English · Original
Chinese · Translation
Formal · Original
You are given a sequence _a_ consisting of _n_ integers. You may partition this sequence into two sequences _b_ and _c_ in such a way that every element belongs exactly to one of these sequences.
Let _B_ be the sum of elements belonging to _b_, and _C_ be the sum of elements belonging to _c_ (if some of these sequences is empty, then its sum is 0). What is the maximum possible value of _B_ - _C_?
## Input
The first line contains one integer _n_ (1 ≤ _n_ ≤ 100) — the number of elements in _a_.
The second line contains _n_ integers _a_1, _a_2, ..., _a__n_ ( - 100 ≤ _a__i_ ≤ 100) — the elements of sequence _a_.
## Output
Print the maximum possible value of _B_ - _C_, where _B_ is the sum of elements of sequence _b_, and _C_ is the sum of elements of sequence _c_.
[samples]
## Note
In the first example we may choose _b_ = {1, 0}, _c_ = { - 2}. Then _B_ = 1, _C_ = - 2, _B_ - _C_ = 3.
In the second example we choose _b_ = {16, 23, 16, 15, 42, 8}, _c_ = {} (an empty sequence). Then _B_ = 120, _C_ = 0, _B_ - _C_ = 120.
给定一个由 #cf_span[n] 个整数组成的序列 #cf_span[a]。你可以将该序列划分为两个序列 #cf_span[b] 和 #cf_span[c]，使得每个元素恰好属于其中一个序列。
设 #cf_span[B] 为属于 #cf_span[b] 的元素之和，#cf_span[C] 为属于 #cf_span[c] 的元素之和（如果某个序列为空，则其和为 #cf_span[0]）。求 #cf_span[B - C] 的最大可能值？
第一行包含一个整数 #cf_span[n] (#cf_span[1 ≤ n ≤ 100]) —— 序列 #cf_span[a] 中的元素个数。
第二行包含 #cf_span[n] 个整数 #cf_span[a1], #cf_span[a2], ..., #cf_span[an] (#cf_span[ - 100 ≤ ai ≤ 100]) —— 序列 #cf_span[a] 的元素。
请输出 #cf_span[B - C] 的最大可能值，其中 #cf_span[B] 是序列 #cf_span[b] 的元素之和，#cf_span[C] 是序列 #cf_span[c] 的元素之和。
在第一个例子中，我们可以选择 #cf_span[b = {1, 0}]，#cf_span[c = { - 2}]。此时 #cf_span[B = 1]，#cf_span[C = - 2]，#cf_span[B - C = 3]。
在第二个例子中，我们选择 #cf_span[b = {16, 23, 16, 15, 42, 8}]，#cf_span[c = {}]（空序列）。此时 #cf_span[B = 120]，#cf_span[C = 0]，#cf_span[B - C = 120]。
## Input
第一行包含一个整数 #cf_span[n] (#cf_span[1 ≤ n ≤ 100]) —— 序列 #cf_span[a] 中的元素个数。第二行包含 #cf_span[n] 个整数 #cf_span[a1], #cf_span[a2], ..., #cf_span[an] (#cf_span[ - 100 ≤ ai ≤ 100]) —— 序列 #cf_span[a] 的元素。
## Output
请输出 #cf_span[B - C] 的最大可能值，其中 #cf_span[B] 是序列 #cf_span[b] 的元素之和，#cf_span[C] 是序列 #cf_span[c] 的元素之和。
[samples]
## Note
在第一个例子中，我们可以选择 #cf_span[b = {1, 0}]，#cf_span[c = { - 2}]。此时 #cf_span[B = 1]，#cf_span[C = - 2]，#cf_span[B - C = 3]。在第二个例子中，我们选择 #cf_span[b = {16, 23, 16, 15, 42, 8}]，#cf_span[c = {}]（空序列）。此时 #cf_span[B = 120]，#cf_span[C = 0]，#cf_span[B - C = 120]。
**Definitions**
Let $ n \in \mathbb{Z} $ be the number of elements.
Let $ A = (a_1, a_2, \dots, a_n) $ be a sequence of integers, where $ a_i \in [-100, 100] $.
Let $ B \subseteq A $ and $ C = A \setminus B $ be a partition of $ A $ into two disjoint subsequences (possibly empty).
Let $ B = \sum_{a_i \in B} a_i $ and $ C = \sum_{a_i \in C} a_i $ be the sums of elements in $ B $ and $ C $, respectively.
**Constraints**
$ 1 \leq n \leq 100 $
**Objective**
Maximize $ B - C $, where $ B + C = S = \sum_{i=1}^n a_i $.
Since $ B - C = B - (S - B) = 2B - S $, the objective is equivalent to:
$$
\max_{B \subseteq A} (2B - S) = 2 \cdot \max_{B \subseteq A} B - S
$$
Thus, the problem reduces to:
$$
\max_{B \subseteq A} (2B - S) = 2 \cdot \left( \max_{B \subseteq A} B \right) - S
$$
Equivalently, maximize $ B - C $ by choosing $ B $ to be the subset of $ A $ with maximum possible sum.
**Final Objective**
$$
\max_{B \subseteq A} (B - (S - B)) = \max_{B \subseteq A} (2B - S)
$$
API Response (JSON)
{
"problem": {
"name": "A. Partition",
"description": {
"content": "You are given a sequence _a_ consisting of _n_ integers. You may partition this sequence into two sequences _b_ and _c_ in such a way that every element belongs exactly to one of these sequences. Let",
"description_type": "Markdown"
},
"platform": "Codeforces",
"limit": {
"time_limit": 1000,
"memory_limit": 262144
},
"difficulty": "None",
"is_remote": true,
"is_sync": true,
"sync_url": null,
"sign": "CF946A"
},
"statements": [
{
"statement_type": "Markdown",
"content": "You are given a sequence _a_ consisting of _n_ integers. You may partition this sequence into two sequences _b_ and _c_ in such a way that every element belongs exactly to one of these sequences.\n\nLet...",
"is_translate": false,
"language": "English"
},
{
"statement_type": "Markdown",
"content": "给定一个由 #cf_span[n] 个整数组成的序列 #cf_span[a]。你可以将该序列划分为两个序列 #cf_span[b] 和 #cf_span[c]，使得每个元素恰好属于其中一个序列。\n\n设 #cf_span[B] 为属于 #cf_span[b] 的元素之和，#cf_span[C] 为属于 #cf_span[c] 的元素之和（如果某个序列为空，则其和为 #cf_span[0]）。求 #c...",
"is_translate": true,
"language": "Chinese"
},
{
"statement_type": "Markdown",
"content": "**Definitions** \nLet $ n \\in \\mathbb{Z} $ be the number of elements. \nLet $ A = (a_1, a_2, \\dots, a_n) $ be a sequence of integers, where $ a_i \\in [-100, 100] $. \n\nLet $ B \\subseteq A $ and $ C = ...",
"is_translate": false,
"language": "Formal"
}
]
}