answer:**Analysis** This question mainly examines the functional characteristics of sequences, properties of sequences, and the maximum value of the sum of the first n terms. It is a basic question. **Solution** Since a_{50}+a_{51}=a_1+a_{100}=0 and d < 0, then a_{50}geqslant 0, a_{51}leqslant 0. Therefore, the first 49 terms of the sequence are greater than 0, the 50th term is greater than or equal to 0, and the 51st term is less than or equal to 0. The terms after the 51st are less than 0. Thus, the sum of the first 50 terms of the sequence is maximized. Hence, the correct choice is boxed{D}.

answer:To solve this problem, we need to determine who has a winning strategy: Alice, who aims to align five white pieces, or Bob, who aims to prevent Alice from achieving this. 1. **Description of Strategy:** - Bob can adopt a strategy based on pairing the cells of an infinite grid. The idea is to systematically pair every cell with another cell. If Alice places a white piece (pawn) on one cell, Bob responds by placing a black piece on its paired cell. 2. **Visualizing the Pairing:** - Imagine the grid as being partitioned into disjoint pairs. This means every cell (denoted as (x, y)) on the grid is matched with another unique cell. - For example, a feasible strategy might be pairing (x, y) with (y, x) or any systematic pairing method such that each cell belongs to exactly one pair. 3. **Bob's Moves:** - Every time Alice places a piece on a cell (a, b), Bob places his piece on the paired cell of (a, b). This ensures that whatever configuration Alice tries to create, Bob has an immediate response that disrupts any potential line of five consecutive white pieces for Alice. 4. **Winning Strategy for Bob:** - By ensuring that he always places a black piece whenever Alice places a white piece and keeping the grid such that Alice can never have five consecutive white pawns in any direction (horizontal, vertical, or diagonal), Bob effectively blocks all possible win configurations for Alice. 5. **Conclusion:** - Given these conditions, Bob can always prevent Alice from creating a line of five consecutive white pieces by systematically following this pairing and response strategy. Thus, from the description of the pairing strategy and ensuring Alice cannot align five pieces, it is evident that: [ boxed{text{Bob possesses a winning strategy.}} ]

answer:Let (z=a+bi(a,b∈R)), Then (z^{2}=(a+bi)^{2}=(a^{2}-b^{2})+2abi), (|z^{2}|= sqrt {(a^{2}-b^{2})^{2}+4a^{2}b^{2}}= sqrt {(a^{2}+b^{2})^{2}}=a^{2}+b^{2}), (|z|^{2}=a^{2}+b^{2}), (∴|z^{2}|=|z|^{2}neq z^{2}). Therefore, the correct answer is: boxed{A}. By setting (z=a+bi(a,b∈R)) and then calculating the values of (|z^{2}|), (|z|^{2}), and (z^{2}), we find the answer. This question tests the multiplication and division operations in the algebraic form of complex numbers and the method of finding the modulus of a complex number, making it a basic question.

answer:To solve the given expression [left(x+yright)^{2}-left(x+yright)left(x-yright)]div 2y, we follow the steps below: 1. Expand the terms inside the brackets: [ begin{align*} & = left(x^2 + 2xy + y^2 - (x^2 - xy + xy - y^2)right) div 2y & = left(x^2 + 2xy + y^2 - x^2 + y^2right) div 2y & = (2y^2 + 2xy) div 2y. end{align*} ] 2. Simplify the expression by dividing each term inside the bracket by 2y: [ begin{align*} & = frac{2y^2}{2y} + frac{2xy}{2y} & = y + x. end{align*} ] Therefore, the simplified expression is boxed{y + x}.