Avoid Half Sum

AtCoder

IDagc065_c

Time2000ms

Memory256MB

Difficulty—

You are given a sequence of non-negative integers $A=(A_1,A_2,\dots,A_N)$ of length $N$. Here, $S=\sum_{i=1}^{N} A_i$ is even. Determine whether there is a pair of sequences of non-negative integers of length $N$, $B=(B_1,B_2,\dots,B_N)$ and $C=(C_1,C_2,\dots,C_N)$, that satisfy the following conditions: * $B_i+C_i=A_i$ for $i=1,2,\dots,N$. * $\sum_{i=1}^{N} X_i \neq \frac{S}{2}$ for every sequence of integers $X=(X_1,X_2,\dots,X_N)$ of length $N$ where $X_i=B_i$ or $X_i=C_i$ for $i=1,2,\dots,N$. Solve $T$ test cases. ## Constraints * $1 \leq T$ * $1 \leq N \leq 2 \times 10^5$ * $1 \leq A_i \leq 10^9$ * $\sum_{i=1}^{N} A_i$ is even. * The sum of $N$ over all test cases in a single input is at most $2 \times 10^5$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $T$ $\mathrm{case}_1$ $\vdots$ $\mathrm{case}_T$ Each case is given in the following format: $N$ $A_1$ $A_2$ $\dots$ $A_N$ [samples]

Samples

Input #1

3
3
1 2 3
6
1 1 2 2 3 3
4
1 1 1000000000 1000000000

Output #1

Yes
No
Yes

For the first test case, $B=(1,1,3)$ and $C=(0,1,0)$ satisfy the conditions.
For the second test case, no pair of $B$ and $C$ satisfies the conditions.

API Response (JSON)

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      "content": "You are given a sequence of non-negative integers $A=(A_1,A_2,\\dots,A_N)$ of length $N$. Here, $S=\\sum_{i=1}^{N} A_i$ is even.\nDetermine whether there is a pair of sequences of non-negative integers o...",
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