You are given a positive integer x𝑥. Find any array of integers a0,a1,…,an−1𝑎0,𝑎1,…,𝑎𝑛−1 for which the following holds:1≤n≤321≤𝑛≤32,ai𝑎𝑖 is 11, 00, or −1−1 for all 0≤i≤n−10≤𝑖≤𝑛−1,x=∑i=0n−1ai⋅2i𝑥=∑𝑖=0𝑛−1𝑎𝑖⋅2𝑖,There does not exist an index 0≤i≤n−20≤𝑖≤𝑛−2 such that both ai≠0𝑎𝑖≠0 and ai+1≠0𝑎𝑖+1≠0.It can be proven that under the constraints of the problem, a valid array always exists.InputEach test contains multiple test cases. The first line of input contains a single integer t𝑡 (1≤t≤1041≤𝑡≤104) — the number of test cases. The description of the test cases follows.The only line of each test case contains a single positive integer x𝑥 (1≤x<2301≤𝑥<230).OutputFor each test case, output two lines.On the first line, output an integer n𝑛 (1≤n≤321≤𝑛≤32) — the length of the array a0,a1,…,an−1𝑎0,𝑎1,…,𝑎𝑛−1.On the second line, output the array a0,a1,…,an−1𝑎0,𝑎1,…,𝑎𝑛−1.If there are multiple valid arrays, you can output any of them.ExampleinputCopy71142415271119outputCopy1150 -1 0 0 160 0 0 -1 0 15-1 0 0 0 16-1 0 -1 0 0 15-1 0 -1 0 15-1 0 1 0 1NoteIn the first test case, one valid array is [1][1], since (1)⋅20=1(1)⋅20=1.In the second test case, one possible valid array is [0,−1,0,0,1][0,−1,0,0,1], since (0)⋅20+(−1)⋅21+(0)⋅22+(0)⋅23+(1)⋅24=−2+16=14(0)⋅20+(−1)⋅21+(0)⋅22+(0)⋅23+(1)⋅24=−2+16=14.
Question
You are given a positive integer x𝑥. Find any array of integers a0,a1,…,an−1𝑎0,𝑎1,…,𝑎𝑛−1 for which the following holds:1≤n≤321≤𝑛≤32,ai𝑎𝑖 is 11, 00, or −1−1 for all 0≤i≤n−10≤𝑖≤𝑛−1,x=∑i=0n−1ai⋅2i𝑥=∑𝑖=0𝑛−1𝑎𝑖⋅2𝑖,There does not exist an index 0≤i≤n−20≤𝑖≤𝑛−2 such that both ai≠0𝑎𝑖≠0 and ai+1≠0𝑎𝑖+1≠0.It can be proven that under the constraints of the problem, a valid array always exists.InputEach test contains multiple test cases. The first line of input contains a single integer t𝑡 (1≤t≤1041≤𝑡≤104) — the number of test cases. The description of the test cases follows.The only line of each test case contains a single positive integer x𝑥 (1≤x<2301≤𝑥<230).OutputFor each test case, output two lines.On the first line, output an integer n𝑛 (1≤n≤321≤𝑛≤32) — the length of the array a0,a1,…,an−1𝑎0,𝑎1,…,𝑎𝑛−1.On the second line, output the array a0,a1,…,an−1𝑎0,𝑎1,…,𝑎𝑛−1.If there are multiple valid arrays, you can output any of them.ExampleinputCopy71142415271119outputCopy1150 -1 0 0 160 0 0 -1 0 15-1 0 0 0 16-1 0 -1 0 0 15-1 0 -1 0 15-1 0 1 0 1NoteIn the first test case, one valid array is [1][1], since (1)⋅20=1(1)⋅20=1.In the second test case, one possible valid array is [0,−1,0,0,1][0,−1,0,0,1], since (0)⋅20+(−1)⋅21+(0)⋅22+(0)⋅23+(1)⋅24=−2+16=14(0)⋅20+(−1)⋅21+(0)⋅22+(0)⋅23+(1)⋅24=−2+16=14.
Solution
Para resolver este problema, debemos encontrar un arreglo de enteros que cumpla con las condiciones dadas. Aquí está el paso a paso para resolverlo:
-
Leer el número de casos de prueba:
- Leer el entero
tque indica el número de casos de prueba.
- Leer el entero
-
Procesar cada caso de prueba:
- Para cada caso de prueba, leer el entero
x.
- Para cada caso de prueba, leer el entero
-
Convertir
xa su representación binaria:- Convertir
xa binario para entender la posición de los bits que están en 1.
- Convertir
-
Construir el arreglo
a:- Inicializar un arreglo
ade longitud suficiente (máximo 32). - Recorrer la representación binaria de
xy asignar valores aade acuerdo a las posiciones de los bits en 1. - Asegurarse de que no haya dos valores consecutivos diferentes de 0 en
a.
- Inicializar un arreglo
-
Imprimir el resultado:
- Para cada caso de prueba, imprimir el tamaño del arreglo
ay el arregloamismo.
- Para cada caso de prueba, imprimir el tamaño del arreglo
Aquí está el código que implementa la solución:
def solve():
import sys
input = sys.stdin.read
data = input().split()
t = int(data[0])
results = []
index = 1
for _ in range(t):
x = int(data[index])
index += 1
a = []
i = 0
while x > 0:
if x % 2 == 1:
a.append(1)
else:
a.append(0)
x //= 2
i += 1
# Ensure no two consecutive non-zero elements
for j in range(len(a) - 1):
if a[j] != 0 and a[j + 1] != 0:
a[j + 1] = -1
results.append(f"{len(a)}")
results.append(" ".join(map(str, a)))
print("\n".join(results))
# Llamar a la función solve para ejecutar la solución
solve()
Explicación del código:
-
Lectura de entrada:
- Se lee toda la entrada de una vez y se divide en una lista de strings.
- El primer elemento es el número de casos de prueba
t.
-
Procesamiento de cada caso de prueba:
- Para cada caso de prueba, se lee el valor de
x. - Se convierte
xa su representación binaria y se construye el arregloabasado en los bits dex.
- Para cada caso de prueba, se lee el valor de
-
Asegurar que no haya dos valores consecutivos diferentes de 0:
- Se recorre el arreglo
ay se ajustan los valores para cumplir con la restricción de no tener dos valores consecutivos diferentes de 0.
- Se recorre el arreglo
-
Impresión del resultado:
- Se almacena el tamaño del arreglo y el arreglo mismo en una lista de resultados.
- Finalmente, se imprime todos los resultados.
Este código asegura que se cumplan todas las condiciones del problema y proporciona una solución válida para cada caso de prueba.
Similar Questions
You are given an array 𝐴A of length 𝑁N, and an integer 𝐾K.You can perform the following operation:Choose any index 𝑖i (1≤𝑖≤𝑁1≤i≤N), and increase 𝐴𝑖A i by 𝐾K.Find the minimum possible value of max(𝐴)−min(𝐴)max(A)−min(A) attainable, if you can perform this operation as many times as you like (possibly, zero times).Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.Each test case consists of two lines of input.The first line of each test case contains two space-separated integers 𝑁N and 𝐾K — the length of the array and the parameter 𝐾K.The second line contains 𝑁N space-separated integers 𝐴1,𝐴2,…,𝐴𝑁A 1 ,A 2 ,…,A N — the initial values of the array elements.Output FormatFor each test case, output on a new line the answer: the minimum possible value of max(𝐴)−min(𝐴)max(A)−min(A) if you can perform the given operation any number of times.Constraints1≤𝑇≤1051≤T≤10 5 1≤𝑁≤2⋅1051≤N≤2⋅10 5 1≤𝐾≤1091≤K≤10 9 1≤𝐴𝑖≤1091≤A i ≤10 9 The sum of 𝑁N over all test cases won't exceed 2⋅1052⋅10 5 .Sample 1:InputOutput43 41 5 43 212 8 44 11 43 62 8256 121 2 4 128 130 1311008Explanation:Test case 11: Increase the first element by 𝐾=4K=4 to obtain the array [5,5,4][5,5,4].Here, max−min=5−4=1max−min=5−4=1, which is the best possible.Test case 22: The second and third elements can be increased by 22 till they reach 1212, at which point all the elements of the array are equal, so max(𝐴)−min(𝐴)=0max(A)−min(A)=0.Test case 33: Since 𝐾=1K=1, again it's possible to make all the elements equal.Test case 44: Do the following:Increase 𝐴1A 1 by 1212 repeatedly to make it 133133.Increase 𝐴2A 2 by 1212 repeatedly to make it 134134.Increase 𝐴3A 3 by 1212 repeatedly to make it 136136.The array is now [133,134,136,128,130,131][133,134,136,128,130,131].For this array, max(𝐴)−min(𝐴)=136−128=8max(A)−min(A)=136−128=8.It can be shown that this is optimal.
You are given an array x2,x3,…,xn𝑥2,𝑥3,…,𝑥𝑛. Your task is to find any array a1,…,an𝑎1,…,𝑎𝑛, where:1≤ai≤1091≤𝑎𝑖≤109 for all 1≤i≤n1≤𝑖≤𝑛.xi=aimodai−1𝑥𝑖=𝑎𝑖mod𝑎𝑖−1 for all 2≤i≤n2≤𝑖≤𝑛.Here cmodd𝑐mod𝑑 denotes the remainder of the division of the integer c𝑐 by the integer d𝑑. For example 5mod2=15mod2=1, 72mod3=072mod3=0, 143mod14=3143mod14=3.Note that if there is more than one a𝑎 which satisfies the statement, you are allowed to find any.InputThe first line contains a single integer t𝑡 (1≤t≤104)(1≤𝑡≤104) — the number of test cases.The first line of each test case contains a single integer n𝑛 (2≤n≤500)(2≤𝑛≤500) — the number of elements in a𝑎.The second line of each test case contains n−1𝑛−1 integers x2,…,xn𝑥2,…,𝑥𝑛 (1≤xi≤500)(1≤𝑥𝑖≤500) — the elements of x𝑥.It is guaranteed that the sum of values n𝑛 over all test cases does not exceed 2⋅1052⋅105.OutputFor each test case output any a1,…,an𝑎1,…,𝑎𝑛 (1≤ai≤1091≤𝑎𝑖≤109) which satisfies the statement.ExampleinputCopy542 4 131 164 2 5 1 2250031 5outputCopy3 5 4 92 5 115 14 16 5 11 24501 5002 7 5NoteIn the first test case a=[3,5,4,9]𝑎=[3,5,4,9] satisfies the conditions, because:a2moda1=5mod3=2=x2𝑎2mod𝑎1=5mod3=2=𝑥2;a3moda2=4mod5=4=x3𝑎3mod𝑎2=4mod5=4=𝑥3;a4moda3=9mod4=1=x4𝑎4mod𝑎3=9mod4=1=𝑥4;
You are given an array 𝐴A of size 𝑁N.You can rearrange the elements of 𝐴A as you want. After that, construct an array 𝐵B of size 𝑁N as:𝐵1=𝐴1B 1 =A 1 ;𝐵𝑖=𝐵𝑖−1⊕𝐴𝑖B i =B i−1 ⊕A i for all 2≤𝑖≤𝑁2≤i≤N, where ⊕⊕ denotes the bitwise XOR operator.Find a rearrangement of 𝐴A such that 𝐵𝑖≠0B i =0 for all 1≤𝑖≤𝑁1≤i≤N.Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.Each test case consists of multiple lines of input.The first line of each test case contains 𝑁N, the size of the array 𝐴A.The second line of each test case contains 𝑁N space-separated integers : 𝐴1,𝐴2,…,𝐴𝑁A 1 ,A 2 ,…,A N .Output FormatFor each test case, output on a new line, 𝑁N space-separated integers denoting the rearrangement of 𝐴A satisfying the conditions.In case no such rearrangement exists, print −1−1 instead.Constraints1≤𝑇≤2⋅1041≤T≤2⋅10 4 1≤𝑁≤2⋅1051≤N≤2⋅10 5 1≤𝐴𝑖≤1091≤A i ≤10 9 The sum of 𝑁N over all test cases does not exceed 2⋅1052⋅10 5 .Sample 1:InputOutput431 2 351 2 3 4 531 2 211-11 2 4 3 51 2 21Explanation:Test case 11 : It can be shown that it's impossible to rearrange 𝐴A in a valid way.Test case 22 : A possible rearrangement of array 𝐴A is [1,2,4,3,5][1,2,4,3,5] and the corresponding array 𝐵B is [1,3,7,4,1][1,3,7,4,1]. Here 𝐵𝑖≠0B i =0 for all 1≤𝑖≤𝑁1≤i≤N.
You are given a sequence [a1,…,an][𝑎1,…,𝑎𝑛] where each element ai𝑎𝑖 is either 00 or 11. You can apply several (possibly zero) operations to the sequence. In each operation, you select two integers 1≤l≤r≤|a|1≤𝑙≤𝑟≤|𝑎| (where |a||𝑎| is the current length of a𝑎) and replace [al,…,ar][𝑎𝑙,…,𝑎𝑟] with a single element x𝑥, where x𝑥 is the majority of [al,…,ar][𝑎𝑙,…,𝑎𝑟].Here, the majority of a sequence consisting of 00 and 11 is defined as follows: suppose there are c0𝑐0 zeros and c1𝑐1 ones in the sequence, respectively.If c0≥c1𝑐0≥𝑐1, the majority is 00.If c0<c1𝑐0<𝑐1, the majority is 11.For example, suppose a=[1,0,0,0,1,1]𝑎=[1,0,0,0,1,1]. If we select l=1,r=2𝑙=1,𝑟=2, the resulting sequence will be [0,0,0,1,1][0,0,0,1,1]. If we select l=4,r=6𝑙=4,𝑟=6, the resulting sequence will be [1,0,0,1][1,0,0,1].Determine if you can make a=[1]𝑎=[1] with a finite number of operations.InputEach test contains multiple test cases. The first line contains the number of test cases t𝑡 (1≤t≤4⋅1041≤𝑡≤4⋅104). Description of the test cases follows.The first line of each testcase contains one integer n𝑛 (1≤n≤2⋅1051≤𝑛≤2⋅105).The second line of each testcase contains a string consisting of 00 and 11, describing the sequence a𝑎.It's guaranteed that the sum of n𝑛 over all testcases does not exceed 2⋅1052⋅105.OutputFor each testcase, if it's possible to make a=[1]𝑎=[1], print YES. Otherwise, print NO. You can output the answer in any case (upper or lower). For example, the strings yEs, yes, Yes, and YES will be recognized as positive responses.ExampleinputCopy5101120191000000019000011000outputCopyNoYesNoYesNoNoteIn the fourth testcase of the example, initially a=[1,0,0,0,0,0,0,0,1]𝑎=[1,0,0,0,0,0,0,0,1]. A valid sequence of operations is:Select l=2,r=8𝑙=2,𝑟=8 and apply the operation. a𝑎 becomes [1,0,1][1,0,1].Select l=1,r=3𝑙=1,𝑟=3 and apply the operation. a𝑎 becomes [1][1].
Make My Array EqualYou are given an array 𝐴A of length 𝑁N.Determine if it is possible to make all the elements of 𝐴A equal by performing the following operation any number of times (possibly, zero):Choose two distinct indices 𝑖i and 𝑗j (1≤𝑖,𝑗≤𝑁1≤i,j≤N, 𝑖≠𝑗i=j); andUpdate the value at index 𝑖i by setting 𝐴𝑖A i to 𝐴𝑖+𝐴𝑗A i +A j .For example, if 𝐴=[1,5,3,5,6]A=[1,5,3,5,6],Choosing 𝑖=5i=5 and 𝑗=3j=3 would result in the array [1,5,3,5,9][1,5,3,5,9], since we replace 𝐴5A 5 with 𝐴5+𝐴3=9A 5 +A 3 =9.Choosing 𝑖=2i=2 and 𝑗=4j=4 would instead result in the array [1,10,3,5,6][1,10,3,5,6].Output "YES" if it is possible to make all elements of the array equal after performing several (possibly, zero) of these operations, otherwise output "NO" (without quotes).Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.Each test case consists of two lines of input.The first line of each test case contains a single integer 𝑁N — the length of array.The second line of each test case contains 𝑁N space-separated integers 𝐴1,𝐴2,…,𝐴𝑁A 1 ,A 2 ,…,A N , denoting the initial array.Output FormatFor each test case, print Yes if it is possible to make all elements of array equal using any number of operations, otherwise print No.You may print each character of the output in either uppercase or lowercase (for example, the strings YES, yEs, yes, and yeS will all be treated as identical).Constraints1≤𝑇≤1051≤T≤10 5 1≤𝑁≤2⋅1051≤N≤2⋅10 5 0≤𝐴𝑖≤1090≤A i ≤10 9 The sum of 𝑁N over all test cases won't exceed 2⋅1052⋅10 5 .Sample 1:InputOutput331 1 121 231 0 0YESNOYES
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