++)
            ans++;
   return ans;
}

function f(n: integer): int64;
var i, j, k: integer;
begin
   result := 0;
   for i := 1 to n do 
      for j := i+1 to n do 
         for k := j+1 to n do
            inc(result);
end;

Using your knowledge of the subject, help Vasya calculate f(n) for given n.

Input file format

Input file contains an integer n.

Output file format

Output file must contain a single integer — f(n).

Constraints

1 ≤ n ≤ 10⁶

Sample tests

1

0

1000

166167000

Problem D. Uncertain Finish ≡

Statement

At the running contest, jury used a new computerized stopwatch system to ensure the most accurate measuring of results. Unfortunately, despite very successfull trials, the system malfunctioned during the actual contest.

Jury was so confident in the new system that it did not use the old mechanical stopwatches. So the only way to determine outcome was to compare visual impressions of the people watching the contest. M such impressions were recorded, each in one of two forms: "runner A finished before runner B" and "runners A and B finished at the same time".

You task is to assign a place p_i to each of N runners, such that

1. If there is a record that A and B finished at the same time, then p_A = p_B.
2. If there is a record that A finished before B, then p_A < p_B.

Additionally, places must be allocated as densely as possible, i.e. 1 ≤ p_i ≤ K for minimum possible value of K.

Input file format

Output file format

Constraints

Sample tests

3 2
3 2 1
2 1 0

2 2 1

Problem E. Repeating digit (easy) ≡

Statement

For given integers P and N you need to find all such values of x < 10^N, that N last digits of x^P are non-zero and equal.

Fortunately, there is not so many numbers showing this property. For example, for P = 2 and N = 2 there exist only 4 of them:

12, 38, 62, 88

Input file format

Output file format

Output the number of existing numbers X, then all these numbers in any order.

Constraints

2 ≤ P ≤ 100

2 ≤ N ≤ 9

Sample tests

3 2

8 
14 42 53 64 77 71 92 99

Problem F. Add one ≡

Statement

Your task is the most trivial one: given non-negative integer N, output N + 1.

The only complication is that the integer is given in an unknown base between 2 and 36 inclusive. Because of that, your program should output all possible distinct answers in lexicographically increasing order.

Input file format

Output file format

Constraints

N contains from 1 to 100 digits

Sample tests

32

33

9

10
A

Problem G. Second Best ≡

Statement

Given the sequence of integers A₁, A₂, …, A_N, find a number A_s such that there exists exactly one A_m > A_s, and for all k ≠ m A_k ≤ A_s.

Input file format

Output file format

Constraints

Sample tests

3
1 2 3

2

4
3 3 2 3

-1

Problem H. Customer support ≡

Statement

Customer support department in an "Incomprehension Amateurs, Ltd" software company has call center for answering users' questions. Support prices are as follows:

So, for example, if user asks the same question three times, first receives incorrect answer, then correct one, and the third time correct answer with explanation, it will cost him 10 + 20 + (40 + 1 * 10) = 80 USD.

Customers are billed monthly according to call log. Company engineers review the log and for each question determine:

1. unique number, so the equivalent questions have same numbers,
2. whether the answer was correct,
3. whether the answer was short or included detailed enough explanation.

Input file format

Output file format

Constraints

Sample tests

1
9834 0 1

10

3
33 1 0
33 0 0
33 1 1

80

Problem I. Shortest Spanning Tree ≡

Statement

Input file format

Output file format

Constraints

Sample tests

3 3
1 2 10
2 3 10
3 1 10

20

Задача J. Табуретки-2 (30 баллов) ≡

Условие

Для изготовления качественной табуретки необходимы 4 ножки одинаковой длины. На табуреткоизготовительную фабрику поступило N ножек, имеющих слегка различающиеся длины L₁, L₂, … L_N. Научно-исследовательский отдел фабрики обнаружил, что выпуск табуреток можно увеличить, если укорачивать некоторые ножки. При этом отпиленная часть выбрасывается.

Требуется определить минимальное количество распилов, необходимых для для изготовления N / 4 качественных табуреток.

Формат входного файла

Входной файл содержит число N, за которым следуют N чисел L_i — длины ножек. Все числа целые.

Формат выходного файла

Ограничения

Примеры тестов

8
18 16 17 17 16 17 17 19

2

Задача K. Домой на электричках ≡

Условие

Одна из команд-участниц олимпиады решила вернуться домой на электричках. При этом ребята хотят попасть домой как можно раньше. К сожалению, не все электрички идут от города, где проводится олимпиада, до станции, на которой живут ребята. И, что еще более обидно, не все электрички, которые идут мимо их станции, останавливаются на ней (равно как вообще, электрички останавливаются далеко не на всех станциях, мимо которых они идут).

Все станции на линии пронумерованы числами от 1 до N. При этом станция номер 1 находится в городе, где проводится олимпиада, и в момент времени 0 ребята приходят на станцию. Станция, на которую нужно попасть ребятам, имеет номер E.

Напишите программу, которая по данному расписанию движения электричек вычисляет минимальное время, за которое ребятам удастся добраться домой.

Формат входного файла

Формат выходного файла

Ограничения

Примеры тестов

5 3
4
2 1 5 2 10
2 2 10 4 15
4 5 0 4 17 3 20 2 35
3 1 2 3 40 4 45

20

Задача L. Ближайшее число ≡

Условие

Дана последовательность из N целых чисел. Для каждого числа вывести ближайшее к нему справа в этой последовательности, которое будет больше него. Для чисел, которым найти ближайшее большее не удалось, вывести сами эти числа.

Формат входного файла

Входной файл содержит целое число N за которым следует N целых чисел a_i - исходная последовательность.

Формат выходного файла

В выходной файл необходимо вывести N целых чисел b_i, таких что b_i является ответом на задачу для числа a_i.

Ограничения

1 ≤ N ≤ 10⁶

|a_i| ≤ 10⁹

Примеры тестов

4 1 2 3 4

2 3 4 4

1
1

1

Problem M. Altitude and visibility ≡

Statement

Consider a sequence of numbers a₁, …, a_N, representing heights. Let's say that element a_i has a visibility radius d if a_i ≥ a_j for all j such that 1 ≤ j ≤ N and |i − j| < d.

You task is to find maximum d_i for every a_i.

Input file format

Input file contains integer N followed by N integers a_i.

Output file format

Output file must contain N integers d_i — maximum visibility for every a_i. If maximum visibility radius for element a_i is unlimited, output d_i = 0.

Constraints

1 ≤ N ≤ 10⁶, 0 ≤ a_i ≤ 10⁹

Sample tests

1
10

0 

6
1 5 2 2 1 4

1 0 1 2 1 4 

Problem N. Topological sorting ≡

Statement

You are to write a program that performs a topological sorting of a directed graph. Graph contains N vertices, numbered with integers from 1 to N, and M edges.

In other words, given a partial order on numbers from 1 to N, your program must produce some linear order on these numbers which does not conflict with the given partial order.

Input file format

These pairs may also be considered comparisons where the first number precedes the second one.

Output file format

Output file must contain a permutation of integers from 1 to N — numbers of vertices sorted in topological order. (That is, representing a linear order.) If ordering is not possible, output file must contain a single number  − 1.

Constraints

Sample tests

4 3
1 2
1 3
3 4

1 3 2 4