This problem is a dynamic programming problem. Here are the steps to solve it: 1. Sort the array in ascending order. This is because we want to maximize the sum, and multiplication yields a higher result than addition when the numbers involved are greater than 2. 2. Initialize a variable S to 0. This will keep track of the maximum possible sum. 3. Iterate over the array. For each element, if it is 1 or 2, add it to S. If it is greater than 2, multiply it with S if S is not 0, otherwise add it to S. This is because multiplying by 0 yields 0, which is not what we want. 4. After iterating over the array, S will contain the maximum possible sum. However, the problem asks for the sum modulo 10^9+7, so compute S modulo 10^9+7 and return this value. Here is a Python solution for the problem: ```python def max_sum(arr): arr.sort() S = 0 mod = 10**9+7 for i in arr: if i <= 2 or S < 2: S += i else: S *= i S %= mod return S ``` This function takes as input a list of integers and returns the maximum possible sum modulo 10^9+7. It first sorts the list in ascending order, then iterates over the list, updating S according to the rules described above. Finally, it returns S modulo 10^9+7.

You are given an array 𝐴A containing 𝑁N integers.Consider the following process:Let 𝑆=0S=0 initially.For each 𝑖i from 11 to 𝑁N in order, update 𝑆S to either (𝑆+𝐴𝑖)(S+A i ) or (𝑆×𝐴𝑖)(S×A i ).That is, either add 𝐴𝑖A i to 𝑆S or multiply 𝑆S by 𝐴𝑖A i .Before performing the process, you're allowed to freely rearrange the elements of 𝐴A as you like.If you choose the rearrangement of 𝐴A and the sequence of operations optimally, what's the maximum possible value of 𝑆S that you can obtain?This maximum value can be very large, so print it modulo 109+710 9 +7.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 number of elements in the array.The second line contains 𝑁N space-separated integers 𝐴1,𝐴2,…,𝐴𝑁A 1 ,A 2 ,…,A N - the elements of the array.Output FormatFor each test case, output on a new line the maximum possible value of 𝑆S, modulo 109+710 9 +7.Constraints1≤𝑇≤1031≤T≤10 3 1≤𝑁≤2⋅1051≤N≤2⋅10 5 1≤𝐴𝑖≤1091≤A i ≤10 9 The sum of 𝑁N over all test cases won't exceed 2⋅1052⋅10 5 .Sample 1:InputOutput244 2 5 231 2 1804Explanation:Test case 11: Choose the rearrangement 𝐴=[2,2,5,4]A=[2,2,5,4]. Then,Add 𝐴1=2A 1 =2 to 𝑆S. Now, 𝑆=2S=2.Add 𝐴2=2A 2 =2 to 𝑆S. Now, 𝑆=4S=4.Multiply 𝑆S by 𝐴3=5A 3 =5. Now, 𝑆=20S=20.Multiply 𝑆S by 𝐴4=4A 4 =4. Now, 𝑆=80S=80.This is the maximum value that can be obtained.Test case 22: Choose any rearrangement and sum up all the numbers to get 1+1+2=41+1+2=4.This is the maximum value that can be obtained.

Question

You are given an array 𝐴A containing 𝑁N integers.Consider the following process:Let 𝑆=0S=0 initially.For each 𝑖i from 11 to 𝑁N in order, update 𝑆S to either (𝑆+𝐴𝑖)(S+A i ) or (𝑆×𝐴𝑖)(S×A i ).That is, either add 𝐴𝑖A i to 𝑆S or multiply 𝑆S by 𝐴𝑖A i .Before performing the process, you're allowed to freely rearrange the elements of 𝐴A as you like.If you choose the rearrangement of 𝐴A and the sequence of operations optimally, what's the maximum possible value of 𝑆S that you can obtain?This maximum value can be very large, so print it modulo 109+710 9 +7.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 number of elements in the array.The second line contains 𝑁N space-separated integers 𝐴1,𝐴2,…,𝐴𝑁A 1 ,A 2 ,…,A N - the elements of the array.Output FormatFor each test case, output on a new line the maximum possible value of 𝑆S, modulo 109+710 9 +7.Constraints1≤𝑇≤1031≤T≤10 3 1≤𝑁≤2⋅1051≤N≤2⋅10 5 1≤𝐴𝑖≤1091≤A i ≤10 9 The sum of 𝑁N over all test cases won't exceed 2⋅1052⋅10 5 .Sample 1:InputOutput244 2 5 231 2 1804Explanation:Test case 11: Choose the rearrangement 𝐴=[2,2,5,4]A=[2,2,5,4]. Then,Add 𝐴1=2A 1 =2 to 𝑆S. Now, 𝑆=2S=2.Add 𝐴2=2A 2 =2 to 𝑆S. Now, 𝑆=4S=4.Multiply 𝑆S by 𝐴3=5A 3 =5. Now, 𝑆=20S=20.Multiply 𝑆S by 𝐴4=4A 4 =4. Now, 𝑆=80S=80.This is the maximum value that can be obtained.Test case 22: Choose any rearrangement and sum up all the numbers to get 1+1+2=41+1+2=4.This is the maximum value that can be obtained.

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Solution

This problem is a dynamic programming problem. Here are the steps to solve it:

Sort the array in ascending order. This is because we want to maximize the sum, and multiplication yields a higher result than addition when the numbers involved are greater than 2.
Initialize a variable S to 0. This will keep track of the maximum possible sum.
Iterate over the array. For each element, if it is 1 or 2, add it to S. If it is greater than 2, multiply it with S if S is not 0, otherwise add it to S. This is because multiplying by 0 yields 0, which is not what we want.
After iterating over the array, S will contain the maximum possible sum. However, the problem asks for the sum modulo 10^9+7, so compute S modulo 10^9+7 and return this value.

Here is a Python solution for the problem:

def max_sum(arr):
    arr.sort()
    S = 0
    mod = 10**9+7
    for i in arr:
        if i <= 2 or S < 2:
            S += i
        else:
            S *= i
        S %= mod
    return S

This function takes as input a list of integers and returns the maximum possible sum modulo 10^9+7. It first sorts the list in ascending order, then iterates over the list, updating S according to the rules described above. Finally, it returns S modulo 10^9+7.

This problem has been solved

Similar Questions

Initially, we had one array, which was a permutation of size n𝑛 (an array of size n𝑛 where each integer from 11 to n𝑛 appears exactly once).We performed q𝑞 operations. During the i𝑖-th operation, we did the following:choose any array we have with at least 22 elements;split it into two non-empty arrays (prefix and suffix);write two integers li𝑙𝑖 and ri𝑟𝑖, where li𝑙𝑖 is the maximum element in the left part which we get after the split, and ri𝑟𝑖 is the maximum element in the right part;remove the array we've chosen from the pool of arrays we can use, and add the two resulting parts into the pool.For example, suppose the initial array was [6,3,4,1,2,5][6,3,4,1,2,5], and we performed the following operations:choose the array [6,3,4,1,2,5][6,3,4,1,2,5] and split it into [6,3][6,3] and [4,1,2,5][4,1,2,5]. Then we write l1=6𝑙1=6 and r1=5𝑟1=5, and the arrays we have are [6,3][6,3] and [4,1,2,5][4,1,2,5];choose the array [4,1,2,5][4,1,2,5] and split it into [4,1,2][4,1,2] and [5][5]. Then we write l2=4𝑙2=4 and r2=5𝑟2=5, and the arrays we have are [6,3][6,3], [4,1,2][4,1,2] and [5][5];choose the array [4,1,2][4,1,2] and split it into [4][4] and [1,2][1,2]. Then we write l3=4𝑙3=4 and r3=2𝑟3=2, and the arrays we have are [6,3][6,3], [4][4], [1,2][1,2] and [5][5].You are given two integers n𝑛 and q𝑞, and two sequences [l1,l2,…,lq][𝑙1,𝑙2,…,𝑙𝑞] and [r1,r2,…,rq][𝑟1,𝑟2,…,𝑟𝑞]. A permutation of size n𝑛 is called valid if we can perform q𝑞 operations and produce the given sequences [l1,l2,…,lq][𝑙1,𝑙2,…,𝑙𝑞] and [r1,r2,…,rq][𝑟1,𝑟2,…,𝑟𝑞].Calculate the number of valid permutations.InputThe first line contains two integers n𝑛 and q𝑞 (1≤q<n≤3⋅1051≤𝑞<𝑛≤3⋅105).The second line contains q𝑞 integers l1,l2,…,lq𝑙1,𝑙2,…,𝑙𝑞 (1≤li≤n1≤𝑙𝑖≤𝑛).The third line contains q𝑞 integers r1,r2,…,rq𝑟1,𝑟2,…,𝑟𝑞 (1≤ri≤n1≤𝑟𝑖≤𝑛).Additional constraint on the input: there exists at least one permutation which can produce the given sequences [l1,l2,…,lq][𝑙1,𝑙2,…,𝑙𝑞] and [r1,r2,…,rq][𝑟1,𝑟2,…,𝑟𝑞].OutputPrint one integer — the number of valid permutations, taken modulo 998244353998244353.ExamplesinputCopy6 36 4 45 5 2outputCopy30inputCopy10 1109outputCopy1814400inputCopy4 124outputCopy8

The two versions of the problem are different. You may want to read both versions. You can make hacks only if both versions are solved.You are given an array a𝑎 of length n𝑛. Start with c=0𝑐=0. Then, for each i𝑖 from 11 to n𝑛 (in increasing order) do exactly one of the following:Option 11: set c𝑐 to c+ai𝑐+𝑎𝑖.Option 22: set c𝑐 to |c+ai||𝑐+𝑎𝑖|, where |x||𝑥| is the absolute value of x𝑥.Let the maximum final value of c𝑐 after the procedure described above be equal to k𝑘. Find k𝑘.InputThe first line contains a single integer t𝑡 (1≤t≤1041≤𝑡≤104) — the number of test cases.The first line of each test case contains a single integer n𝑛 (2≤n≤2⋅1052≤𝑛≤2⋅105).The second line of each case contains n𝑛 integers a1𝑎1, a2𝑎2, a3𝑎3, ……, an𝑎𝑛 (−109≤ai≤109−109≤𝑎𝑖≤109).The sum of n𝑛 over all test cases does not exceed 3⋅1053⋅105.OutputFor each test case, output a single integer — the value of k𝑘.ExampleinputCopy5410 -9 -3 481 4 3 4 1 4 3 43-1 -2 -34-1000000000 1000000000 1000000000 100000000041 9 8 4outputCopy6246400000000022NoteIn the first test case, if we set c𝑐 to its absolute value every time we add to it, we end up with 66. It can be shown that this is the maximum result.In the second test case, taking the absolute value will never change anything, so we can just sum the array without doing anything to get 2424.In the third test case, it is optimal to wait until the end to set c𝑐 to its absolute value, resulting in an answer of 66.

You are given two integer arrays: array a𝑎 of length n𝑛 and array b𝑏 of length n+1𝑛+1.You can perform the following operations any number of times in any order:choose any element of the array a𝑎 and increase it by 11;choose any element of the array a𝑎 and decrease it by 11;choose any element of the array a𝑎, copy it and append the copy to the end of the array a𝑎.Your task is to calculate the minimum number of aforementioned operations (possibly zero) required to transform the array a𝑎 into the array b𝑏. It can be shown that under the constraints of the problem, it is always possible.InputThe first line contains a single integer t𝑡 (1≤t≤1041≤𝑡≤104) — the number of test cases.Each test case consists of three lines:the first line contains a single integer n𝑛 (1≤n≤2⋅1051≤𝑛≤2⋅105);the second line contains n𝑛 integers a1,a2,…,an𝑎1,𝑎2,…,𝑎𝑛 (1≤ai≤1091≤𝑎𝑖≤109);the third line contains n+1𝑛+1 integers b1,b2,…,bn+1𝑏1,𝑏2,…,𝑏𝑛+1 (1≤bi≤1091≤𝑏𝑖≤109).Additional constraint on the input: the sum of n𝑛 over all test cases doesn't exceed 2⋅1052⋅105.OutputFor each test case, print a single integer — the minimum number of operations (possibly zero) required to transform the array a𝑎 into the array b𝑏.ExampleinputCopy3121 323 33 3 344 2 1 22 1 5 2 3outputCopy318NoteIn the first example, you can transform a𝑎 into b𝑏 as follows: [2]→[2,2]→[1,2]→[1,3][2]→[2,2]→[1,2]→[1,3].

You are given an array 𝐴A of length 𝑁N, and a positive integer 𝐾K.It is guaranteed that 1≤𝐴𝑖≤𝐾1≤A i ≤K for every index 𝑖i from 11 to 𝑁N.You can do the following at most once:Choose an index 𝑖i (1≤𝑖≤𝑁1≤i≤N) and a value 𝑥x (1≤𝑥≤𝐾1≤x≤K).Then, set 𝐴𝑖:=𝑥A i :=x.Find the maximum possible value of the sum of adjacent differences of 𝐴A after performing this operation at most once.That is, maximize the quantity∑𝑖=1𝑁−1∣𝐴𝑖−𝐴𝑖+1∣i=1∑N−1 ∣A i −A i+1 ∣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 maximum allowed integer 𝐾K, respectively.The second line contains 𝑁N space-separated integers 𝐴1,𝐴2,…,𝐴𝑁A 1 ,A 2 ,…,A N , the elements of array 𝐴A.Output FormatFor each test case, output on a new line the answer: the maximum possible sum of adjacent differences of 𝐴A after replacing exactly one element.Constraints1≤𝑇≤1001≤T≤1001≤𝑁≤10001≤N≤10001≤𝐾≤2⋅1061≤K≤2⋅10 6 1≤𝐴𝑖≤𝐾1≤A i ≤KThe sum of 𝑁N across all tests won't exceed 10001000.Sample 1:InputOutput32 51 53 87 2 75 2018 3 1 4 1941263Explanation:Test case 11: It's best to leave the array unchanged, giving us a difference of ∣1−5∣=4∣1−5∣=4.Test case 22: It's optimal to set 𝐴2:=1A 2 :=1, giving us the array [7,1,7][7,1,7]. The sum of adjacent differences is 6+6=126+6=12.Test case 33: It's optimal to set 𝐴3:=20A 3 :=20, to obtain [18,3,20,4,19][18,3,20,4,19]. The sum of adjacent differences is 6363.

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].

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