Kids and inequality

Algebra Level 3

$\prod _{ i=1 }^{ 2n }{ \left( { a }_{ i }{ \left\lfloor x \right\rfloor }^{ 2 }+{ b }_{ i }\left\lfloor x \right\rfloor { +c }_{ i } \right) } \ge 0$

Pasha and Masha are playing the game: they're choose non-zero odds $a, b, c$ by rotation (Pasha $\rightarrow { a }_{ 1 }$ , Masha $\rightarrow { b }_{ 1 }$ , Pasha $\rightarrow { c }_{ 1 }$ , Masha $\rightarrow { a }_{ 2}$ and so on) . Pasha is trying to make that all $x$ satisfy the inequality above. Masha is trying to make that this inequality hasn't solution. Who can win anyway for any $n \in \mathbb{N}$ if both of them playing optimally?

Notation: $\left\lfloor \cdot \right\rfloor$ denotes the floor function .

Nobody Pasha Masha

1 solution

Ilya Pavlyuchenko
Apr 17, 2020

Statement I: Masha can prevent Pasha anyway.

Let $\left\lfloor x \right\rfloor = 0$ , then ${ a }_{ j }\left\lfloor x \right\rfloor ^{ 2 }+{ b }_{ j }\left\lfloor x \right\rfloor +{ c }_{ j }={ c }_{ j }$ . If Pasha choose ${ c }_{ 1 } > 0$ then Masha choose ${ c }_{ 2 } < 0$ . And if Pasha choose ${ c }_{ 1 } < 0$ then Masha choose ${ c }_{2 } > 0$ . I next steps Masha just copy value of $c_i$ of Pasha. So $\left\lfloor x \right\rfloor = 0$ is not solution and Pasha isn't winner because $\displaystyle \prod _{ j=1 }^{ 2n }{ { c }_{ j } } =\prod _{ i=1 }^{ n }{ \left( { c }_{ 2i-1 }{ c }_{ 2i } \right) } ={ c }_{ 1 }{ c }_{ 2 }{ \prod _{ i=2 }^{ n }{ \left( { c }_{ 2i-1 }{ c }_{ 2i } \right) } }<0$ .

Statement II: Pasha can prevent Masha anyway.

I claim that Pasha can choose ${ b }_{ 2n }$ at which Masha will not be able to choose ${ c }_{ 2n }$ for winning. (How can I prove it? But I know that proof is based on small movements).

What conclusion can be drawn? If Pasha wants to win, than Masha can prevent him anyway. If Masha wants to win, than Pasha can prevent her anyway. If both wants win, then they will interfere with each other anyway. So, nobody can win in this game.

I think your clain is wrong. Pasha can't prevent by choosing only b 2n. If Masha managed to get all terms except last either >0 or <0 for any x, she need make last term >0 for any x. Then she can chose any positive a 2n and after Pasha choose b 2n calculate minimum of parabola and shift it high enough with c 2n Pasha still prevent Masha's victory but not only by his last choise

Qweros Bistoros - 1 year, 1 month ago

Kids and inequality

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