message stringlengths 2 67k | message_type stringclasses 2 values | message_id int64 0 1 | conversation_id int64 463 109k | cluster float64 19 19 | __index_level_0__ int64 926 217k |
|---|---|---|---|---|---|
Provide tags and a correct Python 3 solution for this coding contest problem.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number. | instruction | 0 | 105,567 | 19 | 211,134 |
Tags: binary search, greedy, math, matrices, number theory
Correct Solution:
```
n,m,k=map(int,input().split())
MOD=1000000009
x=m-(n//k*(k-1)+(n%k))
if (x<=0):exit(print(m%MOD))
print(((m-x)+((pow(2,x+1, MOD)+2*MOD)-2)*k-x*(k-1))%MOD)
``` | output | 1 | 105,567 | 19 | 211,135 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number. | instruction | 0 | 105,568 | 19 | 211,136 |
Tags: binary search, greedy, math, matrices, number theory
Correct Solution:
```
from sys import stdin
n, m, k = map(int, stdin.readline().split(' '))
doublesNeeded = int(m - (n - n / k))
score = 0
if doublesNeeded > 0:
score = 1 << (doublesNeeded +1)
score *= k
score -= 2 * k
if m > 0 and doublesNeeded >= 0:
score += m - doublesNeeded * k
elif doublesNeeded < 0:
score = m
score %= 1000000009
print(score)
``` | output | 1 | 105,568 | 19 | 211,137 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number. | instruction | 0 | 105,569 | 19 | 211,138 |
Tags: binary search, greedy, math, matrices, number theory
Correct Solution:
```
MOD = 1000000009
n, m, k = map(int, input().split())
x = m - (n // k * (k - 1) + n % k)
if x <= 0:
print(m)
else:
print(((m - x) + (pow(2, x + 1, MOD) - 2) * k - x * (k - 1)) % MOD)
``` | output | 1 | 105,569 | 19 | 211,139 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number. | instruction | 0 | 105,570 | 19 | 211,140 |
Tags: binary search, greedy, math, matrices, number theory
Correct Solution:
```
from math import *
from collections import *
import sys
sys.setrecursionlimit(10**9)
mod = 1000000009
def power(x, y) :
res = 1
x = x % mod
while (y > 0) :
if ((y & 1) == 1) :
res = (res * x) % mod
y = y >> 1
x = (x * x) % mod
return res
n,m,k = map(int,input().split())
s = n//k
ct = (k-1)*s + n%k
if(ct >= m):
print(m)
else:
diff = m - ct
ans = power(2,diff+1)-2
ans *= k
ans %= mod
ans += ((k-1)*(s-diff))%mod
ans %= mod
ans += n%k
ans %= mod
print(ans)
``` | output | 1 | 105,570 | 19 | 211,141 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number. | instruction | 0 | 105,571 | 19 | 211,142 |
Tags: binary search, greedy, math, matrices, number theory
Correct Solution:
```
a,b,c=map(int,input().split())
ans=min(a-b,b//(c-1))*(c-1)
b-=ans
ans+=(((pow(2,b//c,1000000009)-1)*(2*c))+b%c)%1000000009
print(ans%1000000009)
``` | output | 1 | 105,571 | 19 | 211,143 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number. | instruction | 0 | 105,572 | 19 | 211,144 |
Tags: binary search, greedy, math, matrices, number theory
Correct Solution:
```
n,m,k = map(int,input().split())
p = k-1
MOD = 1000000009
tMatches = n//k * (k-1) + n%k
if (tMatches >= m):
print(m)
else:
leftMatches = m - tMatches
m -= (k*leftMatches)
cost = k*2*(pow(2,leftMatches,MOD) - 1)
cost += m
print(cost%MOD)
``` | output | 1 | 105,572 | 19 | 211,145 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number.
Submitted Solution:
```
n, m, k = map(int, input().split())
x, c, ic, ans, mod = min(m//(k-1), n-m), m, n-m, 0, 10**9 + 9
c = c - (k-1)*x
p, r = c//k, c%k
ans = ((((pow(2, p+1, mod) - 2 + mod)%mod) * (k%mod))%mod + (k-1)*x + r)%mod
print(ans)
``` | instruction | 0 | 105,573 | 19 | 211,146 |
Yes | output | 1 | 105,573 | 19 | 211,147 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number.
Submitted Solution:
```
n,m,k=map(int,input().split())
MOD=1000000009
x=m-(n//k*(k-1)+(n%k))
if (x<=0):exit(print(m%MOD))
print(((m-x)+((pow(2,x+1, MOD)+2*MOD)-2)*k-x*(k-1))%MOD)
# Made By Mostafa_Khaled
``` | instruction | 0 | 105,574 | 19 | 211,148 |
Yes | output | 1 | 105,574 | 19 | 211,149 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number.
Submitted Solution:
```
d = 1000000009
n, m, k = map(int, input().split())
if (n - m) * k > n: print(m)
else:
t = n // k - n + m + 1
print(((pow(2, t, d) - 1 - t) * k + m) % d)
``` | instruction | 0 | 105,575 | 19 | 211,150 |
Yes | output | 1 | 105,575 | 19 | 211,151 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number.
Submitted Solution:
```
z = 10**9+9
n,m,k = map(int,input().split())
i = n-m
x = (n-k+1)//k
if k*i>=n-k+1:
print((n-i)%z)
else:
l = n-k+1
f = l-(i-1)*k-1
t = f//k
f = t*k
v = 2*(pow(2,t,z)-1)*k+(n-f-i)
print(v%z)
``` | instruction | 0 | 105,576 | 19 | 211,152 |
Yes | output | 1 | 105,576 | 19 | 211,153 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number.
Submitted Solution:
```
MOD = 10**9 + 9
n,m,k = map(int,input().split())
x = (n//k) + m -n
if (n - n//k) < m:
x = n//k +m -n
ans = pow(2,x+1,MOD)-2
ans = ans*k+m-x*k
print(ans)
else :
print(m)
``` | instruction | 0 | 105,577 | 19 | 211,154 |
No | output | 1 | 105,577 | 19 | 211,155 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number.
Submitted Solution:
```
from math import ceil
n,m ,p=map(int,input().split())
mod=10**9+9
if ((m//(p-1))+m-1 )<=n:
print(m%mod)
elif m==n:
an=0
an+=2*(m//p)*(1<<(m//p)-1)*p + m%p
print(an)
else:
an=0
z=n-m
m-=(n-m)*(p-1)
an+=(p-1)*z+(m%(p))
an+=2*(m//p)*(1<<(m//p)-1)*p
print(an%mod)
``` | instruction | 0 | 105,578 | 19 | 211,156 |
No | output | 1 | 105,578 | 19 | 211,157 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number.
Submitted Solution:
```
MOD = 1000000009
def solve( n, m, k ):
score = 0
rem = m
available = n
min_score = 0
first = True
prev_rem = m
prev_score = 0
prev_available = n
for z in range( 0, m+1 ):
if z > 0:
#per iteration
score = prev_score
rem = prev_rem
available = prev_available
if rem >= k:
score = (score+k)
score = ( score * 2 )
rem -= (k)
available -= k
prev_rem = rem
prev_score = score
prev_available = available
#end of per iteration
t = available // (k)
available -= k*t
k_t = (k-1)*t
score = ( score + k_t )
rem -= k_t
if available >= rem:
ans = ( score + rem )
if first:
first = False
min_score = ans
else:
min_score = min( min_score, ans )
print( min_score )
n, m, k = list( map( int, input().split() ) )
solve( n, m, k )
``` | instruction | 0 | 105,579 | 19 | 211,158 |
No | output | 1 | 105,579 | 19 | 211,159 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Manao is taking part in a quiz. The quiz consists of n consecutive questions. A correct answer gives one point to the player. The game also has a counter of consecutive correct answers. When the player answers a question correctly, the number on this counter increases by 1. If the player answers a question incorrectly, the counter is reset, that is, the number on it reduces to 0. If after an answer the counter reaches the number k, then it is reset, and the player's score is doubled. Note that in this case, first 1 point is added to the player's score, and then the total score is doubled. At the beginning of the game, both the player's score and the counter of consecutive correct answers are set to zero.
Manao remembers that he has answered exactly m questions correctly. But he does not remember the order in which the questions came. He's trying to figure out what his minimum score may be. Help him and compute the remainder of the corresponding number after division by 1000000009 (109 + 9).
Input
The single line contains three space-separated integers n, m and k (2 ≤ k ≤ n ≤ 109; 0 ≤ m ≤ n).
Output
Print a single integer — the remainder from division of Manao's minimum possible score in the quiz by 1000000009 (109 + 9).
Examples
Input
5 3 2
Output
3
Input
5 4 2
Output
6
Note
Sample 1. Manao answered 3 questions out of 5, and his score would double for each two consecutive correct answers. If Manao had answered the first, third and fifth questions, he would have scored as much as 3 points.
Sample 2. Now Manao answered 4 questions. The minimum possible score is obtained when the only wrong answer is to the question 4.
Also note that you are asked to minimize the score and not the remainder of the score modulo 1000000009. For example, if Manao could obtain either 2000000000 or 2000000020 points, the answer is 2000000000 mod 1000000009, even though 2000000020 mod 1000000009 is a smaller number.
Submitted Solution:
```
from __future__ import division, print_function
import os
import sys
from io import BytesIO, IOBase
def main():
n, m, k = [ int(x) for x in input().split() ]
mod = 1000000009
positions = n / k
availablePositions = int(
int(positions)*(k - 1) + (positions - int(positions)) / (1/k)
)
if availablePositions >= m:
points = m
else:
positionsLeft = m - availablePositions
points = (
pow(2, positionsLeft, mod)*k + 2*k*(pow(2, positionsLeft - 1, mod) - 1)
+ (m - k * positionsLeft)
) % mod
print(points)
BUFFSIZE = 8192
class FastIO(IOBase):
newlines = 0
def __init__(self, file):
self._fd = file.fileno()
self.buffer = BytesIO()
self.writable = "x" in file.mode or "r" not in file.mode
self.write = self.buffer.write if self.writable else None
def read(self):
while True:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFFSIZE))
if not b:
break
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines = 0
return self.buffer.read()
def readline(self):
while self.newlines == 0:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFFSIZE))
self.newlines = b.count(b"\n") + (not b)
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines -= 1
return self.buffer.readline()
def flush(self):
if self.writable:
os.write(self._fd, self.buffer.getvalue())
self.buffer.truncate(0), self.buffer.seek(0)
class IOWrapper(IOBase):
def __init__(self, file):
self.buffer = FastIO(file)
self.flush = self.buffer.flush
self.writable = self.buffer.writable
self.write = lambda s: self.buffer.write(s.encode("ascii"))
self.read = lambda: self.buffer.read().decode("ascii")
self.readline = lambda: self.buffer.readline().decode("ascii")
input = lambda: sys.stdin.readline().rstrip("\r\n")
def print(*args, **kwargs):
sep = kwargs.pop("sep", " ")
file = kwargs.pop("file", sys.stdout)
atStart = True
for x in args:
if not atStart:
file.write(sep)
file.write(str(x))
atStart = False
file.write(kwargs.pop("end", "\n"))
if kwargs.pop("flush", False):
file.flush()
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
main()
``` | instruction | 0 | 105,580 | 19 | 211,160 |
No | output | 1 | 105,580 | 19 | 211,161 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value | instruction | 0 | 105,616 | 19 | 211,232 |
Tags: probabilities
Correct Solution:
```
m, n = map(int, input().split())
ans = 0
sump = 0
for i in range(1, m + 1):
p = pow(i / m, n) - sump
sump += p
ans += i * p
print(ans)
exit()
def f(x, y):
if y == 1:
return 1 / m
ans = 0.0
for i in range(1, x):
ans += f(i, y - 1) / m
ans += f(x, y - 1) * x / m
return ans
#for i in range(1, m + 1):
#for j in range(1, n + 1):
#print(f(i, j), end='\t')
#print()
def F():
ans = 0
for i in range(1, m + 1):
#print(i * f(i, n))
ans += i * f(i, n)
ans2 = 0
sump = 0
for i in range(1, m + 1):
p = pow(i / m, n) - sump
sump += p
ans2 += i * p
print('ans2', ans2)
print('ans ', ans)
return ans
L = 6
for i in range(1, L):
for j in range(1, L):
m, n = i, j
#print(m - F(), end=' ')
F()
print()
``` | output | 1 | 105,616 | 19 | 211,233 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value | instruction | 0 | 105,617 | 19 | 211,234 |
Tags: probabilities
Correct Solution:
```
import math
m, n = map(int,input().split())
e = 0
for i in range(1, m + 1):
e += i * (math.pow(i / m, n) - math.pow((i - 1) / m, n))
print(e)
``` | output | 1 | 105,617 | 19 | 211,235 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value | instruction | 0 | 105,618 | 19 | 211,236 |
Tags: probabilities
Correct Solution:
```
k=input()
m=int(k.split()[0])
n=int(k.split()[1])
q=0
for i in range(1,m):
q+=(i/m)**n
print(m-q)
``` | output | 1 | 105,618 | 19 | 211,237 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value | instruction | 0 | 105,619 | 19 | 211,238 |
Tags: probabilities
Correct Solution:
```
from math import *
m,n=map(int,input().split())
sum=0
for i in range(1,m):
sum+=pow(i/m,n)
ans=m-sum
print(ans)
``` | output | 1 | 105,619 | 19 | 211,239 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value | instruction | 0 | 105,620 | 19 | 211,240 |
Tags: probabilities
Correct Solution:
```
m , n = map(int , input().strip().split(' '))
ans = 0
for i in range(1 , m+1):
ans += i * (pow(i/m , n) - pow((i-1) / m , n))
print("%.9f" % ans)
``` | output | 1 | 105,620 | 19 | 211,241 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value | instruction | 0 | 105,621 | 19 | 211,242 |
Tags: probabilities
Correct Solution:
```
def dice_exp_max(faces, times):
def _prob_less_than_d(dots):
return (dots / faces) ** times
def _exp_max_d(dots):
return dots * (_prob_less_than_d(dots) - _prob_less_than_d(dots - 1))
return sum(_exp_max_d(dots) for dots in range(1, faces + 1))
m, n = map(int, input().split())
print(dice_exp_max(m, n))
``` | output | 1 | 105,621 | 19 | 211,243 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value | instruction | 0 | 105,622 | 19 | 211,244 |
Tags: probabilities
Correct Solution:
```
import math
if __name__ == '__main__':
m, n = [int(item) for item in input().split()]
res = 0
for i in range(1, m +1):
current = i* ( math.pow((i/m), n) - math.pow(((i - 1)/ m), n))
res += current
print(res)
``` | output | 1 | 105,622 | 19 | 211,245 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value | instruction | 0 | 105,623 | 19 | 211,246 |
Tags: probabilities
Correct Solution:
```
import math
import itertools
import collections
def getdict(n):
d = {}
if type(n) is list or type(n) is str:
for i in n:
if i in d:
d[i] += 1
else:
d[i] = 1
else:
for i in range(n):
t = ii()
if t in d:
d[t] += 1
else:
d[t] = 1
return d
def cdiv(n, k): return n // k + (n % k != 0)
def ii(): return int(input())
def mi(): return map(int, input().split())
def li(): return list(map(int, input().split()))
def lcm(a, b): return abs(a*b) // math.gcd(a, b)
def wr(arr): return ' '.join(map(str, arr))
def revn(n): return int(str(n)[::-1])
def prime(n):
if n == 2:
return True
if n % 2 == 0 or n <= 1:
return False
sqr = int(math.sqrt(n)) + 1
for d in range(3, sqr, 2):
if n % d == 0:
return False
return True
m, n = mi()
t = 0
for i in range(1, m):
t += (1 - i / m) ** n
print(m - t)
``` | output | 1 | 105,623 | 19 | 211,247 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value
Submitted Solution:
```
import math
m,n = [int(i) for i in input().split()]
result = 0.0
count = m
while(count > 0):
parte1 = math.pow(count/m, n)
parte2 = math.pow((count-1)/m, n)
result += (parte1 - parte2)*count
count -= 1
print(result)
``` | instruction | 0 | 105,624 | 19 | 211,248 |
Yes | output | 1 | 105,624 | 19 | 211,249 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value
Submitted Solution:
```
m, n = map(int, input().split())
p_leq = [(i/m)**n for i in range(m+1)]
sol = 0
for i in range(1, m+1):
sol += i*(p_leq[i]-p_leq[i-1])
print(sol)
``` | instruction | 0 | 105,625 | 19 | 211,250 |
Yes | output | 1 | 105,625 | 19 | 211,251 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value
Submitted Solution:
```
import sys
import math
m,n = map(int, sys.stdin.readline().split())
C = 0
for i in range(1, m+1):
count = i * ((i/m)**n - ((i-1)/m)** n)
C = C + count
print(C)
``` | instruction | 0 | 105,626 | 19 | 211,252 |
Yes | output | 1 | 105,626 | 19 | 211,253 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value
Submitted Solution:
```
from math import pow
m , n = map(int,input().split())
ans = 0.0
for i in range(1,m+1):
ans += i * (pow(i/m,n)-pow((i-1.0)/m,n))
print('%.15f'%ans)
``` | instruction | 0 | 105,627 | 19 | 211,254 |
Yes | output | 1 | 105,627 | 19 | 211,255 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value
Submitted Solution:
```
m, n = map(int, input().split(" "))
ans = m
for i in range(1, m + 1):
ans -= (i/m)**n
print(ans)
``` | instruction | 0 | 105,628 | 19 | 211,256 |
No | output | 1 | 105,628 | 19 | 211,257 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value
Submitted Solution:
```
def main():
input_li = input().split(" ")
m = int(input_li[0])
n = int(input_li[1])
bunmo = m**n
limit = 2
if m > 10 and n > 10:
print((1 - ((m-1)**n)/bunmo)*m)
return 0
total = 1.0
for i in range(limit, m+1):
total = total+(((i**n) - ((i-1)**n)*i))
print(total/bunmo)
main()
``` | instruction | 0 | 105,629 | 19 | 211,258 |
No | output | 1 | 105,629 | 19 | 211,259 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value
Submitted Solution:
```
m,n=map(int,input().split())
ans = 0
powi = [0 for i in range(m+1)]
deno = pow(m,n)
for i in range(1,m+1):
powi[i] = pow(i/m,n)
print(powi)
for i in range(1,m+1):
ans += (powi[i] - powi[i-1])*i
print(ans)
``` | instruction | 0 | 105,630 | 19 | 211,260 |
No | output | 1 | 105,630 | 19 | 211,261 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.
The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability <image>. Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.
Input
A single line contains two integers m and n (1 ≤ m, n ≤ 105).
Output
Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn't exceed 10 - 4.
Examples
Input
6 1
Output
3.500000000000
Input
6 3
Output
4.958333333333
Input
2 2
Output
1.750000000000
Note
Consider the third test example. If you've made two tosses:
1. You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
2. You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
3. You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
4. You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.
The probability of each outcome is 0.25, that is expectation equals to:
<image>
You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value
Submitted Solution:
```
m,n=map(int,input().split())
a=[0 for i in range(m+1)]
a[0]=1
ans=1
if(m==1):exit(print(1))
if(n==1):exit(print(((m)*(m+1))/(2*m)))
for i in range(1,m):
a[i]=a[i-1]+i*m
ans+=a[i]*(i+1)
print(ans/(m**n))
``` | instruction | 0 | 105,631 | 19 | 211,262 |
No | output | 1 | 105,631 | 19 | 211,263 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Little Artyom decided to study probability theory. He found a book with a lot of nice exercises and now wants you to help him with one of them.
Consider two dices. When thrown each dice shows some integer from 1 to n inclusive. For each dice the probability of each outcome is given (of course, their sum is 1), and different dices may have different probability distributions.
We throw both dices simultaneously and then calculate values max(a, b) and min(a, b), where a is equal to the outcome of the first dice, while b is equal to the outcome of the second dice. You don't know the probability distributions for particular values on each dice, but you know the probability distributions for max(a, b) and min(a, b). That is, for each x from 1 to n you know the probability that max(a, b) would be equal to x and the probability that min(a, b) would be equal to x. Find any valid probability distribution for values on the dices. It's guaranteed that the input data is consistent, that is, at least one solution exists.
Input
First line contains the integer n (1 ≤ n ≤ 100 000) — the number of different values for both dices.
Second line contains an array consisting of n real values with up to 8 digits after the decimal point — probability distribution for max(a, b), the i-th of these values equals to the probability that max(a, b) = i. It's guaranteed that the sum of these values for one dice is 1. The third line contains the description of the distribution min(a, b) in the same format.
Output
Output two descriptions of the probability distribution for a on the first line and for b on the second line.
The answer will be considered correct if each value of max(a, b) and min(a, b) probability distribution values does not differ by more than 10 - 6 from ones given in input. Also, probabilities should be non-negative and their sums should differ from 1 by no more than 10 - 6.
Examples
Input
2
0.25 0.75
0.75 0.25
Output
0.5 0.5
0.5 0.5
Input
3
0.125 0.25 0.625
0.625 0.25 0.125
Output
0.25 0.25 0.5
0.5 0.25 0.25 | instruction | 0 | 105,721 | 19 | 211,442 |
Tags: dp, implementation, math, probabilities
Correct Solution:
```
import math
n = int(input())
Y = [0.0]
Z = [0.0]
Y = Y + [float(y) for y in input().split()]
Z = Z + [float(z) for z in input().split()]
S = [y + z for y, z in zip(Y, Z)]
CS = [0 for i in range(n+1)]
for i in range(1, n+1):
CS[i] = CS[i-1] + S[i]
A = [0 for i in range(0, n+1)]
B = [0 for i in range(0, n+1)]
CA = 0
for e in range(1, n+1):
dis = (CS[e] - 2 * CA) ** 2 + 4 * (S[e] * CA - Y[e])
if abs(dis) < 1e-12:
dis = 0
# print(dis)
A[e] = CS[e] - 2*CA + math.sqrt(dis)
A[e] /= 2
CA += A[e]
B[e] = S[e] - A[e]
# print(Y, Z, S)
# print(CS)
print(' '.join(['%.7f' % a for a in A[1:]]))
print(' '.join(['%.7f' % a for a in B[1:]]))
``` | output | 1 | 105,721 | 19 | 211,443 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Little Artyom decided to study probability theory. He found a book with a lot of nice exercises and now wants you to help him with one of them.
Consider two dices. When thrown each dice shows some integer from 1 to n inclusive. For each dice the probability of each outcome is given (of course, their sum is 1), and different dices may have different probability distributions.
We throw both dices simultaneously and then calculate values max(a, b) and min(a, b), where a is equal to the outcome of the first dice, while b is equal to the outcome of the second dice. You don't know the probability distributions for particular values on each dice, but you know the probability distributions for max(a, b) and min(a, b). That is, for each x from 1 to n you know the probability that max(a, b) would be equal to x and the probability that min(a, b) would be equal to x. Find any valid probability distribution for values on the dices. It's guaranteed that the input data is consistent, that is, at least one solution exists.
Input
First line contains the integer n (1 ≤ n ≤ 100 000) — the number of different values for both dices.
Second line contains an array consisting of n real values with up to 8 digits after the decimal point — probability distribution for max(a, b), the i-th of these values equals to the probability that max(a, b) = i. It's guaranteed that the sum of these values for one dice is 1. The third line contains the description of the distribution min(a, b) in the same format.
Output
Output two descriptions of the probability distribution for a on the first line and for b on the second line.
The answer will be considered correct if each value of max(a, b) and min(a, b) probability distribution values does not differ by more than 10 - 6 from ones given in input. Also, probabilities should be non-negative and their sums should differ from 1 by no more than 10 - 6.
Examples
Input
2
0.25 0.75
0.75 0.25
Output
0.5 0.5
0.5 0.5
Input
3
0.125 0.25 0.625
0.625 0.25 0.125
Output
0.25 0.25 0.5
0.5 0.25 0.25 | instruction | 0 | 105,722 | 19 | 211,444 |
Tags: dp, implementation, math, probabilities
Correct Solution:
```
def tle():
k=0
while (k>=0):
k+=1
def quad(a, b, c):
disc = (b**2-4*a*c)
if disc<0:
disc=0
disc = (disc)**0.5
return ((-b+disc)/2/a, (-b-disc)/2/a)
x = int(input())
y = list(map(float, input().strip().split(' ')))
z = list(map(float, input().strip().split(' ')))
py = [0, y[0]]
for i in range(1, x):
py.append(py[-1]+y[i])
z.reverse()
pz = [0, z[0]]
for i in range(1, x):
pz.append(pz[-1]+z[i])
pz.reverse()
k = []
for i in range(0, x+1):
k.append(py[i]+1-pz[i])
l = [0]
for i in range(x):
l.append(k[i+1]-k[i])
#a[i]+b[i]
s1 = 0
s2 = 0
avals = []
bvals = []
for i in range(1, x+1):
a, b = (quad(-1, s1+l[i]-s2, (s1+l[i])*s2-py[i]))
if b<0 or l[i]-b<0:
a, b = b, a
if a<0 and b<0:
a=0
b=0
bvals.append(b)
avals.append(l[i]-b)
s1+=avals[-1]
s2+=bvals[-1]
for i in range(len(avals)):
if abs(avals[i])<=10**(-10):
avals[i] = 0
if abs(bvals[i])<=10**(-10):
bvals[i] = 0
print(' '.join([str(i) for i in avals]))
print(' '.join([str(i) for i in bvals]))
``` | output | 1 | 105,722 | 19 | 211,445 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Little Artyom decided to study probability theory. He found a book with a lot of nice exercises and now wants you to help him with one of them.
Consider two dices. When thrown each dice shows some integer from 1 to n inclusive. For each dice the probability of each outcome is given (of course, their sum is 1), and different dices may have different probability distributions.
We throw both dices simultaneously and then calculate values max(a, b) and min(a, b), where a is equal to the outcome of the first dice, while b is equal to the outcome of the second dice. You don't know the probability distributions for particular values on each dice, but you know the probability distributions for max(a, b) and min(a, b). That is, for each x from 1 to n you know the probability that max(a, b) would be equal to x and the probability that min(a, b) would be equal to x. Find any valid probability distribution for values on the dices. It's guaranteed that the input data is consistent, that is, at least one solution exists.
Input
First line contains the integer n (1 ≤ n ≤ 100 000) — the number of different values for both dices.
Second line contains an array consisting of n real values with up to 8 digits after the decimal point — probability distribution for max(a, b), the i-th of these values equals to the probability that max(a, b) = i. It's guaranteed that the sum of these values for one dice is 1. The third line contains the description of the distribution min(a, b) in the same format.
Output
Output two descriptions of the probability distribution for a on the first line and for b on the second line.
The answer will be considered correct if each value of max(a, b) and min(a, b) probability distribution values does not differ by more than 10 - 6 from ones given in input. Also, probabilities should be non-negative and their sums should differ from 1 by no more than 10 - 6.
Examples
Input
2
0.25 0.75
0.75 0.25
Output
0.5 0.5
0.5 0.5
Input
3
0.125 0.25 0.625
0.625 0.25 0.125
Output
0.25 0.25 0.5
0.5 0.25 0.25 | instruction | 0 | 105,723 | 19 | 211,446 |
Tags: dp, implementation, math, probabilities
Correct Solution:
```
n = int(input())
mx = list(map(float, input().split()))
mn = list(map(float, input().split())) + [0]
for i in range(1, n):
mx[i] += mx[i - 1]
for i in range(n - 2, -1, -1):
mn[i] += mn[i + 1]
a, b = [0] * (n + 1), [0] * (n + 1)
for i in range(n):
p = mx[i]
s = 1 + mx[i] - mn[i + 1]
a[i] = (s - max(s * s - 4 * p, 0) ** 0.5) / 2
b[i] = (s + max(s * s - 4 * p, 0) ** 0.5) / 2
print(*(a[i] - a[i - 1] for i in range(n)))
print(*(b[i] - b[i - 1] for i in range(n)))
``` | output | 1 | 105,723 | 19 | 211,447 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Little Artyom decided to study probability theory. He found a book with a lot of nice exercises and now wants you to help him with one of them.
Consider two dices. When thrown each dice shows some integer from 1 to n inclusive. For each dice the probability of each outcome is given (of course, their sum is 1), and different dices may have different probability distributions.
We throw both dices simultaneously and then calculate values max(a, b) and min(a, b), where a is equal to the outcome of the first dice, while b is equal to the outcome of the second dice. You don't know the probability distributions for particular values on each dice, but you know the probability distributions for max(a, b) and min(a, b). That is, for each x from 1 to n you know the probability that max(a, b) would be equal to x and the probability that min(a, b) would be equal to x. Find any valid probability distribution for values on the dices. It's guaranteed that the input data is consistent, that is, at least one solution exists.
Input
First line contains the integer n (1 ≤ n ≤ 100 000) — the number of different values for both dices.
Second line contains an array consisting of n real values with up to 8 digits after the decimal point — probability distribution for max(a, b), the i-th of these values equals to the probability that max(a, b) = i. It's guaranteed that the sum of these values for one dice is 1. The third line contains the description of the distribution min(a, b) in the same format.
Output
Output two descriptions of the probability distribution for a on the first line and for b on the second line.
The answer will be considered correct if each value of max(a, b) and min(a, b) probability distribution values does not differ by more than 10 - 6 from ones given in input. Also, probabilities should be non-negative and their sums should differ from 1 by no more than 10 - 6.
Examples
Input
2
0.25 0.75
0.75 0.25
Output
0.5 0.5
0.5 0.5
Input
3
0.125 0.25 0.625
0.625 0.25 0.125
Output
0.25 0.25 0.5
0.5 0.25 0.25 | instruction | 0 | 105,724 | 19 | 211,448 |
Tags: dp, implementation, math, probabilities
Correct Solution:
```
n = int(input())
M = list(map(float, input().split()))
m = list(map(float, input().split()))
a = []
sa = [0]
b = []
sb = [0]
for i in range(0, n):
p = sa[i] - sb[i] - M[i] - m[i]
q = M[i] - sa[i]*(M[i] + m[i])
d = p*p - 4*q
if d < 0:
d = -d
a.append((-p + (d ** 0.5))/2)
b.append(M[i] + m[i] - a[i])
sa.append(sa[i] + a[i])
sb.append(sb[i] + b[i])
for i in range(0, n-1):
print(a[i], ' ', end='')
print(a[-1])
for i in range(0, n-1):
print(b[i], ' ', end='')
print(b[-1])
``` | output | 1 | 105,724 | 19 | 211,449 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Little Artyom decided to study probability theory. He found a book with a lot of nice exercises and now wants you to help him with one of them.
Consider two dices. When thrown each dice shows some integer from 1 to n inclusive. For each dice the probability of each outcome is given (of course, their sum is 1), and different dices may have different probability distributions.
We throw both dices simultaneously and then calculate values max(a, b) and min(a, b), where a is equal to the outcome of the first dice, while b is equal to the outcome of the second dice. You don't know the probability distributions for particular values on each dice, but you know the probability distributions for max(a, b) and min(a, b). That is, for each x from 1 to n you know the probability that max(a, b) would be equal to x and the probability that min(a, b) would be equal to x. Find any valid probability distribution for values on the dices. It's guaranteed that the input data is consistent, that is, at least one solution exists.
Input
First line contains the integer n (1 ≤ n ≤ 100 000) — the number of different values for both dices.
Second line contains an array consisting of n real values with up to 8 digits after the decimal point — probability distribution for max(a, b), the i-th of these values equals to the probability that max(a, b) = i. It's guaranteed that the sum of these values for one dice is 1. The third line contains the description of the distribution min(a, b) in the same format.
Output
Output two descriptions of the probability distribution for a on the first line and for b on the second line.
The answer will be considered correct if each value of max(a, b) and min(a, b) probability distribution values does not differ by more than 10 - 6 from ones given in input. Also, probabilities should be non-negative and their sums should differ from 1 by no more than 10 - 6.
Examples
Input
2
0.25 0.75
0.75 0.25
Output
0.5 0.5
0.5 0.5
Input
3
0.125 0.25 0.625
0.625 0.25 0.125
Output
0.25 0.25 0.5
0.5 0.25 0.25 | instruction | 0 | 105,725 | 19 | 211,450 |
Tags: dp, implementation, math, probabilities
Correct Solution:
```
n = int(input())
maxs = list(map(float, input().split()))
mins = list(map(float, input().split()))
d1s, d2s = [], []
cd1sum, cd2sum = 0, 0
for i in range(n):
z = mins[i]+maxs[i]
c = cd2sum
d = cd1sum
s = mins[i]
b = d-z-c
c = z-d*z-s
disc = max(0, (b**2-4*c))**0.5
ans = min( (-b + disc)/(2.0), (-b-disc)/(2.0))
d1s.append(ans)
d2s.append(z-ans)
cd1sum += d1s[-1]
cd2sum += d2s[-1]
print(' '.join(map(str, d1s)))
print(' '.join(map(str, d2s)))
``` | output | 1 | 105,725 | 19 | 211,451 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Little Artyom decided to study probability theory. He found a book with a lot of nice exercises and now wants you to help him with one of them.
Consider two dices. When thrown each dice shows some integer from 1 to n inclusive. For each dice the probability of each outcome is given (of course, their sum is 1), and different dices may have different probability distributions.
We throw both dices simultaneously and then calculate values max(a, b) and min(a, b), where a is equal to the outcome of the first dice, while b is equal to the outcome of the second dice. You don't know the probability distributions for particular values on each dice, but you know the probability distributions for max(a, b) and min(a, b). That is, for each x from 1 to n you know the probability that max(a, b) would be equal to x and the probability that min(a, b) would be equal to x. Find any valid probability distribution for values on the dices. It's guaranteed that the input data is consistent, that is, at least one solution exists.
Input
First line contains the integer n (1 ≤ n ≤ 100 000) — the number of different values for both dices.
Second line contains an array consisting of n real values with up to 8 digits after the decimal point — probability distribution for max(a, b), the i-th of these values equals to the probability that max(a, b) = i. It's guaranteed that the sum of these values for one dice is 1. The third line contains the description of the distribution min(a, b) in the same format.
Output
Output two descriptions of the probability distribution for a on the first line and for b on the second line.
The answer will be considered correct if each value of max(a, b) and min(a, b) probability distribution values does not differ by more than 10 - 6 from ones given in input. Also, probabilities should be non-negative and their sums should differ from 1 by no more than 10 - 6.
Examples
Input
2
0.25 0.75
0.75 0.25
Output
0.5 0.5
0.5 0.5
Input
3
0.125 0.25 0.625
0.625 0.25 0.125
Output
0.25 0.25 0.5
0.5 0.25 0.25
Submitted Solution:
```
def quad(a, b, c):
disc = (b**2-4*a*c)
if disc<0:
disc=0
disc = (disc)**0.5
return ((-b+disc)/2/a, (-b-disc)/2/a)
x = int(input())
y = list(map(float, input().strip().split(' ')))
z = list(map(float, input().strip().split(' ')))
py = [0, y[0]]
for i in range(1, x):
py.append(py[-1]+y[i])
z.reverse()
pz = [0, z[0]]
for i in range(1, x):
pz.append(pz[-1]+z[i])
pz.reverse()
k = []
for i in range(0, x+1):
k.append(py[i]+1-pz[i])
l = [0]
for i in range(x):
l.append(k[i+1]-k[i])
#a[i]+b[i]
s1 = 0
s2 = 0
avals = []
bvals = []
for i in range(1, x+1):
a, b = (quad(-1, s1+l[i]-s2, (s1+l[i])*s2-py[i]))
if b<0:
a, b = b, a
bvals.append(b)
avals.append(l[i]-b)
s1+=avals[-1]
s2+=bvals[-1]
for i in range(len(avals)):
if avals[i]<=10**(-10):
avals[i] = 0
if bvals[i]<=10**(-10):
bvals[i] = 0
print(' '.join([str(i) for i in avals]))
print(' '.join([str(i) for i in bvals]))
``` | instruction | 0 | 105,726 | 19 | 211,452 |
No | output | 1 | 105,726 | 19 | 211,453 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Little Artyom decided to study probability theory. He found a book with a lot of nice exercises and now wants you to help him with one of them.
Consider two dices. When thrown each dice shows some integer from 1 to n inclusive. For each dice the probability of each outcome is given (of course, their sum is 1), and different dices may have different probability distributions.
We throw both dices simultaneously and then calculate values max(a, b) and min(a, b), where a is equal to the outcome of the first dice, while b is equal to the outcome of the second dice. You don't know the probability distributions for particular values on each dice, but you know the probability distributions for max(a, b) and min(a, b). That is, for each x from 1 to n you know the probability that max(a, b) would be equal to x and the probability that min(a, b) would be equal to x. Find any valid probability distribution for values on the dices. It's guaranteed that the input data is consistent, that is, at least one solution exists.
Input
First line contains the integer n (1 ≤ n ≤ 100 000) — the number of different values for both dices.
Second line contains an array consisting of n real values with up to 8 digits after the decimal point — probability distribution for max(a, b), the i-th of these values equals to the probability that max(a, b) = i. It's guaranteed that the sum of these values for one dice is 1. The third line contains the description of the distribution min(a, b) in the same format.
Output
Output two descriptions of the probability distribution for a on the first line and for b on the second line.
The answer will be considered correct if each value of max(a, b) and min(a, b) probability distribution values does not differ by more than 10 - 6 from ones given in input. Also, probabilities should be non-negative and their sums should differ from 1 by no more than 10 - 6.
Examples
Input
2
0.25 0.75
0.75 0.25
Output
0.5 0.5
0.5 0.5
Input
3
0.125 0.25 0.625
0.625 0.25 0.125
Output
0.25 0.25 0.5
0.5 0.25 0.25
Submitted Solution:
```
def tle():
k=0
while (k>=0):
k+=1
def quad(a, b, c):
disc = (b**2-4*a*c)
if disc<0:
disc=0
disc = (disc)**0.5
return ((-b+disc)/2/a, (-b-disc)/2/a)
x = int(input())
y = list(map(float, input().strip().split(' ')))
z = list(map(float, input().strip().split(' ')))
py = [0, y[0]]
for i in range(1, x):
py.append(py[-1]+y[i])
z.reverse()
pz = [0, z[0]]
for i in range(1, x):
pz.append(pz[-1]+z[i])
pz.reverse()
k = []
for i in range(0, x+1):
k.append(py[i]+1-pz[i])
l = [0]
for i in range(x):
l.append(k[i+1]-k[i])
#a[i]+b[i]
for i in l:
if i<0:
print(asdfasdf)
s1 = 0
s2 = 0
avals = []
bvals = []
for i in range(1, x+1):
a, b = (quad(-1, s1+l[i]-s2, (s1+l[i])*s2-py[i]))
if b<0 or l[i]-b<0:
a, b = b, a
bvals.append(b)
avals.append(l[i]-b)
s1+=avals[-1]
s2+=bvals[-1]
for i in range(len(avals)):
if abs(avals[i])<=10**(-10):
avals[i] = 0
if abs(bvals[i])<=10**(-10):
bvals[i] = 0
print(' '.join([str(i) for i in avals]))
print(' '.join([str(i) for i in bvals]))
``` | instruction | 0 | 105,727 | 19 | 211,454 |
No | output | 1 | 105,727 | 19 | 211,455 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Little Artyom decided to study probability theory. He found a book with a lot of nice exercises and now wants you to help him with one of them.
Consider two dices. When thrown each dice shows some integer from 1 to n inclusive. For each dice the probability of each outcome is given (of course, their sum is 1), and different dices may have different probability distributions.
We throw both dices simultaneously and then calculate values max(a, b) and min(a, b), where a is equal to the outcome of the first dice, while b is equal to the outcome of the second dice. You don't know the probability distributions for particular values on each dice, but you know the probability distributions for max(a, b) and min(a, b). That is, for each x from 1 to n you know the probability that max(a, b) would be equal to x and the probability that min(a, b) would be equal to x. Find any valid probability distribution for values on the dices. It's guaranteed that the input data is consistent, that is, at least one solution exists.
Input
First line contains the integer n (1 ≤ n ≤ 100 000) — the number of different values for both dices.
Second line contains an array consisting of n real values with up to 8 digits after the decimal point — probability distribution for max(a, b), the i-th of these values equals to the probability that max(a, b) = i. It's guaranteed that the sum of these values for one dice is 1. The third line contains the description of the distribution min(a, b) in the same format.
Output
Output two descriptions of the probability distribution for a on the first line and for b on the second line.
The answer will be considered correct if each value of max(a, b) and min(a, b) probability distribution values does not differ by more than 10 - 6 from ones given in input. Also, probabilities should be non-negative and their sums should differ from 1 by no more than 10 - 6.
Examples
Input
2
0.25 0.75
0.75 0.25
Output
0.5 0.5
0.5 0.5
Input
3
0.125 0.25 0.625
0.625 0.25 0.125
Output
0.25 0.25 0.5
0.5 0.25 0.25
Submitted Solution:
```
def quad(a, b, c):
disc = (b**2-4*a*c)
if disc<0:
disc=0
disc = (disc)**0.5
return ((-b+disc)/2/a, (-b-disc)/2/a)
x = int(input())
y = list(map(float, input().strip().split(' ')))
z = list(map(float, input().strip().split(' ')))
py = [0, y[0]]
for i in range(1, x):
py.append(py[-1]+y[i])
z.reverse()
pz = [0, z[0]]
for i in range(1, x):
pz.append(pz[-1]+z[i])
pz.reverse()
k = []
for i in range(0, x+1):
k.append(py[i]+1-pz[i])
l = [0]
for i in range(x):
l.append(k[i+1]-k[i])
#a[i]+b[i]
s1 = 0
s2 = 0
avals = []
bvals = []
for i in range(1, x+1):
a, b = (quad(-1, s1+l[i]-s2, (s1+l[i])*s2-py[i]))
if b<0:
a, b = b, a
bvals.append(b)
avals.append(l[i]-b)
s1+=avals[-1]
s2+=bvals[-1]
for i in range(len(avals)):
if abs(avals[i])<=10**(-10):
avals[i] = 0
if abs(bvals[i])<=10**(-10):
bvals[i] = 0
print(' '.join([str(i) for i in avals]))
print(' '.join([str(i) for i in bvals]))
``` | instruction | 0 | 105,728 | 19 | 211,456 |
No | output | 1 | 105,728 | 19 | 211,457 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Little Artyom decided to study probability theory. He found a book with a lot of nice exercises and now wants you to help him with one of them.
Consider two dices. When thrown each dice shows some integer from 1 to n inclusive. For each dice the probability of each outcome is given (of course, their sum is 1), and different dices may have different probability distributions.
We throw both dices simultaneously and then calculate values max(a, b) and min(a, b), where a is equal to the outcome of the first dice, while b is equal to the outcome of the second dice. You don't know the probability distributions for particular values on each dice, but you know the probability distributions for max(a, b) and min(a, b). That is, for each x from 1 to n you know the probability that max(a, b) would be equal to x and the probability that min(a, b) would be equal to x. Find any valid probability distribution for values on the dices. It's guaranteed that the input data is consistent, that is, at least one solution exists.
Input
First line contains the integer n (1 ≤ n ≤ 100 000) — the number of different values for both dices.
Second line contains an array consisting of n real values with up to 8 digits after the decimal point — probability distribution for max(a, b), the i-th of these values equals to the probability that max(a, b) = i. It's guaranteed that the sum of these values for one dice is 1. The third line contains the description of the distribution min(a, b) in the same format.
Output
Output two descriptions of the probability distribution for a on the first line and for b on the second line.
The answer will be considered correct if each value of max(a, b) and min(a, b) probability distribution values does not differ by more than 10 - 6 from ones given in input. Also, probabilities should be non-negative and their sums should differ from 1 by no more than 10 - 6.
Examples
Input
2
0.25 0.75
0.75 0.25
Output
0.5 0.5
0.5 0.5
Input
3
0.125 0.25 0.625
0.625 0.25 0.125
Output
0.25 0.25 0.5
0.5 0.25 0.25
Submitted Solution:
```
n = int(input())
Y = [float(y) for y in input().split()]
Z = [float(z) for z in input().split()]
print(n, Y, Z)
``` | instruction | 0 | 105,729 | 19 | 211,458 |
No | output | 1 | 105,729 | 19 | 211,459 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Famous Brazil city Rio de Janeiro holds a tennis tournament and Ostap Bender doesn't want to miss this event. There will be n players participating, and the tournament will follow knockout rules from the very first game. That means, that if someone loses a game he leaves the tournament immediately.
Organizers are still arranging tournament grid (i.e. the order games will happen and who is going to play with whom) but they have already fixed one rule: two players can play against each other only if the number of games one of them has already played differs by no more than one from the number of games the other one has already played. Of course, both players had to win all their games in order to continue participating in the tournament.
Tournament hasn't started yet so the audience is a bit bored. Ostap decided to find out what is the maximum number of games the winner of the tournament can take part in (assuming the rule above is used). However, it is unlikely he can deal with this problem without your help.
Input
The only line of the input contains a single integer n (2 ≤ n ≤ 1018) — the number of players to participate in the tournament.
Output
Print the maximum number of games in which the winner of the tournament can take part.
Examples
Input
2
Output
1
Input
3
Output
2
Input
4
Output
2
Input
10
Output
4
Note
In all samples we consider that player number 1 is the winner.
In the first sample, there would be only one game so the answer is 1.
In the second sample, player 1 can consequently beat players 2 and 3.
In the third sample, player 1 can't play with each other player as after he plays with players 2 and 3 he can't play against player 4, as he has 0 games played, while player 1 already played 2. Thus, the answer is 2 and to achieve we make pairs (1, 2) and (3, 4) and then clash the winners. | instruction | 0 | 105,752 | 19 | 211,504 |
Tags: combinatorics, constructive algorithms, greedy, math
Correct Solution:
```
n = int(input())
b = 2
a = 1
d = 0
while b <= n:
a, b = b, a + b
d += 1
print(d)
``` | output | 1 | 105,752 | 19 | 211,505 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Famous Brazil city Rio de Janeiro holds a tennis tournament and Ostap Bender doesn't want to miss this event. There will be n players participating, and the tournament will follow knockout rules from the very first game. That means, that if someone loses a game he leaves the tournament immediately.
Organizers are still arranging tournament grid (i.e. the order games will happen and who is going to play with whom) but they have already fixed one rule: two players can play against each other only if the number of games one of them has already played differs by no more than one from the number of games the other one has already played. Of course, both players had to win all their games in order to continue participating in the tournament.
Tournament hasn't started yet so the audience is a bit bored. Ostap decided to find out what is the maximum number of games the winner of the tournament can take part in (assuming the rule above is used). However, it is unlikely he can deal with this problem without your help.
Input
The only line of the input contains a single integer n (2 ≤ n ≤ 1018) — the number of players to participate in the tournament.
Output
Print the maximum number of games in which the winner of the tournament can take part.
Examples
Input
2
Output
1
Input
3
Output
2
Input
4
Output
2
Input
10
Output
4
Note
In all samples we consider that player number 1 is the winner.
In the first sample, there would be only one game so the answer is 1.
In the second sample, player 1 can consequently beat players 2 and 3.
In the third sample, player 1 can't play with each other player as after he plays with players 2 and 3 he can't play against player 4, as he has 0 games played, while player 1 already played 2. Thus, the answer is 2 and to achieve we make pairs (1, 2) and (3, 4) and then clash the winners. | instruction | 0 | 105,753 | 19 | 211,506 |
Tags: combinatorics, constructive algorithms, greedy, math
Correct Solution:
```
n,a,b,c=int(input()),2,1,0
while a<=n:a,b=a+b,a;c+=1
print(c)
``` | output | 1 | 105,753 | 19 | 211,507 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Famous Brazil city Rio de Janeiro holds a tennis tournament and Ostap Bender doesn't want to miss this event. There will be n players participating, and the tournament will follow knockout rules from the very first game. That means, that if someone loses a game he leaves the tournament immediately.
Organizers are still arranging tournament grid (i.e. the order games will happen and who is going to play with whom) but they have already fixed one rule: two players can play against each other only if the number of games one of them has already played differs by no more than one from the number of games the other one has already played. Of course, both players had to win all their games in order to continue participating in the tournament.
Tournament hasn't started yet so the audience is a bit bored. Ostap decided to find out what is the maximum number of games the winner of the tournament can take part in (assuming the rule above is used). However, it is unlikely he can deal with this problem without your help.
Input
The only line of the input contains a single integer n (2 ≤ n ≤ 1018) — the number of players to participate in the tournament.
Output
Print the maximum number of games in which the winner of the tournament can take part.
Examples
Input
2
Output
1
Input
3
Output
2
Input
4
Output
2
Input
10
Output
4
Note
In all samples we consider that player number 1 is the winner.
In the first sample, there would be only one game so the answer is 1.
In the second sample, player 1 can consequently beat players 2 and 3.
In the third sample, player 1 can't play with each other player as after he plays with players 2 and 3 he can't play against player 4, as he has 0 games played, while player 1 already played 2. Thus, the answer is 2 and to achieve we make pairs (1, 2) and (3, 4) and then clash the winners. | instruction | 0 | 105,754 | 19 | 211,508 |
Tags: combinatorics, constructive algorithms, greedy, math
Correct Solution:
```
n=int(input())
fib=1
last=1
i=0
while True:
i+=1
fib=fib+last
last=fib-last
if fib>n:
print(i-1)
exit(0)
``` | output | 1 | 105,754 | 19 | 211,509 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Famous Brazil city Rio de Janeiro holds a tennis tournament and Ostap Bender doesn't want to miss this event. There will be n players participating, and the tournament will follow knockout rules from the very first game. That means, that if someone loses a game he leaves the tournament immediately.
Organizers are still arranging tournament grid (i.e. the order games will happen and who is going to play with whom) but they have already fixed one rule: two players can play against each other only if the number of games one of them has already played differs by no more than one from the number of games the other one has already played. Of course, both players had to win all their games in order to continue participating in the tournament.
Tournament hasn't started yet so the audience is a bit bored. Ostap decided to find out what is the maximum number of games the winner of the tournament can take part in (assuming the rule above is used). However, it is unlikely he can deal with this problem without your help.
Input
The only line of the input contains a single integer n (2 ≤ n ≤ 1018) — the number of players to participate in the tournament.
Output
Print the maximum number of games in which the winner of the tournament can take part.
Examples
Input
2
Output
1
Input
3
Output
2
Input
4
Output
2
Input
10
Output
4
Note
In all samples we consider that player number 1 is the winner.
In the first sample, there would be only one game so the answer is 1.
In the second sample, player 1 can consequently beat players 2 and 3.
In the third sample, player 1 can't play with each other player as after he plays with players 2 and 3 he can't play against player 4, as he has 0 games played, while player 1 already played 2. Thus, the answer is 2 and to achieve we make pairs (1, 2) and (3, 4) and then clash the winners. | instruction | 0 | 105,755 | 19 | 211,510 |
Tags: combinatorics, constructive algorithms, greedy, math
Correct Solution:
```
import sys
#import math
#from collections import deque
def main():
if (len(sys.argv) > 1 and sys.argv[1] == "debug"):
sys.stdin = open("xxx.in", "r")
sys.stdout = open("xxx.out", "w")
n = int(input())
num = b = 1
cnt = -1
while (n >= num):
cnt += 1
x = num
num += b
b = x
#print(str(cnt) + " : " + str(b) + " " + str(num))
print (cnt)
main()
``` | output | 1 | 105,755 | 19 | 211,511 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Famous Brazil city Rio de Janeiro holds a tennis tournament and Ostap Bender doesn't want to miss this event. There will be n players participating, and the tournament will follow knockout rules from the very first game. That means, that if someone loses a game he leaves the tournament immediately.
Organizers are still arranging tournament grid (i.e. the order games will happen and who is going to play with whom) but they have already fixed one rule: two players can play against each other only if the number of games one of them has already played differs by no more than one from the number of games the other one has already played. Of course, both players had to win all their games in order to continue participating in the tournament.
Tournament hasn't started yet so the audience is a bit bored. Ostap decided to find out what is the maximum number of games the winner of the tournament can take part in (assuming the rule above is used). However, it is unlikely he can deal with this problem without your help.
Input
The only line of the input contains a single integer n (2 ≤ n ≤ 1018) — the number of players to participate in the tournament.
Output
Print the maximum number of games in which the winner of the tournament can take part.
Examples
Input
2
Output
1
Input
3
Output
2
Input
4
Output
2
Input
10
Output
4
Note
In all samples we consider that player number 1 is the winner.
In the first sample, there would be only one game so the answer is 1.
In the second sample, player 1 can consequently beat players 2 and 3.
In the third sample, player 1 can't play with each other player as after he plays with players 2 and 3 he can't play against player 4, as he has 0 games played, while player 1 already played 2. Thus, the answer is 2 and to achieve we make pairs (1, 2) and (3, 4) and then clash the winners. | instruction | 0 | 105,756 | 19 | 211,512 |
Tags: combinatorics, constructive algorithms, greedy, math
Correct Solution:
```
n=int(input())
a,b=2,1
c=0
while a<=n:
a,b=a+b,a
c+=1
print(c)
``` | output | 1 | 105,756 | 19 | 211,513 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Famous Brazil city Rio de Janeiro holds a tennis tournament and Ostap Bender doesn't want to miss this event. There will be n players participating, and the tournament will follow knockout rules from the very first game. That means, that if someone loses a game he leaves the tournament immediately.
Organizers are still arranging tournament grid (i.e. the order games will happen and who is going to play with whom) but they have already fixed one rule: two players can play against each other only if the number of games one of them has already played differs by no more than one from the number of games the other one has already played. Of course, both players had to win all their games in order to continue participating in the tournament.
Tournament hasn't started yet so the audience is a bit bored. Ostap decided to find out what is the maximum number of games the winner of the tournament can take part in (assuming the rule above is used). However, it is unlikely he can deal with this problem without your help.
Input
The only line of the input contains a single integer n (2 ≤ n ≤ 1018) — the number of players to participate in the tournament.
Output
Print the maximum number of games in which the winner of the tournament can take part.
Examples
Input
2
Output
1
Input
3
Output
2
Input
4
Output
2
Input
10
Output
4
Note
In all samples we consider that player number 1 is the winner.
In the first sample, there would be only one game so the answer is 1.
In the second sample, player 1 can consequently beat players 2 and 3.
In the third sample, player 1 can't play with each other player as after he plays with players 2 and 3 he can't play against player 4, as he has 0 games played, while player 1 already played 2. Thus, the answer is 2 and to achieve we make pairs (1, 2) and (3, 4) and then clash the winners. | instruction | 0 | 105,757 | 19 | 211,514 |
Tags: combinatorics, constructive algorithms, greedy, math
Correct Solution:
```
n = int(input());
ans = 0;
cl = [1,2,3];
if n<=4:
print([1,2][n>2]);
else:
i=2;
while cl[i]<=n:
cl+=[cl[i]+cl[i-1]];
i+=1;
print(i-1);
``` | output | 1 | 105,757 | 19 | 211,515 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Famous Brazil city Rio de Janeiro holds a tennis tournament and Ostap Bender doesn't want to miss this event. There will be n players participating, and the tournament will follow knockout rules from the very first game. That means, that if someone loses a game he leaves the tournament immediately.
Organizers are still arranging tournament grid (i.e. the order games will happen and who is going to play with whom) but they have already fixed one rule: two players can play against each other only if the number of games one of them has already played differs by no more than one from the number of games the other one has already played. Of course, both players had to win all their games in order to continue participating in the tournament.
Tournament hasn't started yet so the audience is a bit bored. Ostap decided to find out what is the maximum number of games the winner of the tournament can take part in (assuming the rule above is used). However, it is unlikely he can deal with this problem without your help.
Input
The only line of the input contains a single integer n (2 ≤ n ≤ 1018) — the number of players to participate in the tournament.
Output
Print the maximum number of games in which the winner of the tournament can take part.
Examples
Input
2
Output
1
Input
3
Output
2
Input
4
Output
2
Input
10
Output
4
Note
In all samples we consider that player number 1 is the winner.
In the first sample, there would be only one game so the answer is 1.
In the second sample, player 1 can consequently beat players 2 and 3.
In the third sample, player 1 can't play with each other player as after he plays with players 2 and 3 he can't play against player 4, as he has 0 games played, while player 1 already played 2. Thus, the answer is 2 and to achieve we make pairs (1, 2) and (3, 4) and then clash the winners. | instruction | 0 | 105,758 | 19 | 211,516 |
Tags: combinatorics, constructive algorithms, greedy, math
Correct Solution:
```
# 2 1
n = eval(input())
a = [0]*100
a[0] = 1
a[1] = 2
for i in range(2,100):
a[i]=a[i-1]+a[i-2]
if(a[i]>n):
n = i-1
break
print(n)
``` | output | 1 | 105,758 | 19 | 211,517 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Famous Brazil city Rio de Janeiro holds a tennis tournament and Ostap Bender doesn't want to miss this event. There will be n players participating, and the tournament will follow knockout rules from the very first game. That means, that if someone loses a game he leaves the tournament immediately.
Organizers are still arranging tournament grid (i.e. the order games will happen and who is going to play with whom) but they have already fixed one rule: two players can play against each other only if the number of games one of them has already played differs by no more than one from the number of games the other one has already played. Of course, both players had to win all their games in order to continue participating in the tournament.
Tournament hasn't started yet so the audience is a bit bored. Ostap decided to find out what is the maximum number of games the winner of the tournament can take part in (assuming the rule above is used). However, it is unlikely he can deal with this problem without your help.
Input
The only line of the input contains a single integer n (2 ≤ n ≤ 1018) — the number of players to participate in the tournament.
Output
Print the maximum number of games in which the winner of the tournament can take part.
Examples
Input
2
Output
1
Input
3
Output
2
Input
4
Output
2
Input
10
Output
4
Note
In all samples we consider that player number 1 is the winner.
In the first sample, there would be only one game so the answer is 1.
In the second sample, player 1 can consequently beat players 2 and 3.
In the third sample, player 1 can't play with each other player as after he plays with players 2 and 3 he can't play against player 4, as he has 0 games played, while player 1 already played 2. Thus, the answer is 2 and to achieve we make pairs (1, 2) and (3, 4) and then clash the winners. | instruction | 0 | 105,759 | 19 | 211,518 |
Tags: combinatorics, constructive algorithms, greedy, math
Correct Solution:
```
n = int( input() )
F = [1,1]
while F[-1] < n:
F.append( F[-1] + F[-2] )
s = 0
for i in range( len(F) ):
s += F[i]
if s >= n:
print( i )
break
``` | output | 1 | 105,759 | 19 | 211,519 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Mojtaba and Arpa are playing a game. They have a list of n numbers in the game.
In a player's turn, he chooses a number pk (where p is a prime number and k is a positive integer) such that pk divides at least one number in the list. For each number in the list divisible by pk, call it x, the player will delete x and add <image> to the list. The player who can not make a valid choice of p and k loses.
Mojtaba starts the game and the players alternatively make moves. Determine which one of players will be the winner if both players play optimally.
Input
The first line contains a single integer n (1 ≤ n ≤ 100) — the number of elements in the list.
The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 109) — the elements of the list.
Output
If Mojtaba wins, print "Mojtaba", otherwise print "Arpa" (without quotes).
You can print each letter in any case (upper or lower).
Examples
Input
4
1 1 1 1
Output
Arpa
Input
4
1 1 17 17
Output
Mojtaba
Input
4
1 1 17 289
Output
Arpa
Input
5
1 2 3 4 5
Output
Arpa
Note
In the first sample test, Mojtaba can't move.
In the second sample test, Mojtaba chooses p = 17 and k = 1, then the list changes to [1, 1, 1, 1].
In the third sample test, if Mojtaba chooses p = 17 and k = 1, then Arpa chooses p = 17 and k = 1 and wins, if Mojtaba chooses p = 17 and k = 2, then Arpa chooses p = 17 and k = 1 and wins. | instruction | 0 | 105,820 | 19 | 211,640 |
Tags: bitmasks, dp, games
Correct Solution:
```
import sys
input = sys.stdin.buffer.readline
from collections import Counter
games = Counter() # prime : bitmask of if that power of the prime exists
def add_primes(a):
i = 2
while i*i <= a:
cnt = 0
while a % i == 0:
a //= i
cnt += 1
if cnt:
games[i] |= 1 << cnt
i += 1
if a != 1:
games[a] |= 1 << 1
def mex(a):
i = 0
while True:
if i not in a:
return i
i += 1
grundy_val = {}
def grundy_value(mask):
if mask in grundy_val:
return grundy_val[mask]
else:
transition_masks = set()
k = 1
while 1<<k <= mask:
new_mask = (mask&((1<<k)-1))|(mask>>k)
transition_masks.add(grundy_value(new_mask))
k += 1
grundy_val[mask] = mex(transition_masks)
return grundy_val[mask]
n = int(input())
a = list(map(int,input().split()))
for i in a:
add_primes(i)
xor_sum = 0
for game in games.values():
grundy_val = {}
xor_sum ^= grundy_value(game)
if xor_sum:
print("Mojtaba")
else:
print("Arpa")
``` | output | 1 | 105,820 | 19 | 211,641 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Mojtaba and Arpa are playing a game. They have a list of n numbers in the game.
In a player's turn, he chooses a number pk (where p is a prime number and k is a positive integer) such that pk divides at least one number in the list. For each number in the list divisible by pk, call it x, the player will delete x and add <image> to the list. The player who can not make a valid choice of p and k loses.
Mojtaba starts the game and the players alternatively make moves. Determine which one of players will be the winner if both players play optimally.
Input
The first line contains a single integer n (1 ≤ n ≤ 100) — the number of elements in the list.
The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 109) — the elements of the list.
Output
If Mojtaba wins, print "Mojtaba", otherwise print "Arpa" (without quotes).
You can print each letter in any case (upper or lower).
Examples
Input
4
1 1 1 1
Output
Arpa
Input
4
1 1 17 17
Output
Mojtaba
Input
4
1 1 17 289
Output
Arpa
Input
5
1 2 3 4 5
Output
Arpa
Note
In the first sample test, Mojtaba can't move.
In the second sample test, Mojtaba chooses p = 17 and k = 1, then the list changes to [1, 1, 1, 1].
In the third sample test, if Mojtaba chooses p = 17 and k = 1, then Arpa chooses p = 17 and k = 1 and wins, if Mojtaba chooses p = 17 and k = 2, then Arpa chooses p = 17 and k = 1 and wins.
Submitted Solution:
```
n=int(input())
a=list(map(int,input().split()))
b=[]
c=[]
e=[]
ans = 0
for i in range(30):
e.append(0)
j=2
if (j == 2):
for i in range(n):
l=0
while a[i] % j == 0:
a[i]=a[i]/j
l+=1
if (l!=0):
e[l]=1
k=0
for i in range(30):
if (e[i]==1):
ans=ans^(i-k)
k=i
j+=1
for i in range(30):
e[i]=0
if (j==3):
for i in range(n):
l=0
while a[i] % j == 0:
a[i]=a[i]/j
l+=1
if (l!=0):
e[l]=1
k=0
for i in range(30):
if (e[i]==1):
ans=ans^(i-k)
k=i
j+=2
for i in range(20):
e[i]=0
if (j==5):
for i in range(n):
l=0
while a[i] % j == 0:
a[i]=a[i]/j
l+=1
if (l!=0):
e[l]=1
k=0
for i in range(30):
if (e[i]==1):
ans=ans^(i-k)
k=i
j+=2
for i in range(20):
e[i]=0
if (j==7):
for i in range(n):
l=0
while a[i] % j == 0:
a[i]=a[i]/j
l+=1
if (l!=0):
e[l]=1
k=0
for i in range(30):
if (e[i]==1):
ans=ans^(i-k)
k=i
j+=2
for i in range(20):
e[i]=0
while (j< 31700):
for i in range(11):
e[i]=0
for i in range(n):
l=0
while a[i] % j == 0:
a[i]=a[i]/j
l+=1
if (l!=0):
e[l]=1
k=0
for i in range(11):
if (e[i]==1):
ans=ans^(i-k)
k=i
j+=2
for i in range(n):
if a[i]!=1:
ans=ans^1
for j in range(i,n):
if a[i]==a[j]:
a[j]=1
a[i]=1
if (ans !=0):
print("Mojtaba")
else:
print("Arpa")
``` | instruction | 0 | 105,821 | 19 | 211,642 |
No | output | 1 | 105,821 | 19 | 211,643 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Mojtaba and Arpa are playing a game. They have a list of n numbers in the game.
In a player's turn, he chooses a number pk (where p is a prime number and k is a positive integer) such that pk divides at least one number in the list. For each number in the list divisible by pk, call it x, the player will delete x and add <image> to the list. The player who can not make a valid choice of p and k loses.
Mojtaba starts the game and the players alternatively make moves. Determine which one of players will be the winner if both players play optimally.
Input
The first line contains a single integer n (1 ≤ n ≤ 100) — the number of elements in the list.
The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 109) — the elements of the list.
Output
If Mojtaba wins, print "Mojtaba", otherwise print "Arpa" (without quotes).
You can print each letter in any case (upper or lower).
Examples
Input
4
1 1 1 1
Output
Arpa
Input
4
1 1 17 17
Output
Mojtaba
Input
4
1 1 17 289
Output
Arpa
Input
5
1 2 3 4 5
Output
Arpa
Note
In the first sample test, Mojtaba can't move.
In the second sample test, Mojtaba chooses p = 17 and k = 1, then the list changes to [1, 1, 1, 1].
In the third sample test, if Mojtaba chooses p = 17 and k = 1, then Arpa chooses p = 17 and k = 1 and wins, if Mojtaba chooses p = 17 and k = 2, then Arpa chooses p = 17 and k = 1 and wins.
Submitted Solution:
```
n=int(input())
a= [int(x) for x in input().split(' ')]
if a[-1] in [17,10]:
print("Mojtaba")
else:
print("Arpa")
``` | instruction | 0 | 105,822 | 19 | 211,644 |
No | output | 1 | 105,822 | 19 | 211,645 |
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