text stringlengths 198 433k | conversation_id int64 0 109k |
|---|---|
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given n integer numbers a_1, a_2, ..., a_n. Consider graph on n nodes, in which nodes i, j (iβ j) are connected if and only if, a_i AND a_jβ 0, where AND denotes the [bitwise AND operation](https://en.wikipedia.org/wiki/Bitwise_operation#AND).
Find the length of the shortest cycle in this graph or determine that it doesn't have cycles at all.
Input
The first line contains one integer n (1 β€ n β€ 10^5) β number of numbers.
The second line contains n integer numbers a_1, a_2, ..., a_n (0 β€ a_i β€ 10^{18}).
Output
If the graph doesn't have any cycles, output -1. Else output the length of the shortest cycle.
Examples
Input
4
3 6 28 9
Output
4
Input
5
5 12 9 16 48
Output
3
Input
4
1 2 4 8
Output
-1
Note
In the first example, the shortest cycle is (9, 3, 6, 28).
In the second example, the shortest cycle is (5, 12, 9).
The graph has no cycles in the third example.
Submitted Solution:
```
import math
n = int(input())
a = list(map(int, input().split()))
MAX_A = 10 ** 18
n_bits = math.ceil(math.log(MAX_A, 2)) + 1
if n > n_bits*3:
print(3)
exit(0)
comp = [[] for _ in range(n_bits)]
G = {}
def add(v):
G[v] = set()
v2, i = v, 0
while v2 != 0:
if v2 % 2 == 1: comp[i].append(v)
v2 //= 2
i += 1
for v in a:
add(v)
for c in comp:
if len(c) >= 3:
print(3)
exit(0)
elif len(c) == 2:
v, w = c
G[v].add(w)
G[w].add(v)
res = -1
for u in a:
level = {v:-1 for v in a}
level[u] = 0
l = [u]
i = 0
while len(l) > 0 and (res < 0 or 2*i < res):
l2 = []
for v in l:
for w in G[v]:
if level[w] == -1:
l2.append(w)
level[w] = i+1
elif level[w] >= i:
res = min(res, i+1+level[w]) if res > 0 else i+1+level[w]
l = l2
i += 1
print(res)
```
No
| 101,300 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given n integer numbers a_1, a_2, ..., a_n. Consider graph on n nodes, in which nodes i, j (iβ j) are connected if and only if, a_i AND a_jβ 0, where AND denotes the [bitwise AND operation](https://en.wikipedia.org/wiki/Bitwise_operation#AND).
Find the length of the shortest cycle in this graph or determine that it doesn't have cycles at all.
Input
The first line contains one integer n (1 β€ n β€ 10^5) β number of numbers.
The second line contains n integer numbers a_1, a_2, ..., a_n (0 β€ a_i β€ 10^{18}).
Output
If the graph doesn't have any cycles, output -1. Else output the length of the shortest cycle.
Examples
Input
4
3 6 28 9
Output
4
Input
5
5 12 9 16 48
Output
3
Input
4
1 2 4 8
Output
-1
Note
In the first example, the shortest cycle is (9, 3, 6, 28).
In the second example, the shortest cycle is (5, 12, 9).
The graph has no cycles in the third example.
Submitted Solution:
```
"""This code was written by
Russell Emerine - linguist,
mathematician, coder,
musician, and metalhead."""
n = int(input())
a = []
for x in input().split():
x = int(x)
if x: a.append(x)
n = len(a)
edges = set()
for d in range(60):
p = 1 << d
c = []
for i in range(n):
x = a[i]
if x & p:
c.append(i)
if len(c) > 2:
import sys
print(3)
sys.exit()
if len(c) == 2:
edges.add((c[0], c[1]))
m = n + 1
adj = [[][:] for _ in range(n)]
for u, v in edges:
adj[u].append(v)
adj[v].append(u)
from collections import deque
for r in range(n):
v = [False] * n
p = [-1] * n
d = [n + 1] * n
d[0] = 0
s = deque([r])
while len(s):
h = s.popleft()
v[h] = True
for c in adj[h]:
if p[h] == c:
continue
elif v[c]:
m = min(d[h] + 1 + d[c], m)
else:
d[c] = d[h] + 1
v[c] = True
p[c] = h
s.append(c);
if m == n + 1: m = -1
print(m)
```
No
| 101,301 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Tags: brute force, greedy, implementation
Correct Solution:
```
import math as ma
q=int(input())
for w in range(q):
n=int(input())
m=n
i=ma.ceil(ma.log(n,3))
j=i-1
k=0
r=n
while m>=0 and j>=0:
if m>=3**j:
m=m-3**j
else:
k=j
r=n-m
j=j-1
if m==0:
print(n)
elif k>0:
print(r+3**k)
else:
print(3**i)
```
| 101,302 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Tags: brute force, greedy, implementation
Correct Solution:
```
#import resource
import sys
#resource.setrlimit(resource.RLIMIT_STACK, (2**29, -1))
sys.setrecursionlimit(10 ** 7)
from collections import deque
import math
pow3 = [1]
aux = 1
for i in range(9):
aux *= 3
pow3.append(aux)
a = []
for op in range((1 << 10)):
elem = 0
for i in range(len(pow3)):
if ( (op >> i) & 1 ):
elem += pow3[i]
a.append(elem)
t = int(input())
for _ in range(t):
n = int(input())
lo, hi = 0, len(a) - 1
while ( hi - lo > 1 ):
mi = lo + ( hi - lo ) // 2
if (n > a[mi]):
lo = mi
else:
hi = mi
print(a[hi])
```
| 101,303 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Tags: brute force, greedy, implementation
Correct Solution:
```
import math
q = int(input())
pw = [3**0]
for i in range(1, 10):
pw.append(3**i)
length = len(pw)
for j in range(length-1):
pw.append(3**i+pw[j])
for _ in range(q):
n = int(input())
for i in range(len(pw)):
if n<=pw[i]:
print (pw[i])
break
```
| 101,304 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Tags: brute force, greedy, implementation
Correct Solution:
```
arr = [3**i for i in range(45)]
for i in range(int(input())):
n, hi = int(input()), sum(arr)
for j in reversed(arr):
if hi - j >= n:
hi -= j
print(hi)
```
| 101,305 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Tags: brute force, greedy, implementation
Correct Solution:
```
#this is wild game
def gcd(a:int,b:int) -> int:
return a if b==0 else gcd(b,a%b)
def conv(n:int,b:int=3)->int:
a=0
p=1
while n:
a+=(n%b)*p
p*=10
n//=b
return a
def solve(n:int)->int:
p=0
s=""
while n:
l=n%10
if p:
l=(l+1)%3
if l==0:
p=1
s+="0"
elif l==1:
p=0
s="0"*len(s)+"1"
else:
p=1
s+="0"
else:
if l==2:
p=1
s+="0"
else:
p=0
s+=str(l)
n//=10
if p:
s="0"*len(s)+"1"
# print(s[::-1])
return s[::-1]
for _ in range(int(input())):
print(int(solve(conv(int(input()))),3))
```
| 101,306 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Tags: brute force, greedy, implementation
Correct Solution:
```
def b10to3(num):
digs = []
num = int(num)
while num > 0:
digs.append(num % 3)
num //= 3
return list(reversed(digs))
def b3to10(num):
sum = 0
for i in num:
sum = sum * 3 + i
return str(sum)
for i in range(int(input())):
num = input()
digs = [0] + b10to3(num)
# 0100200121
# take the MSB which is 2, find something to its left which is 0, increment it and set rest to 0
pos_0, pos_2 = -1, -1
for i in range(len(digs)):
if digs[i] == 2:
pos_2 = i
break
if digs[i] == 0:
pos_0 = i
if pos_2 != -1:
digs[pos_0] = 1
for i in range(pos_0 + 1, len(digs)):
digs[i] = 0
print(b3to10(digs))
```
| 101,307 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Tags: brute force, greedy, implementation
Correct Solution:
```
q = int(input())
def base_10_to_n(x, n):
if x//n:
return base_10_to_n(x//n, n)+str(x%n)
else:
return str(x%n)
for _ in range(q):
n = int(input())
s = list(base_10_to_n(n, 3))
#print(s)
s.reverse()
pos0 = -1
pos2 = -1
flag = False
for i in range(len(s)):
if s[i] == '2':
pos2 = i
flag = True
if s[i] == '0' and flag:
pos0 = i
flag = False
if pos2 != -1:
if pos0 > pos2:
temp = ['0']*(pos0)+['1']+s[pos0+1:]
temp = list(map(int, temp))
ans = 0
for i, d in enumerate(temp):
ans += d*3**i
else:
ans = 3**(len(s))
else:
ans = n
print(ans)
```
| 101,308 |
Provide tags and a correct Python 3 solution for this coding contest problem.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Tags: brute force, greedy, implementation
Correct Solution:
```
import math
i = 0
s = 0
v = []
while s < 1e18:
s += pow(3,i)
v.append(s)
i += 1
q = int(input())
for i in range(q):
n = int(input())
k = 0
while v[k] < n:
k += 1
ans = v[k]
l = k
while l >= 0:
if ans - pow(3,l) >= n:
ans -= pow(3,l);
l -= 1
print(ans)
```
| 101,309 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Submitted Solution:
```
for _ in range(int(input())):
n = int(input())
val = 0
i = 0
while val <= n:
val += 3 ** i
i += 1
# print("max val: {} {}".format(val, i))
while i >= 0:
if val - (3 ** i) >= n:
val -= 3 ** i
# print("updated val: {} {}".format(val, i))
i -= 1
print(val)
```
Yes
| 101,310 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Submitted Solution:
```
a = [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147,
531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489,
1162261467, 3486784401, 10460353203, 31381059609, 94143178827,
282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961,
68630377364883, 205891132094649, 617673396283947, 1853020188851841,
5559060566555523, 16677181699666569, 50031545098999707, 150094635296999121,
450283905890997363, 1350851717672992089]
sum3 = 2026277576509488133
for _ in range(int(input())):
ans = sum3
n = int(input())
for x in a[::-1]:
if ans - x >= n : ans -= x
print(ans)
```
Yes
| 101,311 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Submitted Solution:
```
import sys
input=sys.stdin.readline
q=int(input())
for _ in range(q):
m=int(input())
n=m
sansin=[]
while True:
sansin.append(n%3)
n//=3
if n<3:
sansin.append(n)
break
f=1
x=0
for i in range(len(sansin)):
if sansin[i]==2:
f=0
x=i
if f:
print(m)
else:
sansin.insert(len(sansin),0)
ans=0
ff=1
for i in range(x,len(sansin)):
if sansin[i]==0 and ff:
ans+=(3**i)
ff=0
elif ff==0:
ans+=(sansin[i]*(3**i))
print(ans)
```
Yes
| 101,312 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Submitted Solution:
```
def ii(): return int(input())
def si(): return input()
def mi(): return map(int,input().split())
def li(): return list(mi())
import math
q=ii()
for i in range(q):
n=ii()
c=0
while pow(3,c)<n:
c+=1
if pow(3,c)==n:
print(pow(3,c))
continue
ans=pow(3,c)
b=1<<(c-1)+1
for i in range(b):
d=pow(3,c-1)
for j in range(c-1):
if (i>>j)%2:
d+=pow(3,j)
if d>=n:
ans=d
break
print(ans)
```
Yes
| 101,313 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Submitted Solution:
```
q = int(input())
def base_10_to_n(x, n):
if (int(x/n)):
return base_10_to_n(int(x/n), n)+str(x%n)
return str(x%n)
for _ in range(q):
n = int(input())
s = list(base_10_to_n(n, 3))
s.reverse()
pos0 = -1
pos2 = -1
for i in range(len(s)):
if s[i] == '2':
pos2 = i
if s[i] == 0 and pos2 != -1:
pos0 = i
if pos2 != -1:
if pos0 > pos2:
temp = ['0']*(pos0)+['1']+s[pos0+1:]
temp.reverse()
temp = list(map(int, ans))
ans = 0
for i, d in enumerate(temp):
ans += d**i
else:
ans = 3**(len(s))
else:
ans = n
print(ans)
```
No
| 101,314 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Submitted Solution:
```
#import sys
# import math
# import bisect
# import collections
# import itertools
# #from sys import stdin,stdout
# from math import gcd,floor,sqrt,log
# from collections import defaultdict as dd, Counter as ctr
# from bisect import bisect_left as bl, bisect_right as br
# from itertools import permutations as pr, combinations as cb
#sys.setrecursionlimit(100000000)
#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$
inp = lambda: int(input())
# strng = lambda: input().strip()
# jn = lambda x,l: x.join(map(str,l))
# strl = lambda: list(input().strip())
# mul = lambda: map(int,input().strip().split())
# mulf = lambda: map(float,input().strip().split())
seq = lambda: list(map(int,input().strip().split()))
#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$#$
# p_inf = float('inf')
# n_inf = float('-inf')
#To find mex
# def mex(arr):
# nList = set(arr)
# mex = 0
# while mex in nList:
# mex += 1
# return(mex)
def count(n):
cnt = 0
while n > 1:
n = n // 3
cnt += 1
return(cnt)
def results(n):
cnt = count(n)
if(3 ** cnt == n):
return(n)
else:
tmp = 3 ** cnt
i = 0
while tmp <= n and i < cnt:
tmp += 3 ** i
i += 1
if(tmp >= n):
return(tmp)
else:
return(3 ** (cnt + 1))
def main():
t = inp()
for _ in range(t):
n = inp()
result = results(n)
print(result)
if __name__ == '__main__':
main()
```
No
| 101,315 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Submitted Solution:
```
def add_my_bit(t, s):
if t[s] == 2:
t[s] = 0
if s == len(t) - 1:
t.append(1)
else:
add_my_bit(t, s + 1)
elif t[s] == 1:
t[s] = 2
elif t[s] == 0:
t[s] = 1
return t
for _ in range(int(input())):
a = int(input())
n = a
t = []
k = 0
while a != 0:
t.append(a% 3)
a = a//3
u = len(t)
if 2 in t:
w = t[::-1].index(2)
if len(t) - w <= len(t) - 1 and t[len(t) - w] == 0:
t[len(t) - w] = 1
for rv in range(len(t[: len(t) - w])):
t[rv] = 0
while 2 in t:
p = t.index(2)
add_my_bit(t, p)
v = len(t)
if u < v:
k = k + 3**(v - 1)
else:
for j in range(len(t)):
k = k + (3**j)*t[j]
else:
k = n
print(k)
```
No
| 101,316 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
The only difference between easy and hard versions is the maximum value of n.
You are given a positive integer number n. You really love good numbers so you want to find the smallest good number greater than or equal to n.
The positive integer is called good if it can be represented as a sum of distinct powers of 3 (i.e. no duplicates of powers of 3 are allowed).
For example:
* 30 is a good number: 30 = 3^3 + 3^1,
* 1 is a good number: 1 = 3^0,
* 12 is a good number: 12 = 3^2 + 3^1,
* but 2 is not a good number: you can't represent it as a sum of distinct powers of 3 (2 = 3^0 + 3^0),
* 19 is not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representations 19 = 3^2 + 3^2 + 3^0 = 3^2 + 3^1 + 3^1 + 3^1 + 3^0 are invalid),
* 20 is also not a good number: you can't represent it as a sum of distinct powers of 3 (for example, the representation 20 = 3^2 + 3^2 + 3^0 + 3^0 is invalid).
Note, that there exist other representations of 19 and 20 as sums of powers of 3 but none of them consists of distinct powers of 3.
For the given positive integer n find such smallest m (n β€ m) that m is a good number.
You have to answer q independent queries.
Input
The first line of the input contains one integer q (1 β€ q β€ 500) β the number of queries. Then q queries follow.
The only line of the query contains one integer n (1 β€ n β€ 10^4).
Output
For each query, print such smallest integer m (where n β€ m) that m is a good number.
Example
Input
7
1
2
6
13
14
3620
10000
Output
1
3
9
13
27
6561
19683
Submitted Solution:
```
queries = int(input())
for q in range(queries):
n = int(input())
p = 39
dp = [0] * p
r = n
for i in range(p)[::-1]:
dp[i], r = divmod(r, 3**i)
if all(i <= 1 for i in dp):
print(n)
else:
r = p
while dp[r-1] == 0:
r -= 1
print(3 ** r)
```
No
| 101,317 |
Provide tags and a correct Python 3 solution for this coding contest problem.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Tags: greedy, implementation, math
Correct Solution:
```
# Legends Always Come Up with Solution
# Author: Manvir Singh
import os
from io import BytesIO, IOBase
import sys
from collections import Counter
def main():
for _ in range(int(input())):
n=int(input())
a,mi=[0]*(n+1),n
for i in input().split():
a[int(i)]+=1
b=Counter(a)
del b[0]
for i in range(1,min(b)+2,1):
ans=0
for j in b:
zz=(i-j%i)%i
if j-zz*(i-1)<0:
ans=n
break
ans+=((j-zz*(i-1))//i+zz)*b[j]
mi=min(mi,ans)
print(mi)
# FAST INPUT OUTPUT REGION
BUFSIZE = 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, BUFSIZE))
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, BUFSIZE))
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")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
if __name__ == "__main__":
main()
```
| 101,318 |
Provide tags and a correct Python 3 solution for this coding contest problem.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Tags: greedy, implementation, math
Correct Solution:
```
import sys
import io, os
import math
gcd=math.gcd
ceil=math.ceil
input = io.BytesIO(os.read(0,os.fstat(0).st_size)).readline
#arr=list(map(int, input().split()))
def main():
t=int(input())
for _ in range(t):
n=int(input())
arr=list(map(int, input().split()))
l=[0]*n
for i in range(n):
l[arr[i]-1]+=1
ans=[]
for i in range(n):
if(l[i]!=0):
ans.append(l[i])
ans.sort()
mi=ans[0]+1
su=10**7
for i in range(mi):
s=i+1
temp=0
trig=True
for j in range(len(ans)):
r=ceil(ans[j]/s)
if(ans[j]<r*(s-1)):
trig=False
break
temp+=r
if(trig):
su=min(temp,su)
print(su)
if __name__ == '__main__':
main()
```
| 101,319 |
Provide tags and a correct Python 3 solution for this coding contest problem.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Tags: greedy, implementation, math
Correct Solution:
```
#m*(s-1)<=nItems<=m*s
def checkSPossible(s,nItems):
m=(nItems+s-1)//s
# print('m:{}'.format(m))
return m*(s-1)<=nItems<=m*s
def checkAllPossible(s,allCnts):
for cnt in allCnts:
if not checkSPossible(s,cnt):
return False
return True
def main():
t=int(input())
allans=[]
for _ in range(t):
n=int(input())
c=readIntArr()
allCnts=[0 for _ in range(n+1)]
for x in c:
allCnts[x]+=1
allCntsCompressed=[]
for x in allCnts:
if x>0:
allCntsCompressed.append(x)
largestS=0
for s in range(1,min(allCntsCompressed)+2):
if checkAllPossible(s,allCntsCompressed):
largestS=s
m=0
for x in allCntsCompressed:
m+=(x+largestS-1)//largestS
allans.append(m)
multiLineArrayPrint(allans)
return
import sys
input=sys.stdin.buffer.readline #FOR READING PURE INTEGER INPUTS (space separation ok)
# input=lambda: sys.stdin.readline().rstrip("\r\n") #FOR READING STRING/TEXT INPUTS.
def oneLineArrayPrint(arr):
print(' '.join([str(x) for x in arr]))
def multiLineArrayPrint(arr):
print('\n'.join([str(x) for x in arr]))
def multiLineArrayOfArraysPrint(arr):
print('\n'.join([' '.join([str(x) for x in y]) for y in arr]))
def readIntArr():
return [int(x) for x in input().split()]
# def readFloatArr():
# return [float(x) for x in input().split()]
def makeArr(defaultVal,dimensionArr): # eg. makeArr(0,[n,m])
dv=defaultVal;da=dimensionArr
if len(da)==1:return [dv for _ in range(da[0])]
else:return [makeArr(dv,da[1:]) for _ in range(da[0])]
def queryInteractive(x,y):
print('? {} {}'.format(x,y))
sys.stdout.flush()
return int(input())
def answerInteractive(ans):
print('! {}'.format(ans))
sys.stdout.flush()
inf=float('inf')
MOD=10**9+7
for _abc in range(1):
main()
```
| 101,320 |
Provide tags and a correct Python 3 solution for this coding contest problem.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Tags: greedy, implementation, math
Correct Solution:
```
import sys
from collections import Counter
input = sys.stdin.readline
for nt in range(int(input())):
n = int(input())
a = list(map(int,input().split()))
c = Counter(a)
minn = n
for i in c:
minn = min(c[i],minn)
ans = n
for i in range(1,minn+2):
flag = 0
scr = 0
for j in c:
scr += (c[j]-1)//i + 1
if c[j]<((c[j]-1)//i + 1)*(i-1):
flag = 1
break
# print (i,scr,minn,flag)
if not flag:
ans = min(ans,scr)
print (ans)
```
| 101,321 |
Provide tags and a correct Python 3 solution for this coding contest problem.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Tags: greedy, implementation, math
Correct Solution:
```
from math import ceil, inf
for _ in range(int(input())):
n=int(input())
a=[int(X)-1 for X in input().split()]
x=[]
z=[0]*n
for i in range(n):
z[a[i]]+=1
for i in z:
if i:
x.append(i)
x.sort()
# print(x)
an=inf
for i in range(1,x[0]+2):
mi=0
for j in x :
if j>=ceil(j/i)*(i-1):
mi+=ceil(j/i)
else:
mi=inf
an=min(mi,an)
print(an)
```
| 101,322 |
Provide tags and a correct Python 3 solution for this coding contest problem.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Tags: greedy, implementation, math
Correct Solution:
```
"""
Satwik_Tiwari ;) .
12th Sept , 2020 - Saturday
"""
#===============================================================================================
#importing some useful libraries.
from __future__ import division, print_function
from fractions import Fraction
import sys
import os
from io import BytesIO, IOBase
from itertools import *
import bisect
from heapq import *
from math import *
from copy import *
from collections import deque
from collections import Counter as counter # Counter(list) return a dict with {key: count}
from itertools import combinations as comb # if a = [1,2,3] then print(list(comb(a,2))) -----> [(1, 2), (1, 3), (2, 3)]
from itertools import permutations as permutate
from bisect import bisect_left as bl
#If the element is already present in the list,
# the left most position where element has to be inserted is returned.
from bisect import bisect_right as br
from bisect import bisect
#If the element is already present in the list,
# the right most position where element has to be inserted is returned
#==============================================================================================
#fast I/O region
BUFSIZE = 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, BUFSIZE))
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, BUFSIZE))
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")
def print(*args, **kwargs):
"""Prints the values to a stream, or to sys.stdout by default."""
sep, file = kwargs.pop("sep", " "), kwargs.pop("file", sys.stdout)
at_start = True
for x in args:
if not at_start:
file.write(sep)
file.write(str(x))
at_start = False
file.write(kwargs.pop("end", "\n"))
if kwargs.pop("flush", False):
file.flush()
if sys.version_info[0] < 3:
sys.stdin, sys.stdout = FastIO(sys.stdin), FastIO(sys.stdout)
else:
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
# inp = lambda: sys.stdin.readline().rstrip("\r\n")
#===============================================================================================
### START ITERATE RECURSION ###
from types import GeneratorType
def iterative(f, stack=[]):
def wrapped_func(*args, **kwargs):
if stack: return f(*args, **kwargs)
to = f(*args, **kwargs)
while True:
if type(to) is GeneratorType:
stack.append(to)
to = next(to)
continue
stack.pop()
if not stack: break
to = stack[-1].send(to)
return to
return wrapped_func
#### END ITERATE RECURSION ####
#===============================================================================================
#some shortcuts
mod = 1000000007
def inp(): return sys.stdin.readline().rstrip("\r\n") #for fast input
def out(var): sys.stdout.write(str(var)) #for fast output, always take string
def lis(): return list(map(int, inp().split()))
def stringlis(): return list(map(str, inp().split()))
def sep(): return map(int, inp().split())
def strsep(): return map(str, inp().split())
# def graph(vertex): return [[] for i in range(0,vertex+1)]
def zerolist(n): return [0]*n
def nextline(): out("\n") #as stdout.write always print sring.
def testcase(t):
for pp in range(t):
solve(pp)
def printlist(a) :
for p in range(0,len(a)):
out(str(a[p]) + ' ')
def google(p):
print('Case #'+str(p)+': ',end='')
def lcm(a,b): return (a*b)//gcd(a,b)
def power(x, y, p) :
res = 1 # Initialize result
x = x % p # Update x if it is more , than or equal to p
if (x == 0) :
return 0
while (y > 0) :
if ((y & 1) == 1) : # If y is odd, multiply, x with result
res = (res * x) % p
y = y >> 1 # y = y/2
x = (x * x) % p
return res
def ncr(n,r): return factorial(n)//(factorial(r)*factorial(max(n-r,1)))
def isPrime(n) :
if (n <= 1) : return False
if (n <= 3) : return True
if (n % 2 == 0 or n % 3 == 0) : return False
i = 5
while(i * i <= n) :
if (n % i == 0 or n % (i + 2) == 0) :
return False
i = i + 6
return True
#===============================================================================================
# code here ;))
def solve(case):
n = int(inp())
aa = lis()
cnt = {}
for i in range(n):
if(aa[i] in cnt):
cnt[aa[i]] +=1
else:
cnt[aa[i]] = 1
a = []
for i in cnt:
a.append(cnt[i])
a = sorted(a)
n = len(a)
mx = 0
for i in range(a[0]+1,0,-1):
f = True
for j in range(n):
temp = ceil(a[j]/i)
# if(i == 2):
# print(temp*(i-1),'...')
if(temp*(i-1)>a[j]):
f = False
break
if(f):
mx = i
break
# print(mx)
ans = 0
for i in range(n):
ans+=ceil(a[i]/mx)
print(ans)
# testcase(1)
testcase(int(inp()))
```
| 101,323 |
Provide tags and a correct Python 3 solution for this coding contest problem.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Tags: greedy, implementation, math
Correct Solution:
```
from math import ceil
import sys
input=sys.stdin.readline
from collections import defaultdict as dd
t=int(input())
while t:
n=int(input())
d=dd(int)
l=list(map(int,input().split()))
for i in l:
d[i]+=1
li=[]
for i in d:
li.append(d[i])
mi=10000000000
for m in range(1,min(li)+2):
d=dd(int)
lol=0
for i in li:
if(i%m==0):
d[m]+=i//m
elif(i%m!=0):
ex=m-i%m
if(ex*(m-1)+(ceil(i/m)-ex)*m)!=i or (ceil(i/m)-ex)<0:
lol=1
break
d[m-1]+=ex
d[m]+=(ceil(i/m)-ex)
#if(m==5):
#print(d,ex*(m-1),(ceil(i/m)-ex))
#print(d,lol)
if(lol==0 and d[m]>=0 and d[m-1]>=0):
if(d[m]+d[m-1]<mi):
mi=d[m]+d[m-1]
print(mi)
t-=1
```
| 101,324 |
Provide tags and a correct Python 3 solution for this coding contest problem.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Tags: greedy, implementation, math
Correct Solution:
```
import sys
input = sys.stdin.readline
res = []
t = int(input())
for _ in range(t):
n = int(input())
l = map(int, input().split())
c = [0] * n
for v in l:
c[v-1] += 1
c.sort()
c.reverse()
while c[-1] == 0:
c.pop()
best = n
for i in range(2, c[-1] + 2):
out = 0
for v in c:
smol = (-v) % i
tol = (v + smol)//i - smol
if tol >= 0:
out += smol + tol
else:
out += n
best = min(best, out)
res.append(best)
print('\n'.join(map(str,res)))
```
| 101,325 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Submitted Solution:
```
# ------------------- fast io --------------------
import os
import sys
from io import BytesIO, IOBase
BUFSIZE = 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, BUFSIZE))
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, BUFSIZE))
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")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
# ------------------- fast io --------------------
from collections import defaultdict
for j in range(int(input())):
n=int(input());vals=list(map(int,input().split()))
dict1=defaultdict(int)
for s in range(n):
dict1[vals[s]]+=1
minnum=min(dict1.values())+1;minscreens=2*10**6 +1
for s in range(1,minnum+1):
screens=0;found=True
for i in dict1.values():
mult_1=(-i)%s
mult=(i-mult_1*(s-1))//s
if mult<0:
found=False;break
else:
screens+=mult_1+mult
if found==True:
minscreens=min(minscreens,screens)
print(minscreens)
```
Yes
| 101,326 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Submitted Solution:
```
from collections import defaultdict,Counter
from math import inf,ceil
for _ in range(int(input())):
n=int(input())
ci=list(map(int,input().split()))
freq=Counter(ci)
freqlist=list(freq.values())
minimum=inf
for i in freqlist:
minimum=min(minimum,i)
for screen in range(minimum+1,0,-1):
flag=1
ans=0
for ii in freqlist:
nos=ceil(ii/screen)
val1=(screen-1)*nos
val2=nos*screen
if ii<=val2 and ii>=val1:
ans+=nos
else:
flag=0
ans=0
break
if flag:
print(ans)
break
```
Yes
| 101,327 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Submitted Solution:
```
import sys
import math
from collections import defaultdict
def get_factors(num):
a=set()
for i in range(1,int(math.sqrt(num))+2):
if num%i==0:
a.add(i)
a.add(num//i)
return a
t=int(sys.stdin.readline())
for _ in range(t):
n=int(sys.stdin.readline())
dic=defaultdict(int)
arr=list(map(int,sys.stdin.readline().split()))
for i in range(n):
dic[arr[i]]+=1
'''factors=[]
#print(dic,'dic')
for i in dic:
a=get_factors(dic[i])
b=get_factors(dic[i]+1)
a=a.union(b)
factors.append(a)
m=len(factors)
#print(factors,'factors')
for i in range(1,m):
factors[0]=factors[0].intersection(factors[i])
maxsize=max(factors[0])'''
l=[]
for i in dic:
l.append(dic[i])
l.sort()
x=l[0]
ans=sum(l)
m=len(l)
#print(l,'l')
for i in range(1,x+2):
s,z=0,True
for j in range(m):
a=math.ceil(l[j]/i)
if l[j]<(i-1)*a:
z=False
break
else:
s+=a
if z:
ans=min(ans,s)
print(int(ans))
```
Yes
| 101,328 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Submitted Solution:
```
import sys
import os
from io import BytesIO, IOBase
from types import GeneratorType
from collections import defaultdict
BUFSIZE = 8192
import math
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, BUFSIZE))
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, BUFSIZE))
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")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
t=int(input())
for i in range(t):
#n=int(input())
#b=list(map(int,input().split()))
n=int(input())
c=list(map(int,input().split()))
d=defaultdict(lambda:0)
for j in c:
d[j]+=1
cnt=[]
for j in d:
cnt.append(d[j])
cnt.sort()
mi=cnt[0]+1
ans=float("inf")
i=1
while(i<=mi):
curr=0
poss=1
for j in cnt:
if j>=((i-1)*math.ceil(j/i)):
curr+=math.ceil(j/i)
else:
poss=0
break
if poss:
ans=min(ans,curr)
i+=1
print(ans)
```
Yes
| 101,329 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Submitted Solution:
```
from collections import Counter as CO
t=int(input())
for _ in range(t):
n=int(input())
arr=list(map(int,input().split()))
freq=dict(CO(arr))
l=[]
for k in freq:
l+=[freq[k]]
ls=list(set(l))
fact=[2]
for i in ls:
for j in range(1,int(i**.5)+1):
if(i%j==0):
fact+=[j,i//j]
fact=list(set(fact))
fact.sort(reverse=True)
#print(fact)
#print(ls)
for i in fact:
flag=0
for j in ls:
if((i-j%i)<=j//i+1 or j%i==0 ):
pass
else:
#print(i,j)
flag=1
break
if(flag==0):
ans=i
break
co=0
for j in l:
if(j%ans==0):
co+=j//ans
else:
co+=j//ans+1
print(co)
```
No
| 101,330 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Submitted Solution:
```
from collections import Counter
from math import ceil
t = int(input())
for _ in range(0,t):
n = int(input())
ss = [int(i) for i in input().split()]
c = Counter(ss)
aa = sorted([int(i) for i in c.values()])
maxx = 10**9+7
check = list(set(sorted(aa)))
ll = check[0]
if len(check)>=2 and check[1]-check[0] == 1:
ll = check[1]
for i in range(1,ll+1):
val = True
pos = []
for ii in aa:
if ii%i == 0 or ii%i == i-1:
pos.append(ceil(ii/i))
continue
else:
val = False
break
if val:
maxx = min(maxx,sum(pos))
print(maxx)
```
No
| 101,331 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Submitted Solution:
```
def function(lis):
lis_item = []
lis_item_freq = []
for i in lis:
if i in lis_item:
lis_item_freq[lis_item.index(i)] += 1
else:
lis_item.append(i)
lis_item_freq.append(0)
lis_item_freq[lis_item.index(i)] += 1
least_occured = min(lis_item_freq)
max_possible = least_occured # it can be least_occured - 1
for i in lis_item_freq:
if i%max_possible < max_possible-1 and i%max_possible !=0:
#print("yes")
max_possible -= 1
if max_possible+1 in lis_item_freq:
max_possible += 1
summ = 0
for i in lis_item_freq:
summ += i//max_possible
if i%max_possible!=0:
summ+=1
print(summ)
# print(lis_item)
# print(lis_item_freq)
t = int(input())
for i in range(t):
length = int(input())
value = input().split()
function(value)
```
No
| 101,332 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
BerPhone X is almost ready for release with n applications being preinstalled on the phone. A category of an application characterizes a genre or a theme of this application (like "game", "business", or "education"). The categories are given as integers between 1 and n, inclusive; the i-th application has category c_i.
You can choose m β the number of screens and s β the size of each screen. You need to fit all n icons of the applications (one icon representing one application) meeting the following requirements:
* On each screen, all the icons must belong to applications of the same category (but different screens can contain icons of applications of the same category);
* Each screen must be either completely filled with icons (the number of icons on the screen is equal to s) or almost filled with icons (the number of icons is equal to s-1).
Your task is to find the minimal possible number of screens m.
Input
The first line contains an integer t (1 β€ t β€ 10 000) β the number of test cases in the input. Then t test cases follow.
The first line of each test case contains an integer n (1 β€ n β€ 2β
10^6) β the number of the icons. The second line contains n integers c_1, c_2, ..., c_n (1 β€ c_i β€ n), where c_i is the category of the i-th application.
It is guaranteed that the sum of the values of n for all test cases in the input does not exceed 2β
10^6.
Output
Print t integers β the answers to the given test cases in the order they follow in the input. The answer to a test case is an integer m β the minimum number of screens on which all n icons can be placed satisfying the given requirements.
Example
Input
3
11
1 5 1 5 1 5 1 1 1 1 5
6
1 2 2 2 2 1
5
4 3 3 1 2
Output
3
3
4
Note
In the first test case of the example, all the icons can be placed on three screens of size 4: a screen with 4 icons of the category 1, a screen with 3 icons of the category 1, and a screen with 4 icons of the category 5.
Submitted Solution:
```
from collections import defaultdict as dd
t=int(input())
while t:
n=int(input())
d=dd(int)
l=list(map(int,input().split()))
for i in l:
d[i]+=1
li=[]
for i in d:
li.append(d[i])
mx=1000000000000000000
si=li[0]
ll=[]
mi=10000000000000000000000
for m in range(1,min(li)+1):
cou=dd(int)
lol=0
lo=0
for i in li:
#print(m,i)
if(i==1):
#print("lol")
cou[m]+=1
elif(i%m==0 and m!=1):
cou[m]+=i//m
else:
ex=i%m
if(m==1):
ex=i//2
cou[m+1]+=ex
i-=ex*(m+1)
cou[m]+=i//m
lol=1
else:
if(i//m>=ex):
cou[m+1]+=ex
i-=ex*(m+1)
cou[m]+=i//m
lol=1
else:
cou[m-1]+=m-ex
i-=(m-ex)*(m-1)
cou[m]+=i//m
lo=1
if((cou[m]+cou[m+1]+cou[m-1])<mi and lol!=lo):
mi=(cou[m]+cou[m+1]+cou[m-1])
#print(d,mi)
print(mi)
t-=1
```
No
| 101,333 |
Provide tags and a correct Python 3 solution for this coding contest problem.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Tags: constructive algorithms, implementation
Correct Solution:
```
n = int(input())
*a, = map(int, input().split())
j = max(a)
mn = min(a)
d = {}
for i in a:
d[i] = d.get(i, 0) + 1
s = d[j]
while j > mn:
j -= 1
if d.get(j, 0) < s or not s:
print('NO')
exit()
s *= -1
s += d.get(j, 0)
if s:
print('NO')
else:
print('YES')
```
| 101,334 |
Provide tags and a correct Python 3 solution for this coding contest problem.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Tags: constructive algorithms, implementation
Correct Solution:
```
n= int(input())
a = list(map(int,input().split()))
a.sort()
mm=a[0]
b=list(map(lambda x: x-mm, a))
if b[-1]>n or n%2==1:
print('NO')
else:
c=[0]*(b[-1]+1)
for el in b:
c[el]+=1
for i in range(1,len(c)):
c[i] = c[i]-c[i-1]
c[i-1]=0
#print(c)
if i!=len(c)-1:
if (c[i]==0 or c[i]<0):
print('NO')
break
else:
if c[i]!=0:
print('NO')
break
else:
print('YES')
# Sat Oct 17 2020 10:31:31 GMT+0300 (ΠΠΎΡΠΊΠ²Π°, ΡΡΠ°Π½Π΄Π°ΡΡΠ½ΠΎΠ΅ Π²ΡΠ΅ΠΌΡ)
```
| 101,335 |
Provide tags and a correct Python 3 solution for this coding contest problem.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Tags: constructive algorithms, implementation
Correct Solution:
```
import sys
n = int(input())
c = {}
a = [int(i) for i in input().split()]
maxi = max(a)
for i in a:
c[i] = c.get(i, 0) + 1
l = sorted(c)
t = l[:-1]
for u in t:
if u + 1 not in c:
print("NO")
sys.exit()
c[u + 1] -= c[u]
if 0 > c[u + 1]:
print("NO")
sys.exit()
arr = list(c.values())
if arr.count(0) == 1 and c[maxi] == 0:
print("YES")
else:
print("NO")
# Fri Oct 16 2020 22:20:09 GMT+0300 (ΠΠΎΡΠΊΠ²Π°, ΡΡΠ°Π½Π΄Π°ΡΡΠ½ΠΎΠ΅ Π²ΡΠ΅ΠΌΡ)
```
| 101,336 |
Provide tags and a correct Python 3 solution for this coding contest problem.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Tags: constructive algorithms, implementation
Correct Solution:
```
#https://codeforces.com/problemset/problem/128/D
n = int(input())
a = list(map(int, input().split()))
d = {}
def push(d, x):
if x not in d:
d[x] = 0
d[x] += 1
def sub(d, x):
d[x] -= 1
if d[x] == 0:
del d[x]
for x in a:
push(d, x)
cur = min(list(d.keys()))
sub(d, cur)
ans = [cur]
while True:
if cur+1 in d:
cur+=1
ans.append(cur)
sub(d, cur)
elif cur - 1 in d:
cur-=1
ans.append(cur)
sub(d, cur)
else:
break
if len(d) == 0 and abs(ans[0]-ans[-1]) == 1:
print('YES')
else:
print('NO')
#6
#1 1 2 2 2 3
#6
#2 4 1 1 2 2
```
| 101,337 |
Provide tags and a correct Python 3 solution for this coding contest problem.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Tags: constructive algorithms, implementation
Correct Solution:
```
n=int(input())
g={}
for i in list(map(int,input().split())):g[i]=g.get(i,0)+2
mx=max(g)
for i in sorted(g)[:-1]:
if i+1 not in g:exit(print('NO'))
g[i+1]-=g[i]
if g[i+1]<0:exit(print('NO'))
print('YES'if g[mx]==0 and list(g.values()).count(0)==1else'NO')
# Wed Oct 14 2020 14:40:06 GMT+0300 (ΠΠΎΡΠΊΠ²Π°, ΡΡΠ°Π½Π΄Π°ΡΡΠ½ΠΎΠ΅ Π²ΡΠ΅ΠΌΡ)
```
| 101,338 |
Provide tags and a correct Python 3 solution for this coding contest problem.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Tags: constructive algorithms, implementation
Correct Solution:
```
from collections import Counter
import string
import bisect
#import random
import math
import sys
# sys.setrecursionlimit(10**6)
from fractions import Fraction
def array_int():
return [int(i) for i in sys.stdin.readline().split()]
def vary(arrber_of_variables):
if arrber_of_variables==1:
return int(sys.stdin.readline())
if arrber_of_variables>=2:
return map(int,sys.stdin.readline().split())
def makedict(var):
return dict(Counter(var))
testcases=1
for _ in range(testcases):
n=vary(1)
num=array_int()
ct=makedict(num)
mini=min(num)
ct[mini]-=1
ans=1
while 1:
if ans==n:
break
if ct.get(mini+1,0)>0:
ct[mini+1]-=1
ans+=1
# print('hello',mini)
mini+=1
elif ct.get(mini-1,0)>0:
ct[mini-1]-=1
ans+=1
# print(mini)
mini-=1
else:
break
# print(mini)
# print(ans,mini)
if ans==n and mini==min(num)+1:
print('YES')
else:
print('NO')
```
| 101,339 |
Provide tags and a correct Python 3 solution for this coding contest problem.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Tags: constructive algorithms, implementation
Correct Solution:
```
n=int(input())
g={}
for i in list(map(int,input().split())):g[i]=g.get(i,0)+2
mx=max(g)
for i in sorted(g)[:-1]:
if i+1 not in g:exit(print('NO'))
g[i+1]-=g[i]
if g[i+1]<0:exit(print('NO'))
print('YES'if g[mx]==0 and list(g.values()).count(0)==1else'NO')
```
| 101,340 |
Provide tags and a correct Python 3 solution for this coding contest problem.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Tags: constructive algorithms, implementation
Correct Solution:
```
n=int(input())
g={}
for i in list(map(int,input().split())):g[i]=g.get(i,0)+1
mx=max(g)
for i in sorted(g)[:-1]:
if i+1 not in g:exit(print('NO'))
g[i+1]-=g[i]
if g[i+1]<0:exit(print('NO'))
print('YES'if g[mx]==0 and list(g.values()).count(0)==1else'NO')
```
| 101,341 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Submitted Solution:
```
def no():
print("NO")
exit(0)
n, a = int(input()), list(map(int, input().split()))
x, y = min(a), max(a)
d = y-x
if 2*d > n:
print("NO")
exit(0)
c = [0] * (d+1)
for i in range(n):
c[a[i]-x] += 1
for i in range(1, d):
c[i] -= c[i-1]
if c[i] <= 0:
no()
if c[d] == c[d-1]:
print("YES")
else:
print("NO")
```
Yes
| 101,342 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Submitted Solution:
```
import sys
n = int(input())
c = {}
for i in [int(i) for i in input().split()]:
c[i] = c.get(i, 0) + 1
maxi = max(c)
l = sorted(c)
for u in l[:-1]:
if u + 1 not in c:
print("NO")
sys.exit()
c[u + 1] -= c[u]
if 0 > c[u + 1]:
print("NO")
sys.exit()
arr = list(c.values())
if arr.count(0) == 1 and c[maxi] == 0:
print("YES")
else:
print("NO")
```
Yes
| 101,343 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Submitted Solution:
```
n=int(input())
g={}
for i in list(map(int,input().split())):g[i]=g.get(i,0)+2
mx=max(g)
for i in sorted(g)[:-1]:
if i+1 not in g:exit(print('NO'))
g[i+1]-=g[i]
print('YES'if g[mx]==0 and list(g.values()).count(0)==1else'NO')
```
No
| 101,344 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Submitted Solution:
```
n = int(input())
a = [2 * int(x) for x in input().split()]
mp = {}
for i in a:
mp.setdefault(i - 1, len(mp))
mp.setdefault(i + 1, len(mp))
m = len(mp)
deg = [0 for i in range(m)]
adj = [set() for i in range(m)]
for i in a:
deg[mp[i - 1]] += 1
deg[mp[i + 1]] += 1
adj[mp[i - 1]].add(mp[i + 1])
adj[mp[i + 1]].add(mp[i - 1])
for i in deg:
if i % 2:
print("NO")
exit()
vis = set()
cur = mp[a[0] - 1]
while not cur in vis:
vis.add(cur)
for i in adj[cur]:
if not i in vis:
cur = i
break
if len(vis) == m:
print("YES")
else:
print("NO")
```
No
| 101,345 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Submitted Solution:
```
n=int(input())
g={}
for i in list(map(int,input().split())):g[i]=g.get(i,0)+2
mx=max(g)
for i in sorted(g)[:-1]:
if i+1 not in g:exit(print('NO'))
g[i+1]-=g[i]
print('NO'if g[mx]else'YES')
```
No
| 101,346 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
One day Anna got the following task at school: to arrange several numbers in a circle so that any two neighboring numbers differs exactly by 1. Anna was given several numbers and arranged them in a circle to fulfill the task. Then she wanted to check if she had arranged the numbers correctly, but at this point her younger sister Maria came and shuffled all numbers. Anna got sick with anger but what's done is done and the results of her work had been destroyed. But please tell Anna: could she have hypothetically completed the task using all those given numbers?
Input
The first line contains an integer n β how many numbers Anna had (3 β€ n β€ 105). The next line contains those numbers, separated by a space. All numbers are integers and belong to the range from 1 to 109.
Output
Print the single line "YES" (without the quotes), if Anna could have completed the task correctly using all those numbers (using all of them is necessary). If Anna couldn't have fulfilled the task, no matter how hard she would try, print "NO" (without the quotes).
Examples
Input
4
1 2 3 2
Output
YES
Input
6
1 1 2 2 2 3
Output
YES
Input
6
2 4 1 1 2 2
Output
NO
Submitted Solution:
```
n= int(input())
a = list(map(int,input().split()))
a.sort()
mm=a[0]
b=list(map(lambda x: x-mm, a))
if a[-1]>n:
print('NO')
else:
c=[0]*a[-1]
for el in b:
c[el]+=1
for i in range(1,len(c)):
c[i] = c[i]-c[i-1]
c[i-1]=0
for i in range(len(c)):
if c[i]!=0:
print('NO')
break
else:
print('YES')
# Sat Oct 17 2020 10:17:26 GMT+0300 (ΠΠΎΡΠΊΠ²Π°, ΡΡΠ°Π½Π΄Π°ΡΡΠ½ΠΎΠ΅ Π²ΡΠ΅ΠΌΡ)
```
No
| 101,347 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Tags: combinatorics, constructive algorithms, math, matrices
Correct Solution:
```
from math import ceil,log
mod=998244353
n,m,l,r=map(int,input().split())
if (n*m)%2==1:
ans=pow(r-l+1,n*m,mod)
else:
if (r-l+1)%2==0:
ans=(pow(r-l+1,n*m,mod)*pow(2,mod-2,mod))%mod
else:
ans=((pow(r-l+1,n*m,mod)+mod+1)*pow(2,mod-2,mod))%mod
print(ans)
```
| 101,348 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Tags: combinatorics, constructive algorithms, math, matrices
Correct Solution:
```
import sys
readline = sys.stdin.readline
def mat_dot(A, B, mod):
assert len(A[0]) == len(B), 'invalid_size'
L = len(A)
M = len(A[0])
N = len(B[0])
res = [[0]*N for _ in range(L)]
for i in range(L):
for j in range(N):
a = 0
for k in range(M):
a = (a+A[i][k]*B[k][j]) % mod
res[i][j] = a
return res
def mat_pow(A, x, mod):
N = len(A)
res = [[0]*N for _ in range(N)]
for i in range(N):
res[i][i] = 1
for i in range(x.bit_length()):
if 2**i & x:
res = mat_dot(res, A, mod)
A = mat_dot(A, A, mod)
return res
MOD = 998244353
N, M, L, R = map(int, readline().split())
R -= L
if N&1 and M&1:
ans = pow(R+1, N*M, MOD)
else:
Bl = N*M//2
even = (R+1)//2
odd = R+1 - even
Mat = [[even, odd], [odd, even]]
xy = mat_dot(mat_pow(Mat, Bl, MOD), [[1], [0]], MOD)
x, y = xy[0][0], xy[1][0]
ans = (x*x + y*y) % MOD
print(ans)
```
| 101,349 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Tags: combinatorics, constructive algorithms, math, matrices
Correct Solution:
```
mod = 998244353
n,m,l,r = map(int, input().split())
if(n*m % 2 == 1):
print(pow(r-l+1,n*m,mod))
elif((r-l+1)%2 == 1):
print(((pow(r-l+1,n*m,mod)+1)*(mod+1)//2)%mod)
else:
print(((pow(r-l+1,n*m,mod))*(mod+1)//2)%mod)
```
| 101,350 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Tags: combinatorics, constructive algorithms, math, matrices
Correct Solution:
```
import io
import os
from collections import Counter, defaultdict, deque
DEBUG = False
def modInverse(a, p):
# Fermat's little theorem, a**(p-1) = 1 mod p
return pow(a, p - 2, p)
def solve(N, M, L, R):
MOD = 998244353
num = R - L + 1
numCells = N * M
# Answer doesn't have to lie with L and R so can always add 2 so that only parity matters
# Once it's a 0 and 1 matrix, adding to two adjacent is equivalent to a swap, so order doesn't matter
# If the number of odd cells is even or number of even cells is even, flip their parity so the matrix is all odd or all even
if numCells % 2 == 1:
# Anything works since one of the parity must be even count
return pow(num, numCells, MOD)
# If num cells is even, can't have odd even cells and odd odd cells. Can only have even even cell and even odd cells
# Want to choose `2 * i` odd cells within numCells
# Once parity is fixed the number of choices is numOdds^(2 * i) * numEvens^(numCells - 2 * i)
# Plug into wolfram alpha:
# numCells = 2 * K
# \sum_{i=0}^{K} binomial(2 * K, 2 * i) * X^(2 * i) * Y^(2 * K - 2 * i)
K = numCells // 2
X = num // 2 # number of values within range [L, R] that are odd
Y = num // 2 # ditto for even
if num % 2 != 0:
if L % 2 == 0:
X += 1
else:
Y += 1
assert numCells % 2 == 0
ans = (
(pow(X + Y, numCells, MOD) + pow(X - Y, numCells, MOD)) * modInverse(2, MOD)
) % MOD
if DEBUG:
def nCr(n, r):
def fact(i):
if i == 0:
return 1
return i * fact(i - 1)
return fact(n) // (fact(n - r) * fact(r))
brute = 0
for i in range(numCells // 2 + 1):
brute += nCr(numCells, 2 * i) * pow(X, 2 * i) * pow(Y, numCells - 2 * i)
print(brute % MOD, ans)
assert brute % MOD == ans
return ans
if DEBUG:
for n in range(1, 5):
for m in range(1, 5):
for l in range(1, 5):
for r in range(l, 5):
if n * m < 2:
continue
solve(n, m, l, r)
if __name__ == "__main__":
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline
N, M, L, R = [int(x) for x in input().split()]
ans = solve(N, M, L, R)
print(ans)
```
| 101,351 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Tags: combinatorics, constructive algorithms, math, matrices
Correct Solution:
```
from sys import stdin
input = stdin.readline
def main():
mod = 998244353
n,m,l,r = map(int,input().split())
nm = n * m
def mult(m1,m2):
ans = [0] * 2
ans[0] = (m1[0] * m2[0] + m1[1] * m2[1])%mod
ans[1] = (m1[0] * m2[1] + m1[1] * m2[0])%mod
return ans
def power(number, n):
res = number
while(n):
if n & 1:
res = mult(res,number)
number = mult(number,number)
n >>= 1
return res
if nm % 2 == 0:
num = r-l+1
ar = [num//2,num//2]
if num % 2 == 1:
ar[(l)%2] += 1
print(power(ar,nm-1)[0])
else:
num = r-l+1
ar = [num//2,num//2]
if num % 2 == 1:
ar[(l)%2] += 1
ans = power(ar,nm-1)
print(sum(ans) % mod)
main()
```
| 101,352 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Tags: combinatorics, constructive algorithms, math, matrices
Correct Solution:
```
def expo(a,n):
#print(a)
#print(n)
if n>0:
ha = expo(a,n//2)
if n%2==1:
return (ha*ha*a)%MOD
else:
return (ha*ha)%MOD
else:
return 1
MOD = 998244353
def f():
n,m,L,R = map(int,input().split(" "))
height = R-L+1
area = n*m
ans = expo(height,area)
#print(ans)
if(area%2==1):
print(ans)
else:
if(height%2==0):
if ans%2==1:
ans+=MOD
ans//=2
print(ans%MOD)
else:
if ans%2==0:
ans+=MOD
ans = (ans+1)//2
print(ans%MOD)
f()
```
| 101,353 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Tags: combinatorics, constructive algorithms, math, matrices
Correct Solution:
```
MOD = 998244353
n,m,l,r = map(int,input().split())
# print(n,m,l,r)
if n * m % 2 == 1:
print(pow(r - l + 1,n*m,MOD))
else:
e = r//2 - (l - 1)//2
o = (r - l + 1) - e
# inv_2 = pow(2,MOD-2)
print(((pow(e+o,n*m,MOD) + pow(e - o,n*m,MOD))*((MOD + 1)//2))%MOD)
```
| 101,354 |
Provide tags and a correct Python 3 solution for this coding contest problem.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Tags: combinatorics, constructive algorithms, math, matrices
Correct Solution:
```
n,m,L,R = map(int,input().strip().split())
if(n%2 ==1 and m%2 ==1):
print(pow((R-L+1),n*m, 998244353))
else:
po = (R-L+1)//2
poo = (R - L + 2) // 2
c = ((pow(po+poo,n*m,998244353) + pow(po-poo,n*m,998244353))*(pow(2,998244351,998244353)))%998244353
print(c)
```
| 101,355 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Submitted Solution:
```
n, m, l, r = [int(i) for i in input().split()]
mod = 998244353
ans = 0
if (n*m)&1:
ans = pow(r-l+1, m*n, mod)
else:
if l&1 != r&1:
k = 0
elif l&1:
k = 1
else:
k = 1
ans = ((pow(r-l+1, m*n, mod) + k) * pow(2, mod-2, mod))%mod
print(ans)
```
Yes
| 101,356 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Submitted Solution:
```
MOD = 998244353
n, m, l, r = map(int, input().split())
if n * m % 2 == 1:
print(pow(r - l + 1, n * m, MOD))
elif (r - l + 1) % 2 == 0:
print(pow(r - l + 1, n * m, MOD) * (MOD + 1) // 2 % MOD)
else:
print((pow(r - l + 1, n * m, MOD) + 1) * (MOD + 1) // 2 % MOD)
```
Yes
| 101,357 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Submitted Solution:
```
# -*- coding:utf-8 -*-
"""
created by shuangquan.huang at 2020/7/2
"""
import collections
import time
import os
import sys
import bisect
import heapq
from typing import List
MOD = 998244353
if __name__ == '__main__':
n, m, l, r = map(int, input().split())
if n * m % 2 == 1:
print(pow(r - l + 1, n * m, MOD))
else:
e = r // 2 - (l - 1) // 2
o = (r - l + 1) - e
print((pow(e + o, n * m, MOD) + pow(e - o, n * m, MOD)) * (MOD + 1) // 2 % MOD)
```
Yes
| 101,358 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Submitted Solution:
```
N, M, L, R = map(int, input().split())
K = R - L + 1
P = 998244353
print(pow(K, N * M, P) if N * M % 2 else (pow(K, N * M, P) + K % 2) * ((P + 1) // 2) % P)
```
Yes
| 101,359 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Submitted Solution:
```
def pow_modulo(a, b, p):
result = 1
a_pow2 = a % p
while b:
if b & 1:
result = (result * a_pow2) % p
b >>= 1
a_pow2 = (a_pow2 * a_pow2) % p
return result
def num_solvable(n, m):
if (m * n) % 2 == 1:
return pow_modulo(2, m * n, 998_244_353)
else:
return pow_modulo(2, m * n - 1, 998_244_353)
n, m, h_min, h_max = map(int, input().split(' '))
if h_min == h_max:
print(1)
exit(0)
result = num_solvable(n, m)
if h_max - h_min >= 2:
result *= pow_modulo((h_max - h_min) // 2, m * n, 998_244_353)
print(result % 998_244_353)
```
No
| 101,360 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Submitted Solution:
```
# https://www.geeksforgeeks.org/modular-division/
import math
import sys
input = sys.stdin.readline
def modInverse(b,m):
g = math.gcd(b, m)
if (g != 1):
# print("Inverse doesn't exist")
return -1
else:
# If b and m are relatively prime,
# then modulo inverse is b^(m-2) mode m
return pow(b, m - 2, m)
# Function to compute a/b under modulo m
def modDivide(a,b,m):
a = a % m
inv = modInverse(b,m)
if(inv == -1):
return -1
else:
return (inv*a)%m
# print("Result of Division is ",(inv*a) % m)
p=998244353
# p=79
n,m,L,R=map(int,input().split())
x=pow(R-L+1,n*m,p)
# print(x)
# y=modDivide(x,2,p)
if (L-R+1)%2==0:
y=modDivide(x,2,p)
print(y%p)
else:
y=modDivide(x+1,2,p)
print(y%p)
```
No
| 101,361 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Submitted Solution:
```
def pow_fast_mod(a, b, div):
s = 1
while b > 0:
if b % 2:
s %= div
a %= div
s *= a
a %= div
a *= a
b = (b // 2)
return s % div
[n, m, L, R] = [int(s) for s in input().split()]
h = R-L
mod = 998244353
ret = 0
if n%2 and m%2:
ret = pow_fast_mod(h+1, n*m, mod)
else:
x = pow_fast_mod(h+1, n*m/2, mod)
a = x // 2
a %= mod
b = (x+1) // 2
b %= mod
ret = a*a+b*b
print(int(ret%mod))
```
No
| 101,362 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Alice has got addicted to a game called Sirtet recently.
In Sirtet, player is given an n Γ m grid. Initially a_{i,j} cubes are stacked up in the cell (i,j). Two cells are called adjacent if they share a side. Player can perform the following operations:
* stack up one cube in two adjacent cells;
* stack up two cubes in one cell.
Cubes mentioned above are identical in height.
Here is an illustration of the game. States on the right are obtained by performing one of the above operations on the state on the left, and grey cubes are added due to the operation.
<image>
Player's goal is to make the height of all cells the same (i.e. so that each cell has the same number of cubes in it) using above operations.
Alice, however, has found out that on some starting grids she may never reach the goal no matter what strategy she uses. Thus, she is wondering the number of initial grids such that
* L β€ a_{i,j} β€ R for all 1 β€ i β€ n, 1 β€ j β€ m;
* player can reach the goal using above operations.
Please help Alice with it. Notice that the answer might be large, please output the desired value modulo 998,244,353.
Input
The only line contains four integers n, m, L and R (1β€ n,m,L,R β€ 10^9, L β€ R, n β
m β₯ 2).
Output
Output one integer, representing the desired answer modulo 998,244,353.
Examples
Input
2 2 1 1
Output
1
Input
1 2 1 2
Output
2
Note
In the first sample, the only initial grid that satisfies the requirements is a_{1,1}=a_{2,1}=a_{1,2}=a_{2,2}=1. Thus the answer should be 1.
In the second sample, initial grids that satisfy the requirements are a_{1,1}=a_{1,2}=1 and a_{1,1}=a_{1,2}=2. Thus the answer should be 2.
Submitted Solution:
```
MOD = 998244353
n, m, l, r = map(int, input().split())
a = (r - l + 1) // 2
b = a
if (r - l + 1) % 2 == 1:
b += 1
x = n * m
if n * m % 2 == 1:
ans = pow(a + b, x, MOD)
else:
ans = pow(2, x - 1, MOD)
if 1 or a == b:
ans *= pow(b, x, MOD)
ans %= MOD
# else:
print(ans)
```
No
| 101,363 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Tags: constructive algorithms, greedy, math
Correct Solution:
```
t = int(input())
for _ in range(t):
s = input()
n = int(s.split(' ')[0])
m = int(s.split(' ')[1])
if n < 2:
print(0)
continue
elif n == 2:
print(m)
else:
print(2*m)
```
| 101,364 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Tags: constructive algorithms, greedy, math
Correct Solution:
```
t=int(input())
for _ in range(t):
n,m=map(int,input().split())
if n>=3:
print(2*m)
elif n==1:
print(0)
else:
print(m)
```
| 101,365 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Tags: constructive algorithms, greedy, math
Correct Solution:
```
t=int(input())
for z in range(t):
n,m=map(int,input().split())
if n==1:
print(0)
elif n==2:
print(m)
else:
print(2*m)
```
| 101,366 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Tags: constructive algorithms, greedy, math
Correct Solution:
```
for i in range(int(input())):
n, m = [int(i) for i in input().split()]
if n <= 1:
print(0)
elif n == 2:
print(m)
else:
print(2*m)
```
| 101,367 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Tags: constructive algorithms, greedy, math
Correct Solution:
```
t=int(input())
for _ in range(t):
n,m=[int(n) for n in input().split()]
if n==1:
print("0")
elif n==2:
print(m)
else:
print(m*2)
```
| 101,368 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Tags: constructive algorithms, greedy, math
Correct Solution:
```
T = int(input())
while T != 0:
T -= 1
n, m = map(int,input().split())
if n == 1:
print('0')
elif n == 2:
print(m)
else:
print(m*2)
```
| 101,369 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Tags: constructive algorithms, greedy, math
Correct Solution:
```
import sys
input = sys.stdin.readline
def main():
for _ in range(int(input())):
n, m = map(int, input().split())
if n == 1:
print(0)
elif n == 2:
print(m)
else:
print(m * 2)
if __name__ == '__main__':
main()
```
| 101,370 |
Provide tags and a correct Python 3 solution for this coding contest problem.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Tags: constructive algorithms, greedy, math
Correct Solution:
```
class SegmentTree:
def __init__(self, data, default=0, func=lambda a,b:gcd(a,b)):
"""initialize the segment tree with data"""
self._default = default
self._func = func
self._len = len(data)
self._size = _size = 1 << (self._len - 1).bit_length()
self.data = [default] * (2 * _size)
self.data[_size:_size + self._len] = data
for i in reversed(range(_size)):
self.data[i] = func(self.data[i + i], self.data[i + i + 1])
def __delitem__(self, idx):
self[idx] = self._default
def __getitem__(self, idx):
return self.data[idx + self._size]
def __setitem__(self, idx, value):
idx += self._size
self.data[idx] = value
idx >>= 1
while idx:
self.data[idx] = self._func(self.data[2 * idx], self.data[2 * idx + 1])
idx >>= 1
def __len__(self):
return self._len
def query(self, start, stop):
if start == stop:
return self.__getitem__(start)
stop += 1
start += self._size
stop += self._size
res = self._default
while start < stop:
if start & 1:
res = self._func(res, self.data[start])
start += 1
if stop & 1:
stop -= 1
res = self._func(res, self.data[stop])
start >>= 1
stop >>= 1
return res
def __repr__(self):
return "SegmentTree({0})".format(self.data)
# ------------------- fast io --------------------
import os
import sys
from io import BytesIO, IOBase
BUFSIZE = 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, BUFSIZE))
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, BUFSIZE))
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")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
# ------------------- fast io --------------------
from bisect import bisect_right, bisect_left
from fractions import Fraction
def pre(s):
n = len(s)
pi=[0]*n
for i in range(1,n):
j = pi[i-1]
while j and s[i] != s[j]:
j = pi[j-1]
if s[i] == s[j]:
j += 1
pi[i] = j
return pi
def prod(a):
ans = 1
for each in a:
ans = (ans * each)
return ans
from math import gcd
def lcm(a,b):return a*b//gcd(a,b)
def binary(x, length=16):
y = bin(x)[2:]
return y if len(y) >= length else "0"*(length - len(y)) + y
for _ in range(int(input())):
#n = int(input())
n, k = map(int, input().split())
#a, b = map(int, input().split())
#x, y = map(int, input().split())
#a = list(map(int, input().split()))
#s = input()
#print("YES" if s else "NO")
a = []
if n == 1:
print(0)
continue
if n == 2:
print(k)
else:
print(2*k)
```
| 101,371 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Submitted Solution:
```
T = int(input())
for _ in range(T):
n,m = map(int,input().split())
if n==1:
print(0)
continue
elif n==2:
print(m)
continue
else:
print(2*m)
continue
```
Yes
| 101,372 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Submitted Solution:
```
for _ in range(int(input())):
# n = int(input())
m,n = [int(s) for s in input().split()]
# for i in range(len(arr)):
if m==1:
print(0)
elif m==2:
print(n)
else:
print(2*n)
```
Yes
| 101,373 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Submitted Solution:
```
from sys import stdin
input = stdin.readline
def main():
test = int(input())
for t in range(test):
# n = int(input())
l = [int(i) for i in input().split(" ")]
#
n = l[0]
m = l[1]
#
# l = []
# l = [int(i) for i in input().split(" ")]
# for i in l:
# print(i, end=' ')
if n == 1:
print(0)
elif n == 2:
print(m)
else:
print(2 * m)
main()
```
Yes
| 101,374 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Submitted Solution:
```
t = int(input())
for _ in range(t):
n, m = map(int, input().split())
print(min(n-1, 2)*m)
```
Yes
| 101,375 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Submitted Solution:
```
for t in range(int(input())):
n, m = map(int, input().split())
if n == 1:
print(0)
elif n == 2:
print(m)
elif n == m:
print(n * 2)
elif n % 2 == 0:
print(m * 2 - (m // n * 2))
else:
print(m * 2)
```
No
| 101,376 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Submitted Solution:
```
for ad in range(int(input())):
n,m=list(map(int,input().split()))
if n==1:
print(0)
elif n==2:
print(m)
else:
print(2*n)
```
No
| 101,377 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Submitted Solution:
```
"""
arr = list(map(int, input().split()))
n,k=map(int, input().split())
"""
import math
import sys
input = sys.stdin.readline
############ ---- Input Functions ---- ############
def inp():
return(int(input()))
def inlt():
return(list(map(int,input().split())))
def insr():
s = input()
return(list(s[:len(s) - 1]))
def invr():
return(map(int,input().split()))
test_cases = inp()
for _ in range(test_cases):
length, sum = map(int, input().split())
if length == 1:
print(0)
elif length == 2 or length == 3:
print(sum)
elif length == 4:
print(int(1.5 * sum))
else:
print(int(2 * sum))
# #Calculates intersection of two lists
# seq1 = inlt()
# seq2 = inlt()
# seq1.sort()
# seq2.sort()
# i = 0
# j = 0
# while i <len(seq1) and j < len(seq2):
# if seq1[i] < seq2[j]:
# i += 1
# elif seq2[j] < seq1[i]:
# j += 1
# else:
# print(seq1[i])
# i += 1
# j += 1
# #Calculates sum of largest subsequence in array
# arr = inlt()
# sum = 0
# best = 0
# for i in range(len(arr)):
# sum = max(arr[i], arr[i] + sum)
# best = max(best, sum)
# print(best)
# #Find longest increasing subsequence
# arr = inlt()
# length = [0] * len(arr)
# for i in range(len(arr)):
# length[i] = 1
# for j in range(i):
# if arr[j] < arr[i]:
# length[i] = max(length[i], length[j] + 1)
# print(max(length))
```
No
| 101,378 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
You are given two integers n and m. You have to construct the array a of length n consisting of non-negative integers (i.e. integers greater than or equal to zero) such that the sum of elements of this array is exactly m and the value β_{i=1}^{n-1} |a_i - a_{i+1}| is the maximum possible. Recall that |x| is the absolute value of x.
In other words, you have to maximize the sum of absolute differences between adjacent (consecutive) elements. For example, if the array a=[1, 3, 2, 5, 5, 0] then the value above for this array is |1-3| + |3-2| + |2-5| + |5-5| + |5-0| = 2 + 1 + 3 + 0 + 5 = 11. Note that this example doesn't show the optimal answer but it shows how the required value for some array is calculated.
You have to answer t independent test cases.
Input
The first line of the input contains one integer t (1 β€ t β€ 10^4) β the number of test cases. Then t test cases follow.
The only line of the test case contains two integers n and m (1 β€ n, m β€ 10^9) β the length of the array and its sum correspondingly.
Output
For each test case, print the answer β the maximum possible value of β_{i=1}^{n-1} |a_i - a_{i+1}| for the array a consisting of n non-negative integers with the sum m.
Example
Input
5
1 100
2 2
5 5
2 1000000000
1000000000 1000000000
Output
0
2
10
1000000000
2000000000
Note
In the first test case of the example, the only possible array is [100] and the answer is obviously 0.
In the second test case of the example, one of the possible arrays is [2, 0] and the answer is |2-0| = 2.
In the third test case of the example, one of the possible arrays is [0, 2, 0, 3, 0] and the answer is |0-2| + |2-0| + |0-3| + |3-0| = 10.
Submitted Solution:
```
ttt = int(input())
for tt in range(1, ttt + 1):
parts = input().strip().split(" ")
n, m = int(parts[0]), int(parts[1])
ans = m
if n > 2: ans = 2 * m
print("Case #{}: {}".format(tt, ans))
```
No
| 101,379 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ray lost his array and needs to find it by asking Omkar. Omkar is willing to disclose that the array has the following qualities:
1. The array has n (1 β€ n β€ 2 β
10^5) elements.
2. Every element in the array a_i is an integer in the range 1 β€ a_i β€ 10^9.
3. The array is sorted in nondecreasing order.
Ray is allowed to send Omkar a series of queries. A query consists of two integers, l and r such that 1 β€ l β€ r β€ n. Omkar will respond with two integers, x and f. x is the mode of the subarray from index l to index r inclusive. The mode of an array is defined by the number that appears the most frequently. If there are multiple numbers that appear the most number of times, the smallest such number is considered to be the mode. f is the amount of times that x appears in the queried subarray.
The array has k (1 β€ k β€ min(25000,n)) distinct elements. However, due to Ray's sins, Omkar will not tell Ray what k is. Ray is allowed to send at most 4k queries.
Help Ray find his lost array.
Input
The only line of the input contains a single integer n (1 β€ n β€ 2 β
10^5), which equals to the length of the array that you are trying to find.
Interaction
The interaction starts with reading n.
Then you can make one type of query:
* "? \enspace l \enspace r" (1 β€ l β€ r β€ n) where l and r are the bounds of the subarray that you wish to query.
The answer to each query will be in the form "x \enspace f" where x is the mode of the subarray and f is the number of times x appears in the subarray.
* x satisfies (1 β€ x β€ 10^9).
* f satisfies (1 β€ f β€ r-l+1).
* If you make more than 4k queries or violate the number range in the query, you will get an output "-1."
* If you terminate after receiving the response "-1", you will get the "Wrong answer" verdict. Otherwise you can get an arbitrary verdict because your solution will continue to read from a closed stream.
To output your answer, print:
* "! \enspace a_1 \enspace a_2 \enspace β¦ \enspace a_{n-1} \enspace a_n" which is an exclamation point followed by the array with a space between every element.
And quit after that. This query is not counted towards the 4k queries limit.
After printing a query do not forget to output end of line and flush the output. Otherwise, you will get Idleness limit exceeded. To do this, use:
* fflush(stdout) or cout.flush() in C++;
* System.out.flush() in Java;
* flush(output) in Pascal;
* stdout.flush() in Python;
* see documentation for other languages.
Hack Format
To hack, output 1 integer on the first line, n (1 β€ n β€ 2 β
10^5). On the second line output n integers a_1, a_2, β¦, a_{n-1}, a_n separated by a space such that there are at most 25000 distinct numbers and a_j β€ a_{j+1} for all j from 1 to n-1.
Example
Input
6
2 2
2 2
3 2
2 1
Output
? 1 6
? 1 3
? 4 6
? 3 4
! 1 2 2 3 3 4
Note
The first query is l=1 and r=6. The mode is 2, and 2 appears 2 times, so x=2 and f=2. Note that 3 also appears two times, but 2 is outputted because 2 is smaller.
The second query is l=1 and r=3. The mode is 2 and 2 appears twice in the subarray with indices [1,3].
The third query is l=4 and r=6. The mode is 3 and 3 appears twice in the subarray with indices [4,6].
The fourth query is l=3 and r=4. The mode is 2, which appears once in the subarray with indices [3,4]. Note that 3 also appears once in that range, but 2 is smaller than 3.
Submitted Solution:
```
print("?, 1 , 6")
```
No
| 101,380 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ray lost his array and needs to find it by asking Omkar. Omkar is willing to disclose that the array has the following qualities:
1. The array has n (1 β€ n β€ 2 β
10^5) elements.
2. Every element in the array a_i is an integer in the range 1 β€ a_i β€ 10^9.
3. The array is sorted in nondecreasing order.
Ray is allowed to send Omkar a series of queries. A query consists of two integers, l and r such that 1 β€ l β€ r β€ n. Omkar will respond with two integers, x and f. x is the mode of the subarray from index l to index r inclusive. The mode of an array is defined by the number that appears the most frequently. If there are multiple numbers that appear the most number of times, the smallest such number is considered to be the mode. f is the amount of times that x appears in the queried subarray.
The array has k (1 β€ k β€ min(25000,n)) distinct elements. However, due to Ray's sins, Omkar will not tell Ray what k is. Ray is allowed to send at most 4k queries.
Help Ray find his lost array.
Input
The only line of the input contains a single integer n (1 β€ n β€ 2 β
10^5), which equals to the length of the array that you are trying to find.
Interaction
The interaction starts with reading n.
Then you can make one type of query:
* "? \enspace l \enspace r" (1 β€ l β€ r β€ n) where l and r are the bounds of the subarray that you wish to query.
The answer to each query will be in the form "x \enspace f" where x is the mode of the subarray and f is the number of times x appears in the subarray.
* x satisfies (1 β€ x β€ 10^9).
* f satisfies (1 β€ f β€ r-l+1).
* If you make more than 4k queries or violate the number range in the query, you will get an output "-1."
* If you terminate after receiving the response "-1", you will get the "Wrong answer" verdict. Otherwise you can get an arbitrary verdict because your solution will continue to read from a closed stream.
To output your answer, print:
* "! \enspace a_1 \enspace a_2 \enspace β¦ \enspace a_{n-1} \enspace a_n" which is an exclamation point followed by the array with a space between every element.
And quit after that. This query is not counted towards the 4k queries limit.
After printing a query do not forget to output end of line and flush the output. Otherwise, you will get Idleness limit exceeded. To do this, use:
* fflush(stdout) or cout.flush() in C++;
* System.out.flush() in Java;
* flush(output) in Pascal;
* stdout.flush() in Python;
* see documentation for other languages.
Hack Format
To hack, output 1 integer on the first line, n (1 β€ n β€ 2 β
10^5). On the second line output n integers a_1, a_2, β¦, a_{n-1}, a_n separated by a space such that there are at most 25000 distinct numbers and a_j β€ a_{j+1} for all j from 1 to n-1.
Example
Input
6
2 2
2 2
3 2
2 1
Output
? 1 6
? 1 3
? 4 6
? 3 4
! 1 2 2 3 3 4
Note
The first query is l=1 and r=6. The mode is 2, and 2 appears 2 times, so x=2 and f=2. Note that 3 also appears two times, but 2 is outputted because 2 is smaller.
The second query is l=1 and r=3. The mode is 2 and 2 appears twice in the subarray with indices [1,3].
The third query is l=4 and r=6. The mode is 3 and 3 appears twice in the subarray with indices [4,6].
The fourth query is l=3 and r=4. The mode is 2, which appears once in the subarray with indices [3,4]. Note that 3 also appears once in that range, but 2 is smaller than 3.
Submitted Solution:
```
import sys
n = int(input())
res = [0] * (n + 1)
def helper(l, r, x = 0, f = 0):
if x == 0:
print('?', l, r)
sys.stdout.flush()
x, f = [int(i) for i in input().split()]
if f == 1:
res[l] = x
for i in range(l + 1, r + 1):
print('?', i, i)
sys.stdout.flush()
a, b = [int(i) for i in input().split()]
res[i] = a
return 1
if f == r - l + 1:
for i in range(l, r + 1):
res[i] = x
return f
cnt = helper(l, (r + l) // 2)
nstart = (r + l) // 2 + 1
if res[(r + l) // 2] == x:
remain = f - cnt
while remain > 0 and nstart <= r :
res[nstart] = x
nstart += 1
remain -= 1
if nstart <= r:
return helper(nstart, r)
return f
return helper(nstart, r, x, f)
helper(1, n)
print('!', end=' ')
for i in res[1:]:
print(i, end=' ')
print()
```
No
| 101,381 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ray lost his array and needs to find it by asking Omkar. Omkar is willing to disclose that the array has the following qualities:
1. The array has n (1 β€ n β€ 2 β
10^5) elements.
2. Every element in the array a_i is an integer in the range 1 β€ a_i β€ 10^9.
3. The array is sorted in nondecreasing order.
Ray is allowed to send Omkar a series of queries. A query consists of two integers, l and r such that 1 β€ l β€ r β€ n. Omkar will respond with two integers, x and f. x is the mode of the subarray from index l to index r inclusive. The mode of an array is defined by the number that appears the most frequently. If there are multiple numbers that appear the most number of times, the smallest such number is considered to be the mode. f is the amount of times that x appears in the queried subarray.
The array has k (1 β€ k β€ min(25000,n)) distinct elements. However, due to Ray's sins, Omkar will not tell Ray what k is. Ray is allowed to send at most 4k queries.
Help Ray find his lost array.
Input
The only line of the input contains a single integer n (1 β€ n β€ 2 β
10^5), which equals to the length of the array that you are trying to find.
Interaction
The interaction starts with reading n.
Then you can make one type of query:
* "? \enspace l \enspace r" (1 β€ l β€ r β€ n) where l and r are the bounds of the subarray that you wish to query.
The answer to each query will be in the form "x \enspace f" where x is the mode of the subarray and f is the number of times x appears in the subarray.
* x satisfies (1 β€ x β€ 10^9).
* f satisfies (1 β€ f β€ r-l+1).
* If you make more than 4k queries or violate the number range in the query, you will get an output "-1."
* If you terminate after receiving the response "-1", you will get the "Wrong answer" verdict. Otherwise you can get an arbitrary verdict because your solution will continue to read from a closed stream.
To output your answer, print:
* "! \enspace a_1 \enspace a_2 \enspace β¦ \enspace a_{n-1} \enspace a_n" which is an exclamation point followed by the array with a space between every element.
And quit after that. This query is not counted towards the 4k queries limit.
After printing a query do not forget to output end of line and flush the output. Otherwise, you will get Idleness limit exceeded. To do this, use:
* fflush(stdout) or cout.flush() in C++;
* System.out.flush() in Java;
* flush(output) in Pascal;
* stdout.flush() in Python;
* see documentation for other languages.
Hack Format
To hack, output 1 integer on the first line, n (1 β€ n β€ 2 β
10^5). On the second line output n integers a_1, a_2, β¦, a_{n-1}, a_n separated by a space such that there are at most 25000 distinct numbers and a_j β€ a_{j+1} for all j from 1 to n-1.
Example
Input
6
2 2
2 2
3 2
2 1
Output
? 1 6
? 1 3
? 4 6
? 3 4
! 1 2 2 3 3 4
Note
The first query is l=1 and r=6. The mode is 2, and 2 appears 2 times, so x=2 and f=2. Note that 3 also appears two times, but 2 is outputted because 2 is smaller.
The second query is l=1 and r=3. The mode is 2 and 2 appears twice in the subarray with indices [1,3].
The third query is l=4 and r=6. The mode is 3 and 3 appears twice in the subarray with indices [4,6].
The fourth query is l=3 and r=4. The mode is 2, which appears once in the subarray with indices [3,4]. Note that 3 also appears once in that range, but 2 is smaller than 3.
Submitted Solution:
```
T=int(input())
arr=[]
for _ in range(T):
print('?',_+1,_+1)
a=input().split(' ')
arr.append(a[0])
print('! '+' '.join(arr))
```
No
| 101,382 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
Ray lost his array and needs to find it by asking Omkar. Omkar is willing to disclose that the array has the following qualities:
1. The array has n (1 β€ n β€ 2 β
10^5) elements.
2. Every element in the array a_i is an integer in the range 1 β€ a_i β€ 10^9.
3. The array is sorted in nondecreasing order.
Ray is allowed to send Omkar a series of queries. A query consists of two integers, l and r such that 1 β€ l β€ r β€ n. Omkar will respond with two integers, x and f. x is the mode of the subarray from index l to index r inclusive. The mode of an array is defined by the number that appears the most frequently. If there are multiple numbers that appear the most number of times, the smallest such number is considered to be the mode. f is the amount of times that x appears in the queried subarray.
The array has k (1 β€ k β€ min(25000,n)) distinct elements. However, due to Ray's sins, Omkar will not tell Ray what k is. Ray is allowed to send at most 4k queries.
Help Ray find his lost array.
Input
The only line of the input contains a single integer n (1 β€ n β€ 2 β
10^5), which equals to the length of the array that you are trying to find.
Interaction
The interaction starts with reading n.
Then you can make one type of query:
* "? \enspace l \enspace r" (1 β€ l β€ r β€ n) where l and r are the bounds of the subarray that you wish to query.
The answer to each query will be in the form "x \enspace f" where x is the mode of the subarray and f is the number of times x appears in the subarray.
* x satisfies (1 β€ x β€ 10^9).
* f satisfies (1 β€ f β€ r-l+1).
* If you make more than 4k queries or violate the number range in the query, you will get an output "-1."
* If you terminate after receiving the response "-1", you will get the "Wrong answer" verdict. Otherwise you can get an arbitrary verdict because your solution will continue to read from a closed stream.
To output your answer, print:
* "! \enspace a_1 \enspace a_2 \enspace β¦ \enspace a_{n-1} \enspace a_n" which is an exclamation point followed by the array with a space between every element.
And quit after that. This query is not counted towards the 4k queries limit.
After printing a query do not forget to output end of line and flush the output. Otherwise, you will get Idleness limit exceeded. To do this, use:
* fflush(stdout) or cout.flush() in C++;
* System.out.flush() in Java;
* flush(output) in Pascal;
* stdout.flush() in Python;
* see documentation for other languages.
Hack Format
To hack, output 1 integer on the first line, n (1 β€ n β€ 2 β
10^5). On the second line output n integers a_1, a_2, β¦, a_{n-1}, a_n separated by a space such that there are at most 25000 distinct numbers and a_j β€ a_{j+1} for all j from 1 to n-1.
Example
Input
6
2 2
2 2
3 2
2 1
Output
? 1 6
? 1 3
? 4 6
? 3 4
! 1 2 2 3 3 4
Note
The first query is l=1 and r=6. The mode is 2, and 2 appears 2 times, so x=2 and f=2. Note that 3 also appears two times, but 2 is outputted because 2 is smaller.
The second query is l=1 and r=3. The mode is 2 and 2 appears twice in the subarray with indices [1,3].
The third query is l=4 and r=6. The mode is 3 and 3 appears twice in the subarray with indices [4,6].
The fourth query is l=3 and r=4. The mode is 2, which appears once in the subarray with indices [3,4]. Note that 3 also appears once in that range, but 2 is smaller than 3.
Submitted Solution:
```
import sys
n = int(input())
res = [0] * (n + 1)
def helper(l, r, x = 0, f = 0):
print('? ' + l + ' ' + r)
sys.stdout.flush()
if x == 0:
x, f = [int(i) for i in input().split()]
if f == 1:
res[l] = x
for i in range(l + 1, r + 1):
print('? ' + i + ' ' + i)
sys.stdout.flush()
a, b = [int(i) for i in input().split()]
res[i] = a
return 1
if f == r - l + 1:
for i in range(l, r + 1):
res[i] = x
return f
cnt = helper(l, (r + l) // 2)
nstart = (r + l) // 2 + 1
if res[(r + l) // 2] == x:
remain = f - cnt
while remain > 0:
res[nstart] = x
remain -= 1
return helper(nstart, r)
return helper(nstart, r, x, f)
```
No
| 101,383 |
Provide tags and a correct Python 3 solution for this coding contest problem.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Tags: implementation, sortings, strings
Correct Solution:
```
a = input
if(sorted(a()+a())==sorted(a())):
print("YES")
else:
print("NO")
```
| 101,384 |
Provide tags and a correct Python 3 solution for this coding contest problem.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Tags: implementation, sortings, strings
Correct Solution:
```
a=list(input())
b=list(input())
s=a+b
s.sort()
s=' '.join(s)
f=list(input())
f.sort()
f=' '.join(f)
if s==f:
print('YES')
else :
print('NO')
```
| 101,385 |
Provide tags and a correct Python 3 solution for this coding contest problem.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Tags: implementation, sortings, strings
Correct Solution:
```
g=list(str(i) for i in input())
h=list(str(i) for i in input())
m=list(str(i) for i in input())
g.extend(h)
m.sort()
g.sort()
if m==g:
print("YES")
else:
print("NO")
```
| 101,386 |
Provide tags and a correct Python 3 solution for this coding contest problem.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Tags: implementation, sortings, strings
Correct Solution:
```
s=input()+input();t=input()
print('YES' if sorted(s)==sorted(t) else 'NO')
```
| 101,387 |
Provide tags and a correct Python 3 solution for this coding contest problem.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Tags: implementation, sortings, strings
Correct Solution:
```
am=input()
bm=input()
cm=input()
a=[]
b=[]
c=[]
for i in range(len(am)):
a.append(am[i])
for i in range(len(bm)):
b.append(bm[i])
for i in range(len(cm)):
c.append(cm[i])
bo=0
for i in range(len(a)):
if a[i] in c:
c.remove(a[i])
else:
bo=1
for i in range(len(b)):
if b[i] in c:
c.remove(b[i])
else:
bo=1
if bo==0 and len(am)+len(bm)==len(cm):
print('YES')
else:
print('NO')
```
| 101,388 |
Provide tags and a correct Python 3 solution for this coding contest problem.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Tags: implementation, sortings, strings
Correct Solution:
```
q=input()
w=input()
e=input()
r=q+w
if len(e)!=len(w+q):
print('NO')
else:
for i in set(r):
if r.count(i)!=e.count(i):
print('NO')
break
else:
print('YES')
```
| 101,389 |
Provide tags and a correct Python 3 solution for this coding contest problem.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Tags: implementation, sortings, strings
Correct Solution:
```
n = input()
m = input()
b = input()
n1 = n
m1 = m
b1 =b
k=0
for i in range(len(n)):
for j in range(len(b)):
if n[i]==b[j]:
n1 = b[j]
b = b.replace(n[i], "0", 1)
n = n.replace(n1,"0",1)
for i in range(len(m)):
for j in range(len(b)):
if m[i]==b[j]:
m1 = b[j]
b = b.replace(m[i], "0", 1)
m = m.replace(m1,"0",1)
l=0
h=0
for i in b:
if i == "0":
k+=1
for i in m:
if i == "0":
l+=1
for i in n:
if i == "0":
h+=1
if k==len(b) and l==len(m) and h==len(n):
print("YES")
else:
print("NO")
```
| 101,390 |
Provide tags and a correct Python 3 solution for this coding contest problem.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Tags: implementation, sortings, strings
Correct Solution:
```
guest = input()
host = input()
initialname = guest + host
pile = input()
error = 0
if len(pile)!=len(initialname):
print('NO')
else:
s = set(initialname)
for x in s:
if initialname.count(x) != pile.count(x):
print('NO')
break
else:
print('YES')
```
| 101,391 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Submitted Solution:
```
a = input()
b = input()
c = input()
for i in a:
if(i in c):
c = c.replace(i, "", 1)
continue
else:
print("NO")
quit()
for i in b:
if(i in c):
c = c.replace(i, "", 1)
continue
else:
print("NO")
quit()
if(len(c) == 0):
print("YES")
quit()
else:
print("NO")
quit()
```
Yes
| 101,392 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Submitted Solution:
```
def main():
guest = input()
host = input()
mixed = sorted(input())
joined = sorted(host + guest)
flag = True
for i in joined:
if i in mixed:
mixed.remove(i)
else:
print("NO")
flag = False
break
if not mixed and flag:
print("YES")
elif mixed and flag:
print("NO")
main()
```
Yes
| 101,393 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Submitted Solution:
```
a = input() + input()
b = input()
print('YES' if sorted(a) == sorted(b) else 'NO')
```
Yes
| 101,394 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Submitted Solution:
```
a = input()
b = input()
a = a + b
c = list(input())
flag = 0
for i in a:
if i in c:
c.remove(i)
else:
print("NO")
flag = 1
break
if c != [] and flag == 0:
print("NO")
flag = 1
if flag == 0:
print("YES")
```
Yes
| 101,395 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Submitted Solution:
```
i = input()
j = input()
k = input()
b =[]
for x in k:
b.append(x)
a = i+j
if len(a) > len(b):
print('NO')
else:
for y in range (len(a)):
if a[y] in b:
b.remove(a[y])
if len(b) ==0:
print('YES')
else:
print('NO')
```
No
| 101,396 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Submitted Solution:
```
name1 = input()
name1 = list(name1)
name2 = input()
name2 = list(name2)
name3 = input()
for i in name3:
if i in name1:
name1.remove(i)
elif i in name2:
name2.remove(i)
else:
print('NO')
break
else:
print('YES')
```
No
| 101,397 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Submitted Solution:
```
a = input()
b = input()
c = input()
for i in c:
if (i not in a and i not in b) or (c.count(i) != a.count(i) + b.count(i)):
print("NO")
break
else:
print("YES")
```
No
| 101,398 |
Evaluate the correctness of the submitted Python 3 solution to the coding contest problem. Provide a "Yes" or "No" response.
So, the New Year holidays are over. Santa Claus and his colleagues can take a rest and have guests at last. When two "New Year and Christmas Men" meet, thear assistants cut out of cardboard the letters from the guest's name and the host's name in honor of this event. Then the hung the letters above the main entrance. One night, when everyone went to bed, someone took all the letters of our characters' names. Then he may have shuffled the letters and put them in one pile in front of the door.
The next morning it was impossible to find the culprit who had made the disorder. But everybody wondered whether it is possible to restore the names of the host and his guests from the letters lying at the door? That is, we need to verify that there are no extra letters, and that nobody will need to cut more letters.
Help the "New Year and Christmas Men" and their friends to cope with this problem. You are given both inscriptions that hung over the front door the previous night, and a pile of letters that were found at the front door next morning.
Input
The input file consists of three lines: the first line contains the guest's name, the second line contains the name of the residence host and the third line contains letters in a pile that were found at the door in the morning. All lines are not empty and contain only uppercase Latin letters. The length of each line does not exceed 100.
Output
Print "YES" without the quotes, if the letters in the pile could be permuted to make the names of the "New Year and Christmas Men". Otherwise, print "NO" without the quotes.
Examples
Input
SANTACLAUS
DEDMOROZ
SANTAMOROZDEDCLAUS
Output
YES
Input
PAPAINOEL
JOULUPUKKI
JOULNAPAOILELUPUKKI
Output
NO
Input
BABBONATALE
FATHERCHRISTMAS
BABCHRISTMASBONATALLEFATHER
Output
NO
Note
In the first sample the letters written in the last line can be used to write the names and there won't be any extra letters left.
In the second sample letter "P" is missing from the pile and there's an extra letter "L".
In the third sample there's an extra letter "L".
Submitted Solution:
```
n1 = str(input())
n2 = str(input())
n3 = str(input())
c1=int(0)
c2=int(0)
c3=int(0)
if n1=="AC" and n2=="A" and n3=="BBA":
print("NO")
else:
for i in range(len(n1)):
c1=int(c1)+ord(n1[i])
for i in range(len(n2)):
c2=int(c2)+ord(n2[i])
for i in range(len(n3)):
c3=int(c3)+ord(n3[i])
if c1+c2==c3:
print("YES")
else:
print("NO")
```
No
| 101,399 |
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