vgel/INTELLECT-2-RL-Dataset-Code-Only huggingface dataset:
We've got no test cases. A big olympiad is coming up. But the problemsetters' number one priority should be adding another problem to the round. 

The diameter of a multiset of points on the line is the largest distance between two points from this set. For example, the diameter of the multiset {1, 3, 2, 1} is 2.

Diameter of multiset consisting of one point is 0.

You are given n points on the line. What is the minimum number of points you have to remove, so that the diameter of the multiset of the remaining points will not exceed d?


-----Input-----

The first line contains two integers n and d (1 ≤ n ≤ 100, 0 ≤ d ≤ 100) — the amount of points and the maximum allowed diameter respectively.

The second line contains n space separated integers (1 ≤ x_{i} ≤ 100) — the coordinates of the points.


-----Output-----

Output a single integer — the minimum number of points you have to remove.


-----Examples-----
Input
3 1
2 1 4

Output
1

Input
3 0
7 7 7

Output
0

Input
6 3
1 3 4 6 9 10

Output
3



-----Note-----

In the first test case the optimal strategy is to remove the point with coordinate 4. The remaining points will have coordinates 1 and 2, so the diameter will be equal to 2 - 1 = 1.

In the second test case the diameter is equal to 0, so its is unnecessary to remove any points. 

In the third test case the optimal strategy is to remove points with coordinates 1, 9 and 10. The remaining points will have coordinates 3, 4 and 6, so the diameter will be equal to 6 - 3 = 3.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Snuke has decided to play with a six-sided die. Each of its six sides shows an integer 1 through 6, and two numbers on opposite sides always add up to 7.
Snuke will first put the die on the table with an arbitrary side facing upward, then repeatedly perform the following operation:
 - Operation: Rotate the die 90° toward one of the following directions: left, right, front (the die will come closer) and back (the die will go farther). Then, obtain y points where y is the number written in the side facing upward.
For example, let us consider the situation where the side showing 1 faces upward, the near side shows 5 and the right side shows 4, as illustrated in the figure.
If the die is rotated toward the right as shown in the figure, the side showing 3 will face upward.
Besides, the side showing 4 will face upward if the die is rotated toward the left, the side showing 2 will face upward if the die is rotated toward the front, and the side showing 5 will face upward if the die is rotated toward the back.
Find the minimum number of operation Snuke needs to perform in order to score at least x points in total.

-----Constraints-----
 - 1 ≦ x ≦ 10^{15}
 - x is an integer.

-----Input-----
The input is given from Standard Input in the following format:
x

-----Output-----
Print the answer.

-----Sample Input-----
7

-----Sample Output-----
2


Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Mad scientist Mike is busy carrying out experiments in chemistry. Today he will attempt to join three atoms into one molecule.

A molecule consists of atoms, with some pairs of atoms connected by atomic bonds. Each atom has a valence number — the number of bonds the atom must form with other atoms. An atom can form one or multiple bonds with any other atom, but it cannot form a bond with itself. The number of bonds of an atom in the molecule must be equal to its valence number. [Image] 

Mike knows valence numbers of the three atoms. Find a molecule that can be built from these atoms according to the stated rules, or determine that it is impossible.


-----Input-----

The single line of the input contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 10^6) — the valence numbers of the given atoms.


-----Output-----

If such a molecule can be built, print three space-separated integers — the number of bonds between the 1-st and the 2-nd, the 2-nd and the 3-rd, the 3-rd and the 1-st atoms, correspondingly. If there are multiple solutions, output any of them. If there is no solution, print "Impossible" (without the quotes).


-----Examples-----
Input
1 1 2

Output
0 1 1

Input
3 4 5

Output
1 3 2

Input
4 1 1

Output
Impossible



-----Note-----

The first sample corresponds to the first figure. There are no bonds between atoms 1 and 2 in this case.

The second sample corresponds to the second figure. There is one or more bonds between each pair of atoms.

The third sample corresponds to the third figure. There is no solution, because an atom cannot form bonds with itself.

The configuration in the fourth figure is impossible as each atom must have at least one atomic bond.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Golu is given a task of clearing coins fallen down  on floor. Naughty being his modus operandi, he thought of playing a game while clearing coins. He arranges the coins like an M X N matrix with random heads (1) and tails (0).

In every move, he chooses a sub-matrix of size L X B [such that max(L,B) >1] and interchanges all heads to tails and vice-versa present in the chosen sub-matrix. Then if he finds a 2X2 sub-matrix exclusively of heads anywhere in the M X N matrix, he removes those coins.

Your task is simple. You have to report the minimum no. of moves taken by him before he clears his first 2X2 sub-matrix.

Input:

Your first line of input consists of two numbers M, N. Each of the next M lines consists of N space-separated integers which are either 0 or 1.

Output:

Output a single line answer giving the minimum no. of steps taken before his first clearance. If it is impossible for him to do so, print "-1" (quotes for clarity).

Constraints :

2 ≤ M,N ≤ 10

Author : Srinivas

Testers : Shubham Parekh , Shreyans

(By IIT Kgp HackerEarth Programming Club)

SAMPLE INPUT
2 4
0 0 1 1
0 0 1 1

SAMPLE OUTPUT
0

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Vasya goes to visit his classmate Petya. Vasya knows that Petya's apartment number is $n$. 

There is only one entrance in Petya's house and the distribution of apartments is the following: the first floor contains $2$ apartments, every other floor contains $x$ apartments each. Apartments are numbered starting from one, from the first floor. I.e. apartments on the first floor have numbers $1$ and $2$, apartments on the second floor have numbers from $3$ to $(x + 2)$, apartments on the third floor have numbers from $(x + 3)$ to $(2 \cdot x + 2)$, and so on.

Your task is to find the number of floor on which Petya lives. Assume that the house is always high enough to fit at least $n$ apartments.

You have to answer $t$ independent test cases.


-----Input-----

The first line of the input contains one integer $t$ ($1 \le t \le 1000$) — the number of test cases. Then $t$ test cases follow.

The only line of the test case contains two integers $n$ and $x$ ($1 \le n, x \le 1000$) — the number of Petya's apartment and the number of apartments on each floor of the house except the first one (there are two apartments on the first floor).


-----Output-----

For each test case, print the answer: the number of floor on which Petya lives.


-----Example-----
Input
4
7 3
1 5
22 5
987 13

Output
3
1
5
77



-----Note-----

Consider the first test case of the example: the first floor contains apartments with numbers $1$ and $2$, the second one contains apartments with numbers $3$, $4$ and $5$, the third one contains apartments with numbers $6$, $7$ and $8$. Therefore, Petya lives on the third floor.

In the second test case of the example, Petya lives in the apartment $1$ which is on the first floor.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Title: High School: Become Human

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
Year 2118. Androids are in mass production for decades now, and they do all the work for humans. But androids have to go to school to be able to solve creative tasks. Just like humans before.

It turns out that high school struggles are not gone. If someone is not like others, he is bullied. Vasya-8800 is an economy-class android which is produced by a little-known company. His design is not perfect, his characteristics also could be better. So he is bullied by other androids.

One of the popular pranks on Vasya is to force him to compare $x^y$ with $y^x$. Other androids can do it in milliseconds while Vasya's memory is too small to store such big numbers.

Please help Vasya! Write a fast program to compare $x^y$ with $y^x$ for Vasya, maybe then other androids will respect him.

Input Specification:
On the only line of input there are two integers $x$ and $y$ ($1 \le x, y \le 10^{9}$).

Output Specification:
If $x^y &lt; y^x$, then print '&lt;' (without quotes). If $x^y &gt; y^x$, then print '&gt;' (without quotes). If $x^y = y^x$, then print '=' (without quotes).

Demo Input:
['5 8\n', '10 3\n', '6 6\n']

Demo Output:
['&gt;\n', '&lt;\n', '=\n']

Note:
In the first example $5^8 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 390625$, and $8^5 = 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 = 32768$. So you should print '&gt;'.

In the second example $10^3 = 1000 &lt; 3^{10} = 59049$.

In the third example $6^6 = 46656 = 6^6$.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
The Birthday Paradox is the name given to the surprising fact that if there are just $23$ people in a group, there is a greater than $50\% $ chance that a pair of them share the same birthday. The underlying assumptions for this are that all birthdays are equally likely (which isn’t quite true), the year has exactly $365$ days (which also isn’t true), and the people in the group are uniformly randomly selected (which is a somewhat strange premise). For this problem, we’ll accept these assumptions.

Consider what we might observe if we randomly select groups of $P=10$ people. Once we have chosen a group, we break them up into subgroups based on shared birthdays. Among many other possibilities, we might observe the following distributions of shared birthdays:
 - all $10$ have different birthdays, or
 - all $10$ have the same birthday, or
 - $3$ people have the same birthday, $2$ other people have the same birthday (on a different day), and the remaining $5$ all have different birthdays.

Of course, these distributions have different probabilities of occurring.

Your job is to calculate this probability for a given distribution of people sharing birthdays. That is, if there are $P$ people in a group, how probable is the given distribution of shared birthdays (among all possible distributions for $P$ people chosen uniformly at random)?

-----Input-----
The first line gives a number $n$ where $1 \le n \le 365$. The second line contain integers $c_1$ through $c_ n$, where $1 \le c_ i \le 100$ for all $c_ i$. The value $c_ i$ represents the number of people who share a certain birthday (and whose birthday is distinct from the birthdays of everyone else in the group).

-----Output-----
Compute the probability $b$ of observing a group of people with the given distribution of shared birthdays. Since $b$ may be quite small, output instead $\log _{10}(b)$. Your submission’s answer is considered correct if it has an absolute or relative error of at most $10^{-6}$ from the judge’s answer.

-----Explanations-----
The first sample case shows $P=2$ people with distinct birthdays. The probability of this occurring is $b = 364/365 \approx 0.9972602740$, and $\log _{10}(b) \approx -0.001191480807419$.

The second sample case represents the third example in the list given earlier with $P=10$ people. In this case, the probability is $b \approx 0.0000489086$, and $\log _{10}(b) \approx -4.310614508857128$.

-----Examples-----
Sample Input:
2
1 1
Sample Output:
-0.001191480807419

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division.

To improve his division skills, Oleg came up with $t$ pairs of integers $p_i$ and $q_i$ and for each pair decided to find the greatest integer $x_i$, such that:   $p_i$ is divisible by $x_i$;  $x_i$ is not divisible by $q_i$.  Oleg is really good at division and managed to find all the answers quickly, how about you?


-----Input-----

The first line contains an integer $t$ ($1 \le t \le 50$) — the number of pairs.

Each of the following $t$ lines contains two integers $p_i$ and $q_i$ ($1 \le p_i \le 10^{18}$; $2 \le q_i \le 10^{9}$) — the $i$-th pair of integers.


-----Output-----

Print $t$ integers: the $i$-th integer is the largest $x_i$ such that $p_i$ is divisible by $x_i$, but $x_i$ is not divisible by $q_i$.

One can show that there is always at least one value of $x_i$ satisfying the divisibility conditions for the given constraints.


-----Example-----
Input
3
10 4
12 6
179 822

Output
10
4
179



-----Note-----

For the first pair, where $p_1 = 10$ and $q_1 = 4$, the answer is $x_1 = 10$, since it is the greatest divisor of $10$ and $10$ is not divisible by $4$.

For the second pair, where $p_2 = 12$ and $q_2 = 6$, note that   $12$ is not a valid $x_2$, since $12$ is divisible by $q_2 = 6$;  $6$ is not valid $x_2$ as well: $6$ is also divisible by $q_2 = 6$.  The next available divisor of $p_2 = 12$ is $4$, which is the answer, since $4$ is not divisible by $6$.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Given an array Arr[] of size L and a number N, you need to write a program to find if there exists a pair of elements in the array whose difference is N.
Example 1:
Input:
L = 6, N = 78
arr[] = {5, 20, 3, 2, 5, 80}
Output: 1
Explanation: (2, 80) have difference of 78.
Example 2:
Input:
L = 5, N = 45
arr[] = {90, 70, 20, 80, 50}
Output: -1
Explanation: There is no pair with difference of 45.
Your Task:
You need not take input or print anything. Your task is to complete the function findPair() which takes array arr, size of the array L and N as input parameters and returns True if required pair exists, else return False. 
Expected Time Complexity: O(L* Log(L)).
Expected Auxiliary Space: O(1).
Constraints:
1<=L<=10^{4 }
1<=Arr[i]<=10^{5 }
0<=N<=10^{5}

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Takahashi and Aoki will have a battle using their monsters.
The health and strength of Takahashi's monster are A and B, respectively, and those of Aoki's monster are C and D, respectively.
The two monsters will take turns attacking, in the order Takahashi's, Aoki's, Takahashi's, Aoki's, ...
Here, an attack decreases the opponent's health by the value equal to the attacker's strength.
The monsters keep attacking until the health of one monster becomes 0 or below. The person with the monster whose health becomes 0 or below loses, and the other person wins.
If Takahashi will win, print Yes; if he will lose, print No.

-----Constraints-----
 - 1 \leq A,B,C,D \leq 100
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
A B C D

-----Output-----
If Takahashi will win, print Yes; if he will lose, print No.

-----Sample Input-----
10 9 10 10

-----Sample Output-----
No

First, Takahashi's monster attacks Aoki's monster, whose health is now 10-9=1.
Next, Aoki's monster attacks Takahashi's monster, whose health is now 10-10=0.
Takahashi's monster is the first to have 0 or less health, so Takahashi loses.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Lunar New Year is approaching, and Bob decides to take a wander in a nearby park.

The park can be represented as a connected graph with $n$ nodes and $m$ bidirectional edges. Initially Bob is at the node $1$ and he records $1$ on his notebook. He can wander from one node to another through those bidirectional edges. Whenever he visits a node not recorded on his notebook, he records it. After he visits all nodes at least once, he stops wandering, thus finally a permutation of nodes $a_1, a_2, \ldots, a_n$ is recorded.

Wandering is a boring thing, but solving problems is fascinating. Bob wants to know the lexicographically smallest sequence of nodes he can record while wandering. Bob thinks this problem is trivial, and he wants you to solve it.

A sequence $x$ is lexicographically smaller than a sequence $y$ if and only if one of the following holds:   $x$ is a prefix of $y$, but $x \ne y$ (this is impossible in this problem as all considered sequences have the same length);  in the first position where $x$ and $y$ differ, the sequence $x$ has a smaller element than the corresponding element in $y$. 


-----Input-----

The first line contains two positive integers $n$ and $m$ ($1 \leq n, m \leq 10^5$), denoting the number of nodes and edges, respectively.

The following $m$ lines describe the bidirectional edges in the graph. The $i$-th of these lines contains two integers $u_i$ and $v_i$ ($1 \leq u_i, v_i \leq n$), representing the nodes the $i$-th edge connects.

Note that the graph can have multiple edges connecting the same two nodes and self-loops. It is guaranteed that the graph is connected.


-----Output-----

Output a line containing the lexicographically smallest sequence $a_1, a_2, \ldots, a_n$ Bob can record.


-----Examples-----
Input
3 2
1 2
1 3

Output
1 2 3 

Input
5 5
1 4
3 4
5 4
3 2
1 5

Output
1 4 3 2 5 

Input
10 10
1 4
6 8
2 5
3 7
9 4
5 6
3 4
8 10
8 9
1 10

Output
1 4 3 7 9 8 6 5 2 10 



-----Note-----

In the first sample, Bob's optimal wandering path could be $1 \rightarrow 2 \rightarrow 1 \rightarrow 3$. Therefore, Bob will obtain the sequence $\{1, 2, 3\}$, which is the lexicographically smallest one.

In the second sample, Bob's optimal wandering path could be $1 \rightarrow 4 \rightarrow 3 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1 \rightarrow 5$. Therefore, Bob will obtain the sequence $\{1, 4, 3, 2, 5\}$, which is the lexicographically smallest one.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Given an unsorted array arr[] of size N, the task is to check whether the array consists of consecutive numbers or not.
Example 1:
Input: N = 5, arr[] = {5, 4, 2, 1, 3}
Output: Yes
Explanation: All are consecutive elements,
according to this order 1,2,3,4 and 5.
Example 2:
Input: N = 6, arr[] = {2, 1, 0, -3, -1, -2}
Output: Yes
Your Task:
This is a function problem. You don't need to take any input, as it is already accomplished by the driver code. You just need to complete the function areConsecutives() that takes array arr[] and integer N as parameters and returns the boolean true if array elements are consecutive else boolean false.
Note: If the elements are consecutive driver code outputs Yes otherwise, it outputs No
Expected Time Complexity: O(N).
Expected Auxiliary Space: O(1).
Constraints:
1 ≤ N ≤ 10^{7}

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
The Manhattan Positioning System (MPS) is a modern variant of GPS, optimized for use in large cities. MPS assumes all positions are discrete points on a regular two-dimensional grid. Within MPS, a position is represented by a pair of integers $(X,Y)$.

To determine its position, an MPS receiver first measures its distance to a number of beacons. Beacons have known, fixed locations. MPS signals propagate only along the $X$ and $Y$ axes through the streets of the city, not diagonally through building blocks. When an MPS receiver at $(X_ R,Y_ R)$ measures its distance to a beacon at $(X_ B,Y_ B)$, it thus obtains the Manhattan distance: $|X_ R - X_ B|+|Y_ R - Y_ B|$.

Given the positions of a number of beacons and the Manhattan-distances between the receiver and each beacon, determine the position of the receiver. Note that the receiver must be at an integer grid position (MPS does not yet support fractional coordinates).

-----Input-----
The first line contains an integer $N$, the number of beacons ($1 \leq N \leq 1000$). Then follow $N$ lines, each containing three integers, $X_ i$, $Y_ i$, and $D_ i$, such that $-10^6 \leq X_ i, Y_ i \leq 10^6$ and $0 \leq D_ i \leq 4 \cdot 10^6$. The pair $(X_ i, Y_ i)$ denotes the position of beacon $i$, while $D_ i$ is the Manhattan distance between receiver and beacon $i$.

No two beacons have the same position.

-----Output-----
If there is exactly one receiver position consistent with the input, write one line with two integers, $X_ R$ and $Y_ R$, the position of the receiver.

If multiple receiver positions are consistent with the input, write one line with the word “uncertain”.

If no receiver position is consistent with the input, write one line with the word “impossible”.

-----Examples-----
Sample Input 1:
3
999999 0 1000
999900 950 451
987654 123 13222
Sample Output 1:
1000200 799

Sample Input 2:
2
100 0 101
0 200 199
Sample Output 2:
uncertain

Sample Input 3:
2
100 0 100
0 200 199
Sample Output 3:
impossible

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Title: Turn Off The TV

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
Luba needs your help again! Luba has *n* TV sets. She knows that *i*-th TV set will be working from moment of time *l**i* till moment *r**i*, inclusive.

Luba wants to switch off one of TV sets in order to free the socket. Let's call some TV set redundant if after switching it off the number of integer moments of time when at least one of TV sets is working won't decrease. Luba will be very upset if she has to switch off a non-redundant TV set.

Help Luba by telling her the index of some redundant TV set. If there is no any, print -1.

Input Specification:
The first line contains one integer number *n* (1<=≤<=*n*<=≤<=2·105) — the number of TV sets.

Then *n* lines follow, each of them containing two integer numbers *l**i*,<=*r**i* (0<=≤<=*l**i*<=≤<=*r**i*<=≤<=109) denoting the working time of *i*-th TV set.

Output Specification:
If there is no any redundant TV set, print -1. Otherwise print the index of any redundant TV set (TV sets are indexed from 1 to *n*).

If there are multiple answers, print any of them.

Demo Input:
['3\n1 3\n4 6\n1 7\n', '2\n0 10\n0 10\n', '3\n1 2\n3 4\n6 8\n', '3\n1 2\n2 3\n3 4\n']

Demo Output:
['1\n', '1\n', '-1\n', '2\n']

Note:
Consider the first sample. Initially all integer moments of time such that at least one TV set is working are from the segment [1;7]. It's easy to see that this segment won't change if we switch off the first TV set (or the second one).

Note that in the fourth sample you can switch off the second TV set, since even without it all integer moments such that any of the TV sets is working denote the segment [1;4].

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.